Optimizers#

Check out How to specify algorithms and algorithm specific options to see how to select an algorithm and specify algo_options when using maximize or minimize.

Optimizers from scipy#

estimagic supports most scipy algorithms and scipy is automatically installed when you install estimagic.

scipy_lbfgsb
"scipy_lbfgsb"

Minimize a scalar function of one or more variables using the L-BFGS-B algorithm.

The optimizer is taken from scipy, which calls the Fortran code written by the original authors of the algorithm. The Fortran code includes the corrections and improvements that were introduced in a follow up paper.

lbfgsb is a limited memory version of the original bfgs algorithm, that deals with lower and upper bounds via an active set approach.

The lbfgsb algorithm is well suited for differentiable scalar optimization problems with up to several hundred parameters.

It is a quasi-newton line search algorithm. At each trial point it evaluates the criterion function and its gradient to find a search direction. It then approximates the hessian using the stored history of gradients and uses the hessian to calculate a candidate step size. Then it uses a gradient based line search algorithm to determine the actual step length. Since the algorithm always evaluates the gradient and criterion function jointly, the user should provide a criterion_and_derivative function that exploits the synergies in the calculation of criterion and gradient.

The lbfgsb algorithm is almost perfectly scale invariant. Thus, it is not necessary to scale the parameters.

  • convergence.relative_criterion_tolerance (float): Stop when the relative improvement between two iterations is smaller than this. More formally, this is expressed as

\[\begin{split}\frac{(f^k - f^{k+1})}{\\max{{|f^k|, |f^{k+1}|, 1}}} \leq \text{relative_criterion_tolerance}\end{split}\]
  • convergence.absolute_gradient_tolerance (float): Stop if all elements of the projected gradient are smaller than this.

  • stopping.max_criterion_evaluations (int): If the maximum number of function evaluation is reached, the optimization stops but we do not count this as convergence.

  • stopping.max_iterations (int): If the maximum number of iterations is reached, the optimization stops, but we do not count this as convergence.

  • limited_memory_storage_length (int): Maximum number of saved gradients used to approximate the hessian matrix.

scipy_slsqp
"scipy_slsqp"

Minimize a scalar function of one or more variables using the SLSQP algorithm.

SLSQP stands for Sequential Least Squares Programming.

SLSQP is a line search algorithm. It is well suited for continuously differentiable scalar optimization problems with up to several hundred parameters.

The optimizer is taken from scipy which wraps the SLSQP optimization subroutine originally implemented by [algo_1].

Note

SLSQP’s general nonlinear constraints are not supported yet by estimagic.

  • convergence.absolute_criterion_tolerance (float): Precision goal for the value of f in the stopping criterion.

  • stopping.max_iterations (int): If the maximum number of iterations is reached, the optimization stops, but we do not count this as convergence.

scipy_neldermead
"scipy_neldermead"

Minimize a scalar function using the Nelder-Mead algorithm.

The Nelder-Mead algorithm is a direct search method (based on function comparison) and is often applied to nonlinear optimization problems for which derivatives are not known. Unlike most modern optimization methods, the Nelder–Mead heuristic can converge to a non-stationary point, unless the problem satisfies stronger conditions than are necessary for modern methods.

Nelder-Mead is never the best algorithm to solve a problem but rarely the worst. Its popularity is likely due to historic reasons and much larger than its properties warrant.

The argument initial_simplex is not supported by estimagic as it is not compatible with estimagic’s handling of constraints.

  • stopping.max_iterations (int): If the maximum number of iterations is reached, the optimization stops, but we do not count this as convergence.

  • stopping.max_criterion_evaluations (int): If the maximum number of function evaluation is reached, the optimization stops but we do not count this as convergence.

  • convergence.absolute_params_tolerance (float): Absolute difference in parameters between iterations that is tolerated to declare convergence. As no relative tolerances can be passed to Nelder-Mead, estimagic sets a non zero default for this.

  • convergence.absolute_criterion_tolerance (float): Absolute difference in the criterion value between iterations that is tolerated to declare convergence. As no relative tolerances can be passed to Nelder-Mead, estimagic sets a non zero default for this.

  • adaptive (bool): Adapt algorithm parameters to dimensionality of problem. Useful for high-dimensional minimization ([algo_2], p. 259-277). scipy’s default is False.

scipy_powell
"scipy_powell"

Minimize a scalar function using the modified Powell method.

Warning

In our benchmark using a quadratic objective function, the Powell algorithm did not find the optimum very precisely (less than 4 decimal places). If you require high precision, you should refine an optimum found with Powell with another local optimizer.

The criterion function need not be differentiable.

Powell’s method is a conjugate direction method, minimizing the function by a bi-directional search in each parameter’s dimension.

The argument direc, which is the initial set of direction vectors and which is part of the scipy interface is not supported by estimagic because it is incompatible with how estimagic handles constraints.

  • convergence.relative_params_tolerance (float): Stop when the relative movement between parameter vectors is smaller than this.

  • convergence.relative_criterion_tolerance (float): Stop when the relative improvement between two iterations is smaller than this. More formally, this is expressed as

    \[\begin{split}\frac{(f^k - f^{k+1})}{\\max{{\{|f^k|, |f^{k+1}|, 1\}}}} \leq \text{relative_criterion_tolerance}\end{split}\]
  • stopping.max_criterion_evaluations (int): If the maximum number of function evaluation is reached, the optimization stops but we do not count thisas convergence.

  • stopping.max_iterations (int): If the maximum number of iterations is reached, the optimization stops, but we do not count this as convergence.

scipy_bfgs
"scipy_bfgs"

Minimize a scalar function of one or more variables using the BFGS algorithm.

BFGS stands for Broyden-Fletcher-Goldfarb-Shanno algorithm. It is a quasi-Newton method that can be used for solving unconstrained nonlinear optimization problems.

BFGS is not guaranteed to converge unless the function has a quadratic Taylor expansion near an optimum. However, BFGS can have acceptable performance even for non-smooth optimization instances.

  • convergence.absolute_gradient_tolerance (float): Stop if all elements of the gradient are smaller than this.

  • stopping.max_iterations (int): If the maximum number of iterations is reached, the optimization stops, but we do not count this as convergence.

  • norm (float): Order of the vector norm that is used to calculate the gradient’s “score” that is compared to the gradient tolerance to determine convergence. Default is infinite which means that the largest entry of the gradient vector is compared to the gradient tolerance.

scipy_conjugate_gradient
"scipy_conjugate_gradient"

Minimize a function using a nonlinear conjugate gradient algorithm.

The conjugate gradient method finds functions’ local optima using just the gradient.

This conjugate gradient algorithm is based on that of Polak and Ribiere, detailed in [algo_3], pp. 120-122.

Conjugate gradient methods tend to work better when:

  • the criterion has a unique global minimizing point, and no local minima or other stationary points.

  • the criterion is, at least locally, reasonably well approximated by a quadratic function.

  • the criterion is continuous and has a continuous gradient.

  • the gradient is not too large, e.g., has a norm less than 1000.

  • The initial guess is reasonably close to the criterion’s global minimizer.

  • convergence.absolute_gradient_tolerance (float): Stop if all elements of the gradient are smaller than this.

  • stopping.max_iterations (int): If the maximum number of iterations is reached, the optimization stops, but we do not count this as convergence.

  • norm (float): Order of the vector norm that is used to calculate the gradient’s “score” that is compared to the gradient tolerance to determine convergence. Default is infinite which means that the largest entry of the gradient vector is compared to the gradient tolerance.

scipy_newton_cg
"scipy_newton_cg"

Minimize a scalar function using Newton’s conjugate gradient algorithm.

Warning

In our benchmark using a quadratic objective function, the truncated newton algorithm did not find the optimum very precisely (less than 4 decimal places). If you require high precision, you should refine an optimum found with Powell with another local optimizer.

Newton’s conjugate gradient algorithm uses an approximation of the Hessian to find the minimum of a function. It is practical for small and large problems (see [algo_3], p. 140).

Newton-CG methods are also called truncated Newton methods. This function differs scipy_truncated_newton because

  • scipy_newton_cg’s algorithm is written purely in Python using NumPy and scipy while scipy_truncated_newton’s algorithm calls a C function.

  • scipy_newton_cg’s algorithm is only for unconstrained minimization while scipy_truncated_newton’s algorithm supports bounds.

Conjugate gradient methods tend to work better when:

  • the criterion has a unique global minimizing point, and no local minima or other stationary points.

  • the criterion is, at least locally, reasonably well approximated by a quadratic function.

  • the criterion is continuous and has a continuous gradient.

  • the gradient is not too large, e.g., has a norm less than 1000.

  • The initial guess is reasonably close to the criterion’s global minimizer.

  • convergence.relative_params_tolerance (float): Stop when the relative movement between parameter vectors is smaller than this. Newton CG uses the average relative change in the parameters for determining the convergence.

  • stopping.max_iterations (int): If the maximum number of iterations is reached, the optimization stops, but we do not count this as convergence.

scipy_cobyla
"scipy_cobyla"

Minimize a scalar function of one or more variables using the COBYLA algorithm.

COBYLA stands for Constrained Optimization By Linear Approximation. It is derivative-free and supports nonlinear inequality and equality constraints.

Note

Cobyla’s general nonlinear constraints is not supported yet by estimagic.

Scipy’s implementation wraps the FORTRAN implementation of the algorithm.

For more information on COBYLA see [algo_4], [algo_5] and [algo_6].

  • stopping.max_iterations (int): If the maximum number of iterations is reached, the optimization stops, but we do not count this as convergence.

  • convergence.relative_params_tolerance (float): Stop when the relative movement between parameter vectors is smaller than this. In case of COBYLA this is a lower bound on the size of the trust region and can be seen as the required accuracy in the variables but this accuracy is not guaranteed.

  • trustregion.initial_radius (float): Initial value of the trust region radius. Since a linear approximation is likely only good near the current simplex, the linear program is given the further requirement that the solution, which will become the next evaluation point must be within a radius RHO_j from x_j. RHO_j only decreases, never increases. The initial RHO_j is the trustregion.initial_radius. In this way COBYLA’s iterations behave like a trust region algorithm.

scipy_truncated_newton
"scipy_truncated_newton"

Minimize a scalar function using truncated Newton algorithm.

This function differs from scipy_newton_cg because

  • scipy_newton_cg’s algorithm is written purely in Python using NumPy and scipy while scipy_truncated_newton’s algorithm calls a C function.

  • scipy_newton_cg’s algorithm is only for unconstrained minimization while scipy_truncated_newton’s algorithm supports bounds.

Conjugate gradient methods tend to work better when:

  • the criterion has a unique global minimizing point, and no local minima or other stationary points.

  • the criterion is, at least locally, reasonably well approximated by a quadratic function.

  • the criterion is continuous and has a continuous gradient.

  • the gradient is not too large, e.g., has a norm less than 1000.

  • The initial guess is reasonably close to the criterion’s global minimizer.

estimagic does not support the scale nor offset argument as they are not compatible with the way estimagic handles constraints. It also does not support messg_num which is an additional way to control the verbosity of the optimizer.

  • func_min_estimate (float): Minimum function value estimate. Defaults to 0. stopping_max_iterations (int): If the maximum number of iterations is reached, the optimization stops, but we do not count this as convergence.

  • stopping.max_criterion_evaluations (int): If the maximum number of function evaluation is reached, the optimization stops but we do not count this as convergence.

  • convergence.absolute_params_tolerance (float): Absolute difference in parameters between iterations after scaling that is tolerated to declare convergence.

  • convergence.absolute_criterion_tolerance (float): Absolute difference in the criterion value between iterations after scaling that is tolerated to declare convergence.

  • convergence.absolute_gradient_tolerance (float): Stop if the value of the projected gradient (after applying x scaling factors) is smaller than this. If convergence.absolute_gradient_tolerance < 0.0, convergence.absolute_gradient_tolerance is set to 1e-2 * sqrt(accuracy).

  • max_hess_evaluations_per_iteration (int): Maximum number of hessian*vector evaluations per main iteration. If max_hess_evaluations == 0, the direction chosen is - gradient. If max_hess_evaluations < 0, max_hess_evaluations is set to max(1,min(50,n/2)) where n is the length of the parameter vector. This is also the default.

  • max_step_for_line_search (float): Maximum step for the line search. It may be increased during the optimization. If too small, it will be set to 10.0. By default we use scipy’s default.

  • line_search_severity (float): Severity of the line search. If < 0 or > 1, set to 0.25. Estimagic defaults to scipy’s default.

  • finitie_difference_precision (float): Relative precision for finite difference calculations. If <= machine_precision, set to sqrt(machine_precision). Estimagic defaults to scipy’s default.

  • criterion_rescale_factor (float): Scaling factor (in log10) used to trigger criterion rescaling. If 0, rescale at each iteration. If a large value, never rescale. If < 0, rescale is set to 1.3. Estimagic defaults to scipy’s default.

scipy_trust_constr
"scipy_trust_constr"

Minimize a scalar function of one or more variables subject to constraints.

Warning

In our benchmark using a quadratic objective function, the trust_constr algorithm did not find the optimum very precisely (less than 4 decimal places). If you require high precision, you should refine an optimum found with Powell with another local optimizer.

Note

Its general nonlinear constraints’ handling is not supported yet by estimagic.

It switches between two implementations depending on the problem definition. It is the most versatile constrained minimization algorithm implemented in SciPy and the most appropriate for large-scale problems. For equality constrained problems it is an implementation of Byrd-Omojokun Trust-Region SQP method described in [algo_7] and in [algo_8], p. 549. When inequality constraints are imposed as well, it switches to the trust-region interior point method described in [algo_9]. This interior point algorithm in turn, solves inequality constraints by introducing slack variables and solving a sequence of equality-constrained barrier problems for progressively smaller values of the barrier parameter. The previously described equality constrained SQP method is used to solve the subproblems with increasing levels of accuracy as the iterate gets closer to a solution.

It approximates the Hessian using the Broyden-Fletcher-Goldfarb-Shanno (BFGS) Hessian update strategy.

  • convergence.absolute_gradient_tolerance (float): Tolerance for termination by the norm of the Lagrangian gradient. The algorithm will terminate when both the infinity norm (i.e., max abs value) of the Lagrangian gradient and the constraint violation are smaller than the convergence.absolute_gradient_tolerance. For this algorithm we use scipy’s gradient tolerance for trust_constr. This smaller tolerance is needed for the sum of squares tests to pass.

  • stopping.max_iterations (int): If the maximum number of iterations is reached, the optimization stops, but we do not count this as convergence.

  • convergence.relative_params_tolerance (float): Tolerance for termination by the change of the independent variable. The algorithm will terminate when the radius of the trust region used in the algorithm is smaller than the convergence.relative_params_tolerance.

  • trustregion.initial_radius (float): Initial value of the trust region radius. The trust radius gives the maximum distance between solution points in consecutive iterations. It reflects the trust the algorithm puts in the local approximation of the optimization problem. For an accurate local approximation the trust-region should be large and for an approximation valid only close to the current point it should be a small one. The trust radius is automatically updated throughout the optimization process, with trustregion_initial_radius being its initial value.

scipy_ls_dogbox
"scipy_ls_dogbox"

Minimize a nonlinear least squares problem using a rectangular trust region method.

Typical use case is small problems with bounds. Not recommended for problems with rank-deficient Jacobian.

The algorithm supports the following options:

  • convergence.relative_criterion_tolerance (float): Stop when the relative improvement between two iterations is below this.

  • convergence.relative_gradient_tolerance (float): Stop when the gradient, divided by the absolute value of the criterion function is smaller than this.

  • stopping.max_criterion_evaluations (int): If the maximum number of function evaluation is reached, the optimization stops but we do not count this as convergence.

  • tr_solver (str): Method for solving trust-region subproblems, relevant only for ‘trf’ and ‘dogbox’ methods.

    • ‘exact’ is suitable for not very large problems with dense Jacobian matrices. The computational complexity per iteration is comparable to a singular value decomposition of the Jacobian matrix.

    • ‘lsmr’ is suitable for problems with sparse and large Jacobian matrices. It uses the iterative procedure scipy.sparse.linalg.lsmr for finding a solution of a linear least-squares problem and only requires matrix-vector product evaluations. If None (default), the solver is chosen based on the type of Jacobian returned on the first iteration.

  • tr_solver_options (dict): Keyword options passed to trust-region solver.

    • tr_solver='exact': tr_options are ignored.

    • tr_solver='lsmr': options for scipy.sparse.linalg.lsmr.

scipy_ls_trf
"scipy_ls_trf"

Minimize a nonlinear least squares problem using a trustregion reflective method.

Trust Region Reflective algorithm, particularly suitable for large sparse problems with bounds. Generally robust method.

The algorithm supports the following options:

  • convergence.relative_criterion_tolerance (float): Stop when the relative improvement between two iterations is below this.

  • convergence.relative_gradient_tolerance (float): Stop when the gradient, divided by the absolute value of the criterion function is smaller than this.

  • stopping.max_criterion_evaluations (int): If the maximum number of function evaluation is reached, the optimization stops but we do not count this as convergence.

  • tr_solver (str): Method for solving trust-region subproblems, relevant only for ‘trf’ and ‘dogbox’ methods.

    • ‘exact’ is suitable for not very large problems with dense Jacobian matrices. The computational complexity per iteration is comparable to a singular value decomposition of the Jacobian matrix.

    • ‘lsmr’ is suitable for problems with sparse and large Jacobian matrices. It uses the iterative procedure scipy.sparse.linalg.lsmr for finding a solution of a linear least-squares problem and only requires matrix-vector product evaluations. If None (default), the solver is chosen based on the type of Jacobian returned on the first iteration.

  • tr_solver_options (dict): Keyword options passed to trust-region solver.

    • tr_solver='exact': tr_options are ignored.

    • tr_solver='lsmr': options for scipy.sparse.linalg.lsmr.

scipy_ls_lm
"scipy_ls_lm"

Minimize a nonlinear least squares problem using a Levenberg-Marquardt method.

Does not handle bounds and sparse Jacobians. Usually the most efficient method for small unconstrained problems.

The algorithm supports the following options:

  • convergence.relative_criterion_tolerance (float): Stop when the relative improvement between two iterations is below this.

  • convergence.relative_gradient_tolerance (float): Stop when the gradient, divided by the absolute value of the criterion function is smaller than this.

  • stopping.max_criterion_evaluations (int): If the maximum number of function evaluation is reached, the optimization stops but we do not count this as convergence.

  • tr_solver (str): Method for solving trust-region subproblems, relevant only for ‘trf’ and ‘dogbox’ methods.

    • ‘exact’ is suitable for not very large problems with dense Jacobian matrices. The computational complexity per iteration is comparable to a singular value decomposition of the Jacobian matrix.

    • ‘lsmr’ is suitable for problems with sparse and large Jacobian matrices. It uses the iterative procedure scipy.sparse.linalg.lsmr for finding a solution of a linear least-squares problem and only requires matrix-vector product evaluations. If None (default), the solver is chosen based on the type of Jacobian returned on the first iteration.

  • tr_solver_options (dict): Keyword options passed to trust-region solver.

    • tr_solver='exact': tr_options are ignored.

    • tr_solver='lsmr': options for scipy.sparse.linalg.lsmr.

scipy_basinhopping
"scipy_basinhopping"

Find the global minimum of a function using the basin-hopping algorithm which combines a global stepping algorithm with local minimization at each step.

Basin-hopping is a two-phase method that combines a global stepping algorithm with local minimization at each step. Designed to mimic the natural process of energy minimization of clusters of atoms, it works well for similar problems with “funnel-like, but rugged” energy landscapes.

This is mainly supported for completeness. Consider estimagic’s built in multistart optimization for a similar approach that can run multiple optimizations in parallel, supports all local algorithms in estimagic (as opposed to just those from scipy) and allows for a better visualization of the multistart history.

When provided the derivative is passed to the local minimization method.

The algorithm supports the following options:

  • local_algorithm (str/callable): Any scipy local minimizer: valid options are. “Nelder-Mead”. “Powell”. “CG”. “BFGS”. “Newton-CG”. “L-BFGS-B”. “TNC”. “COBYLA”.

“SLSQP”. “trust-constr”. “dogleg”. “trust-ncg”. “trust-exact”. “trust-krylov”. or a custom function for local minimization, default is “L-BFGS-B”. - n_local_optimizations: (int) The number local optimizations. Default is 100 as in scipy’s default. - temperature: (float) Controls the randomness in the optimization process. Higher the temperatures the larger jumps in function value will be accepted. Default is 1.0 as in scipy’s default. - stepsize: (float) Maximum step size. Default is 0.5 as in scipy’s default. - local_algo_options: (dict) Additional keyword arguments for the local

minimizer. Check the documentation of the local scipy algorithms for details on what is supported.

  • take_step: (callable) Replaces the default step-taking routine. Default is None as in scipy’s default.

  • accept_test: (callable) Define a test to judge the acception of steps. Default is None as in scipy’s default.

  • interval: (int) Determined how often the step size is updated. Default is 50 as in scipy’s default.

  • convergence.n_unchanged_iterations: (int) Number of iterations the global

minimum estimate stays the same to stops the algorithm. Default is None as in scipy’s default. - seed: (None, int, numpy.random.Generator,numpy.random.RandomState)Default is None as in scipy’s default. - target_accept_rate: (float) Adjusts the step size. Default is 0.5 as in scipy’s default. - stepwise_factor: (float) Step size multiplier upon each step. Lies between (0,1), default is 0.9 as in scipy’s default.

scipy_brute
"scipy_brute"

Find the global minimum of a fuction over a given range by brute force.

Brute force evaluates the criterion at each point and that is why better suited for problems with very few parameters.

The start values are not actually used because the grid is only defined by bounds. It is still necessary for estimagic to infer the number and format of the parameters.

Due to the parallelization, this algorithm cannot collect a history of parameters and criterion evaluations.

The algorithm supports the following options:

  • n_grid_points (int): the number of grid points to use for the brute force search. Default is 20 as in scipy.

  • polishing_function (callable): Function to seek a more precise minimum near brute-force’ best gridpoint taking brute-force’s result at initial guess as a positional argument. Default is None providing no polishing.

  • n_cores (int): The number of cores on which the function is evaluated in

parallel. Default 1. - batch_evaluator (str or callable). An estimagic batch evaluator. Default ‘joblib’.

scipy_differential_evolution
"scipy_differential_evolution"

Find the global minimum of a multivariate function using differential evolution (DE). DE is a gradient-free method.

Due to estimagic’s general parameter format the integrality and vectorized arguments are not supported.

The algorithm supports the following options:

  • strategy (str): Measure of quality to improve a candidate solution, can be one of the following keywords - ‘best1bin’ - ‘best1exp’ - ‘rand1exp’ - ‘randtobest1exp’ - ‘currenttobest1exp’ - ‘best2exp’ - ‘rand2exp’ - ‘randtobest1bin’ - ‘currenttobest1bin’ - ‘best2bin’ - ‘rand2bin’ - ‘rand1bin’

,default is ‘best1bin’. - stopping.max_iterations (int): The maximum number of criterion evaluations without polishing is(stopping.max_iterations + 1) * population_size * number of parameters) - population_size_multiplier (int): A multiplier setting the population size. The number of individuals in the population is population_size * number of parameters. The default 15. - convergence.relative_criterion_tolerance (float): Default 0.01. - mutation_constant (float/tuple): The differential weight denoted by F in literature. Should be within 0 and 2. The tuple form is used to specify (min, max) dithering which can help speed convergence. Default is (0.5, 1). - recombination_constant (float): The crossover probability or CR in the literature determines the probability that two solution vectors will be combined to produce a new solution vector. Should be between 0 and 1. The default is 0.7. - seed (int): DE is stochastic. Define a seed for reproducability. - polish (bool): Uses scipy’s L-BFGS-B for unconstrained problems and trust-constr for constrained problems to slightly improve the minimization. Default is True. - sampling_method (str/np.array): Specify the sampling method for the initial population. It can be one of the following options

  • “latinhypercube”

  • “sobol”

  • “halton”

  • “random”

  • an array specifying the initial population of shape (total population size, number of parameters). The initial population is clipped to bounds before use. Default is ‘latinhypercube’

  • convergence.absolute_criterion_tolerance (float): CONVERGENCE_SECOND_BEST_ABSOLUTE_CRITERION_TOLERANCE

  • n_cores (int): The number of cores on which the function is evaluated in

parallel. Default 1. - batch_evaluator (str or callable). An estimagic batch evaluator. Default ‘joblib’.

scipy_shgo
"scipy_shgo"

Find the global minimum of a fuction using simplicial homology global optimization.

The algorithm supports the following options:

  • local_algorithm (str): The local optimization algorithm to be used. Only

COBYLA and SLSQP supports constraints. Valid options are “Nelder-Mead”. “Powell”. “CG”. “BFGS”. “Newton-CG”. “L-BFGS-B”. “TNC”. “COBYLA”. “SLSQP”. “trust-constr”. “dogleg”. “trust-ncg”. “trust-exact”. “trust-krylov” or a custom function for local minimization, default is “L-BFGS-B”. - local_algo_options: (dict) Additional keyword arguments for the local

minimizer. Check the documentation of the local scipy algorithms for details on what is supported.

  • n_sampling_points (int): Specify the number of sampling points to construct the simplical complex.

  • n_simplex_iterations (int): Number of iterations to construct the simplical complex. Default is 1 as in scipy.

  • sampling_method (str/callable): The method to use for sampling the search space. Default ‘simplicial’.

  • max_sampling_evaluations (int): The maximum number of evaluations of the criterion function in the sampling phase.

  • convergence.minimum_criterion_value (float): Specify the global minimum when it is known. Default is - **np.inf. For maximization problems, flip the sign.

  • convergence.minimum_criterion_tolerance (float): Specify the relative error between the current best minimum and the supplied global criterion_minimum allowed. Default is scipy’s default, 1e-4.

  • stopping.max_iterations (int): The maximum number of iterations.

  • stopping.max_criterion_evaluations (int): The maximum number of criterion

evaluations. - stopping.max_processing_time (int): The maximum time allowed for the optimization. - minimum_homology_group_rank_differential (int): The minimum difference in the rank of the homology group between iterations. - symmetry (bool): Specify whether the criterion contains symetric variables. - minimize_every_iteration ()bool: Specify whether the gloabal sampling points are passed to the local algorithm in every iteration. - max_local_minimizations_per_iteration (int): The maximum number of local optimizations per iteration. Default False, i.e. no limit. - infinity_constraints (bool): Specify whether to save the sampling points outside the feasible domain. Default is True.

scipy_dual_annealing
"scipy_dual_annealing"

Find the global minimum of a function using dual annealing for continuous variables.

The algorithm supports the following options:

  • stopping.max_iterations (int): Specify the maximum number of global searh iterations.

  • local_algorithm (str): The local optimization algorithm to be used. valid

options are. “Nelder-Mead”. “Powell”. “CG”. “BFGS”. “Newton-CG”. “L-BFGS-B”. “TNC”. “COBYLA”. “SLSQP”. “trust-constr”. “dogleg”. “trust-ncg”. “trust-exact”. “trust-krylov”. Default “L-BFGS-B”. - local_algo_options: (dict) Additional keyword arguments for the local

minimizer. Check the documentation of the local scipy algorithms for details on what is supported.

  • initial_temperature (float): The temparature algorithm starts with. The higher values lead to a wider search space. The range is (0.01, 5.e4] and default is 5230.0.

  • restart_temperature_ratio (float): Reanneling starts when the algorithm is decreased to initial_temperature * restart_temperature_ratio. Default is 2e-05.

  • visit (float): Specify the thickness of visiting distribution’s tails. Range is (1, 3] and default is scipy’s default, 2.62.

  • accept (float): Controls the probability of acceptance. Range is (-1e4, -5] and default is scipy’s default, -5.0. Smaller values lead to lower acceptance probability.

  • stopping.max_criterion_evaluations (int): soft limit for the number of criterion evaluations.

  • seed (int, None or RNG): Dual annealing is a stochastic process. Seed or

random number generator. Default None. - no_local_search (bool): Specify whether to apply a traditional Generalized Simulated Annealing with no local search. Default is False.

scipy_direct
"scipy_direct"

Find the global minimum of a function using dividing rectangles method. It is not necessary to provide an initial guess.

The algorithm supports the following options:

  • eps (float): Specify the minimum difference of the criterion values between the current best hyperrectangle and the next potentially best hyperrectangle to be divided determining the trade off between global and local search. Default is 1e-6 differing from scipy’s default 1e-4.

  • stopping_max_criterion_evaluations (int/None): Maximum number of criterion evaluations allowed. Default is None which caps the number of evaluations at 1000 * number of dimentions automatically.

  • stopping_max_iterations (int): Maximum number of iterations allowed.

  • locally_biased (bool): Determine whether to use the locally biased variant of the algorithm DIRECT_L. Default is True.

  • convergence.minimum_criterion_value (float): Specify the global minimum when it is known. Default is - **np.inf. For maximization problems, flip the sign.

  • convergence.minimum_criterion_tolerance (float): Specify the relative error between the current best minimum and the supplied global criterion_minimum allowed. Default is scipy’s default, 1e-4.

  • volume_hyperrectangle_tolerance (float): Specify the smallest volume of the hyperrectangle containing the lowest criterion value allowed. Range is (0,1). Default is 1e-16.

  • length_hyperrectangle_tolerance (float): Depending on locally_biased it can refer to normalized side (True) or diagonal (False) length of the hyperrectangle containing the lowest criterion value. Range is (0,1). Default is scipy’s default, 1e-6.

Own optimizers#

We implement a few algorithms from scratch. They are currently considered experimental.

bhhh
"bhhh"

Minimize a likelihood function using the BHHH algorithm.

BHHH ([algo_10]) can - and should ONLY - be used for minimizing (or maximizing) a likelihood. It is similar to the Newton-Raphson algorithm, but replaces the Hessian matrix with the outer product of the gradient. This approximation is based on the information matrix equality ([algo_11]) and is thus only vaid when minimizing (or maximizing) a likelihood.

The criterion function func() should return a dictionary with at least the entry {"contributions": array_or_pytree} where array_or_pytree contains the likelihood contributions of each individual.

bhhh supports the following options:

  • convergence_absolute_gradient_tolerance (float): Stopping criterion for the gradient tolerance. Default is 1e-8.

  • stopping_max_iterations (int): Maximum number of iterations. If reached, terminate. Default is 200.

neldermead_parallel
"neldermead_parallel"

Minimize a function using the neldermead_parallel algorithm.

This is a parallel Nelder-Mead algorithm following Lee D., Wiswall M., A parallel implementation of the simplex function minimization routine, Computational Economics, 2007.

The algorithm was implemented by Jacek Barszczewski

The algorithm supports the following options:

  • init_simplex_method (string or callable): Name of the method to create initial simplex or callable which takes as an argument initial value of parameters and returns initial simplex as j+1 x j array, where j is length of x. The default is “gao_han”.

  • n_cores (int): Degree of parallization. The default is 1 (no parallelization).

  • adaptive (bool): Adjust parameters of Nelder-Mead algorithm to account for simplex size. The default is True.

  • stopping.max_iterations (int): Maximum number of algorithm iterations. The default is STOPPING_MAX_ITERATIONS.

  • convergence.absolute_criterion_tolerance (float): maximal difference between function value evaluated on simplex points. The default is CONVERGENCE_SECOND_BEST_ABSOLUTE_CRITERION_TOLERANCE.

  • convergence.absolute_params_tolerance (float): maximal distance between points in the simplex. The default is CONVERGENCE_SECOND_BEST_ABSOLUTE_PARAMS_TOLERANCE.

  • batch_evaluator (string or callable): See Batch evaluators for

    details. Default “joblib”.

pounders
"pounders"

Minimize a function using the POUNDERS algorithm.

POUNDERs ([algo_12], [algo_13], GitHub repository)

can be a useful tool for economists who estimate structural models using indirect inference, because unlike commonly used algorithms such as Nelder-Mead, POUNDERs is tailored for minimizing a non-linear sum of squares objective function, and therefore may require fewer iterations to arrive at a local optimum than Nelder-Mead.

The criterion function func() should return a dictionary with the following fields:

  1. "value": The sum of squared (potentially weighted) errors.

  2. "root_contributions": An array containing the root (weighted) contributions.

Scaling the problem is necessary such that bounds correspond to the unit hypercube \([0, 1]^n\). For unconstrained problems, scale each parameter such that unit changes in parameters result in similar order-of-magnitude changes in the criterion value(s).

pounders supports the following options:

  • convergence_absolute_gradient_tolerance: Convergence tolerance for the absolute gradient norm. Stop if norm of the gradient is less than this. Default is 1e-8.

  • convergence_relative_gradient_tolerance: Convergence tolerance for the relative gradient norm. Stop if norm of the gradient relative to the criterion value is less than this. Default is 1-8.

  • convergence_scaled_gradient_tolerance: Convergence tolerance for the scaled gradient norm. Stop if norm of the gradient divided by norm of the gradient at the initial parameters is less than this. Disabled, i.e. set to False, by default.

  • max_interpolation_points (int): Maximum number of interpolation points. Default is 2 * n + 1, where n is the length of the parameter vector.

  • stopping_max_iterations (int): Maximum number of iterations. If reached, terminate. Default is 2000.

  • trustregion_initial_radius (float): Delta, initial trust-region radius. 0.1 by default.

  • trustregion_minimal_radius (float): Minimal trust-region radius. 1e-6 by default.

  • trustregion_maximal_radius (float): Maximal trust-region radius. 1e6 by default.

  • trustregion_shrinking_factor_not_successful (float): Shrinking factor of the trust-region radius in case the solution vector of the suproblem is not accepted, but the model is fully linear (i.e. “valid”). Defualt is 0.5.

  • trustregion_expansion_factor_successful (float): Shrinking factor of the trust-region radius in case the solution vector of the suproblem is accepted. Default is 2.

  • theta1 (float): Threshold for adding the current x candidate to the model. Function argument to find_affine_points(). Default is 1e-5.

  • theta2 (float): Threshold for adding the current x candidate to the model. Argument to get_interpolation_matrices_residual_model(). Default is 1e-4.

  • trustregion_threshold_successful (float): First threshold for accepting the solution vector of the subproblem as the best x candidate. Default is 0.

  • trustregion_threshold_very_successful (float): Second threshold for accepting the solution vector of the subproblem as the best x candidate. Default is 0.1.

  • c1 (float): Treshold for accepting the norm of our current x candidate. Function argument to find_affine_points() for the case where input array model_improving_points is zero.

  • c2 (int): Treshold for accepting the norm of our current x candidate. Equal to 10 by default. Argument to find_affine_points() in case the input array model_improving_points is not zero.

  • trustregion_subproblem_solver (str): Solver to use for the trust-region subproblem. Two internal solvers are supported: - “bntr”: Bounded Newton Trust-Region (default, supports bound constraints) - “gqtpar”: (does not support bound constraints)

  • trustregion_subsolver_options (dict): Options dictionary containing the stopping criteria for the subproblem. It takes different keys depending on the type of subproblem solver used. With the exception of the stopping criterion “maxiter”, which is always included.

    If the subsolver “bntr” is used, the dictionary also contains the tolerance levels “gtol_abs”, “gtol_rel”, and “gtol_scaled”. Moreover, the “conjugate_gradient_method” can be provided. Available conjugate gradient methods are: - “cg”. In this case, two additional stopping criteria are “gtol_abs_cg” and “gtol_rel_cg” - “steihaug-toint” - “trsbox” (default)

    If the subsolver “gqtpar” is employed, the two stopping criteria are “k_easy” and “k_hard”.

    None of the dictionary keys need to be specified by default, but can be.

  • batch_evaluator (str or callable): Name of a pre-implemented batch evaluator (currently “joblib” and “pathos_mp”) or callable with the same interface as the estimagic batch_evaluators. Default is “joblib”.

  • n_cores (int): Number of processes used to parallelize the function evaluations. Default is 1.

Optimizers from the Toolkit for Advanced Optimization (TAO)#

We wrap the pounders algorithm from the Toolkit of Advanced optimization. To use it you need to have petsc4py installed.

tao_pounders
"tao_pounders"

Minimize a function using the POUNDERs algorithm.

POUNDERs ([algo_12], [algo_13], GitHub repository)

can be a useful tool for economists who estimate structural models using indirect inference, because unlike commonly used algorithms such as Nelder-Mead, POUNDERs is tailored for minimizing a non-linear sum of squares objective function, and therefore may require fewer iterations to arrive at a local optimum than Nelder-Mead.

The criterion function func() should return a dictionary with the following fields:

  1. "value": The sum of squared (potentially weighted) errors.

  2. "root_contributions": An array containing the root (weighted) contributions.

Scaling the problem is necessary such that bounds correspond to the unit hypercube \([0, 1]^n\). For unconstrained problems, scale each parameter such that unit changes in parameters result in similar order-of-magnitude changes in the criterion value(s).

POUNDERs has several convergence criteria. Let \(X\) be the current parameter vector, \(X_0\) the initial parameter vector, \(g\) the gradient, and \(f\) the criterion function.

absolute_gradient_tolerance stops the optimization if the norm of the gradient falls below \(\epsilon\).

\[||g(X)|| < \epsilon\]

relative_gradient_tolerance stops the optimization if the norm of the gradient relative to the criterion value falls below \(epsilon\).

\[\frac{||g(X)||}{|f(X)|} < \epsilon\]

scaled_gradient_tolerance stops the optimization if the norm of the gradient is lower than some fraction \(epsilon\) of the norm of the gradient at the initial parameters.

\[\frac{||g(X)||}{||g(X0)||} < \epsilon\]
  • convergence.absolute_gradient_tolerance (float): Stop if norm of gradient is less than this. If set to False the algorithm will not consider convergence.absolute_gradient_tolerance.

  • convergence.relative_gradient_tolerance (float): Stop if relative norm of gradient is less than this. If set to False the algorithm will not consider convergence.relative_gradient_tolerance.

  • convergence.scaled_gradient_tolerance (float): Stop if scaled norm of gradient is smaller than this. If set to False the algorithm will not consider convergence.scaled_gradient_tolerance.

  • trustregion.initial_radius (float): Initial value of the trust region radius. It must be \(> 0\).

  • stopping.max_iterations (int): Alternative Stopping criterion. If set the routine will stop after the number of specified iterations or after the step size is sufficiently small. If the variable is set the default criteria will all be ignored.

Optimizers from the Numerical Algorithms Group (NAG)#

We wrap two algorithms from the numerical algorithms group. To use them, you need to install each of them separately:

  • pip install DFO-LS

  • pip install Py-BOBYQA

nag_dfols
"nag_dfols"

Minimize a function with least squares structure using DFO-LS.

The DFO-LS algorithm [algo_14] is designed to solve the nonlinear least-squares minimization problem (with optional bound constraints). Remember to cite [algo_14] when using DF-OLS in addition to estimagic.

\[\begin{split}\min_{x\in\mathbb{R}^n} &\quad f(x) := \sum_{i=1}^{m}r_{i}(x)^2 \\ \text{s.t.} &\quad \text{lower_bounds} \leq x \leq \text{upper_bounds}\end{split}\]

The \(r_{i}\) are called root contributions in estimagic.

DFO-LS is a derivative-free optimization algorithm, which means it does not require the user to provide the derivatives of f(x) or \(r_{i}(x)\), nor does it attempt to estimate them internally (by using finite differencing, for instance).

There are two main situations when using a derivative-free algorithm (such as DFO-LS) is preferable to a derivative-based algorithm (which is the vast majority of least-squares solvers):

  1. If the residuals are noisy, then calculating or even estimating their derivatives may be impossible (or at least very inaccurate). By noisy, we mean that if we evaluate \(r_{i}(x)\) multiple times at the same value of x, we get different results. This may happen when a Monte Carlo simulation is used, for instance.

  2. If the residuals are expensive to evaluate, then estimating derivatives (which requires n evaluations of each \(r_{i}(x)\) for every point of interest x) may be prohibitively expensive. Derivative-free methods are designed to solve the problem with the fewest number of evaluations of the criterion as possible.

To read the detailed documentation of the algorithm click here.

There are four possible convergence criteria:

  1. when the lower trust region radius is shrunk below a minimum (convergence.minimal_trustregion_radius_tolerance).

  2. when the improvements of iterations become very small (convergence.slow_progress). This is very similar to relative_criterion_tolerance but convergence.slow_progress is more general allowing to specify not only the threshold for convergence but also a period over which the improvements must have been very small.

  3. when a sufficient reduction to the criterion value at the start parameters has been reached, i.e. when \(\frac{f(x)}{f(x_0)} \leq \text{convergence.scaled_criterion_tolerance}\)

  4. when all evaluations on the interpolation points fall within a scaled version of the noise level of the criterion function. This is only applicable if the criterion function is noisy. You can specify this criterion with convergence.noise_corrected_criterion_tolerance.

DF-OLS supports resetting the optimization and doing a fast start by starting with a smaller interpolation set and growing it dynamically. For more information see their detailed documentation and [algo_14].

  • clip_criterion_if_overflowing (bool): see The default algorithm options. convergence.minimal_trustregion_radius_tolerance (float): see The default algorithm options.

  • convergence.noise_corrected_criterion_tolerance (float): Stop when the evaluations on the set of interpolation points all fall within this factor of the noise level. The default is 1, i.e. when all evaluations are within the noise level. If you want to not use this criterion but still flag your criterion function as noisy, set this tolerance to 0.0.

    Warning

    Very small values, as in most other tolerances don’t make sense here.

  • convergence.scaled_criterion_tolerance (float): Terminate if a point is reached where the ratio of the criterion value to the criterion value at the start params is below this value, i.e. if \(f(x_k)/f(x_0) \leq \text{convergence.scaled_criterion_tolerance}\). Note this is deactivated unless the lowest mathematically possible criterion value (0.0) is actually achieved.

  • convergence.slow_progress (dict): Arguments for converging when the evaluations over several iterations only yield small improvements on average, see see The default algorithm options for details.

  • initial_directions (str): see The default algorithm options.

  • interpolation_rounding_error (float): see The default algorithm options.

  • noise_additive_level (float): Used for determining the presence of noise and the convergence by all interpolation points being within noise level. 0 means no additive noise. Only multiplicative or additive is supported.

  • noise_multiplicative_level (float): Used for determining the presence of noise and the convergence by all interpolation points being within noise level. 0 means no multiplicative noise. Only multiplicative or additive is supported.

  • noise_n_evals_per_point (callable): How often to evaluate the criterion function at each point. This is only applicable for criterion functions with noise, when averaging multiple evaluations at the same point produces a more accurate value. The input parameters are the upper_trustregion_radius (\(\Delta\)), the lower_trustregion_radius (\(\rho\)), how many iterations the algorithm has been running for, n_iterations and how many resets have been performed, n_resets. The function must return an integer. Default is no averaging (i.e. noise_n_evals_per_point(...) = 1).

  • random_directions_orthogonal (bool): see The default algorithm options.

  • stopping.max_criterion_evaluations (int): see The default algorithm options.

  • threshold_for_safety_step (float): see The default algorithm options.

  • trustregion.expansion_factor_successful (float): see The default algorithm options.

  • trustregion.expansion_factor_very_successful (float): see The default algorithm options.

  • trustregion.fast_start_options (dict): see The default algorithm options.

  • trustregion.initial_radius (float): Initial value of the trust region radius.

  • trustregion.method_to_replace_extra_points (str): If replacing extra points in successful iterations, whether to use geometry improving steps or the momentum method. Can be “geometry_improving” or “momentum”.

  • trustregion.n_extra_points_to_replace_successful (int): The number of extra points (other than accepting the trust region step) to replace. Useful when trustregion.n_interpolation_points > len(x) + 1.

  • trustregion.n_interpolation_points (int): The number of interpolation points to use. The default is len(x) + 1. If using resets, this is the number of points to use in the first run of the solver, before any resets.

  • trustregion.precondition_interpolation (bool): see The default algorithm options.

  • trustregion.shrinking_factor_not_successful (float): see The default algorithm options.

  • trustregion.shrinking_factor_lower_radius (float): see The default algorithm options.

  • trustregion.shrinking_factor_upper_radius (float): see The default algorithm options.

  • trustregion.threshold_successful (float): Share of the predicted improvement that has to be achieved for a trust region iteration to count as successful.

  • trustregion.threshold_very_successful (float): Share of the predicted improvement that has to be achieved for a trust region iteration to count as very successful.

nag_pybobyqa
"nag_pybobyqa"

Minimize a function using the BOBYQA algorithm.

BOBYQA ([algo_15], [algo_16], [algo_17]) is a derivative-free trust-region method. It is designed to solve nonlinear local minimization problems.

Remember to cite [algo_15] and [algo_16] when using pybobyqa in addition to estimagic. If you take advantage of the seek_global_optimum option, cite [algo_17] additionally.

There are two main situations when using a derivative-free algorithm like BOBYQA is preferable to derivative-based algorithms:

  1. The criterion function is not deterministic, i.e. if we evaluate the criterion function multiple times at the same parameter vector we get different results.

  2. The criterion function is very expensive to evaluate and only finite differences are available to calculate its derivative.

The detailed documentation of the algorithm can be found here.

There are four possible convergence criteria:

  1. when the trust region radius is shrunk below a minimum. This is approximately equivalent to an absolute parameter tolerance.

  2. when the criterion value falls below an absolute, user-specified value, the optimization terminates successfully.

  3. when insufficient improvements have been gained over a certain number of iterations. The (absolute) threshold for what constitutes an insufficient improvement, how many iterations have to be insufficient and with which iteration to compare can all be specified by the user.

  4. when all evaluations on the interpolation points fall within a scaled version of the noise level of the criterion function. This is only applicable if the criterion function is noisy.

  • clip_criterion_if_overflowing (bool): see The default algorithm options.

  • convergence.criterion_value (float): Terminate successfully if the criterion value falls below this threshold. This is deactivated (i.e. set to -inf) by default.

  • convergence.minimal_trustregion_radius_tolerance (float): Minimum allowed value of the trust region radius, which determines when a successful termination occurs.

  • convergence.noise_corrected_criterion_tolerance (float): Stop when the evaluations on the set of interpolation points all fall within this factor of the noise level. The default is 1, i.e. when all evaluations are within the noise level. If you want to not use this criterion but still flag your criterion function as noisy, set this tolerance to 0.0.

    Warning

    Very small values, as in most other tolerances don’t make sense here.

  • convergence.slow_progress (dict): Arguments for converging when the evaluations over several iterations only yield small improvements on average, see see The default algorithm options for details.

  • initial_directions (str)``: see The default algorithm options.

  • interpolation_rounding_error (float): see The default algorithm options.

  • noise_additive_level (float): Used for determining the presence of noise and the convergence by all interpolation points being within noise level. 0 means no additive noise. Only multiplicative or additive is supported.

  • noise_multiplicative_level (float): Used for determining the presence of noise and the convergence by all interpolation points being within noise level. 0 means no multiplicative noise. Only multiplicative or additive is supported.

  • noise_n_evals_per_point (callable): How often to evaluate the criterion function at each point. This is only applicable for criterion functions with noise, when averaging multiple evaluations at the same point produces a more accurate value. The input parameters are the upper_trustregion_radius (delta), the lower_trustregion_radius (rho), how many iterations the algorithm has been running for, n_iterations and how many resets have been performed, n_resets. The function must return an integer. Default is no averaging (i.e. noise_n_evals_per_point(...) = 1).

  • random_directions_orthogonal (bool): see The default algorithm options.

  • seek_global_optimum (bool): whether to apply the heuristic to escape local minima presented in [algo_17]. Only applies for noisy criterion functions.

  • stopping.max_criterion_evaluations (int): see The default algorithm options.

  • threshold_for_safety_step (float): see The default algorithm options.

  • trustregion.expansion_factor_successful (float): see The default algorithm options.

  • trustregion.expansion_factor_very_successful (float): see The default algorithm options.

  • trustregion.initial_radius (float): Initial value of the trust region radius.

  • trustregion.minimum_change_hession_for_underdetermined_interpolation (bool): Whether to solve the underdetermined quadratic interpolation problem by minimizing the Frobenius norm of the Hessian, or change in Hessian.

  • trustregion.n_interpolation_points (int): The number of interpolation points to use. With $n=len(x)$ the default is $2n+1$ if the criterion is not noisy. Otherwise, it is set to $(n+1)(n+2)/2)$.

    Larger values are particularly useful for noisy problems. Py-BOBYQA requires

    \[n + 1 \leq \text{trustregion.n_interpolation_points} \leq (n+1)(n+2)/2.\]
  • trustregion.precondition_interpolation (bool): see The default algorithm options.

  • trustregion.reset_options (dict): Options for resetting the optimization, see The default algorithm options for details.

  • trustregion.shrinking_factor_not_successful (float): see The default algorithm options.

  • trustregion.shrinking_factor_upper_radius (float): see The default algorithm options.

  • trustregion.shrinking_factor_lower_radius (float): see The default algorithm options.

  • trustregion.threshold_successful (float): see The default algorithm options.

  • trustregion.threshold_very_successful (float): see The default algorithm options.

PYGMO2 Optimizers#

Please cite [algo_18] in addition to estimagic when using pygmo. estimagic supports the following pygmo2 optimizers.

pygmo_gaco
"pygmo_gaco"

Minimize a scalar function using the generalized ant colony algorithm.

The version available through pygmo is an generalized version of the original ant colony algorithm proposed by [algo_19].

This algorithm can be applied to box-bounded problems.

Ant colony optimization is a class of optimization algorithms modeled on the actions of an ant colony. Artificial “ants” (e.g. simulation agents) locate optimal solutions by moving through a parameter space representing all possible solutions. Real ants lay down pheromones directing each other to resources while exploring their environment. The simulated “ants” similarly record their positions and the quality of their solutions, so that in later simulation iterations more ants locate better solutions.

The generalized ant colony algorithm generates future generations of ants by using a multi-kernel gaussian distribution based on three parameters (i.e., pheromone values) which are computed depending on the quality of each previous solution. The solutions are ranked through an oracle penalty method.

  • population_size (int): Size of the population. If None, it’s twice the number of parameters but at least 64.

  • batch_evaluator (str or Callable): Name of a pre-implemented batch evaluator (currently ‘joblib’ and ‘pathos_mp’) or Callable with the same interface as the estimagic batch_evaluators. See Batch evaluators.

  • n_cores (int): Number of cores to use.

  • seed (int): seed used by the internal random number generator.

  • discard_start_params (bool): If True, the start params are not guaranteed to be part of the initial population. This saves one criterion function evaluation that cannot be done in parallel with other evaluations. Default False.

  • stopping.max_iterations (int): Number of generations to evolve.

  • kernel_size (int): Number of solutions stored in the solution archive.

  • speed_parameter_q (float): This parameter manages the convergence speed towards the found minima (the smaller the faster). In the pygmo documentation it is referred to as $q$. It must be positive and can be larger than 1. The default is 1.0 until threshold is reached. Then it is set to 0.01.

  • oracle (float): oracle parameter used in the penalty method.

  • accuracy (float): accuracy parameter for maintaining a minimum penalty function’s values distances.

  • threshold (int): when the iteration counter reaches the threshold the convergence speed is set to 0.01 automatically. To deactivate this effect set the threshold to stopping.max_iterations which is the largest allowed value.

  • speed_of_std_values_convergence (int): parameter that determines the convergence speed of the standard deviations. This must be an integer (n_gen_mark in pygmo and pagmo).

  • stopping.max_n_without_improvements (int): if a positive integer is assigned here, the algorithm will count the runs without improvements, if this number exceeds the given value, the algorithm will be stopped.

  • stopping.max_criterion_evaluations (int): maximum number of function evaluations.

  • focus (float): this parameter makes the search for the optimum greedier and more focused on local improvements (the higher the greedier). If the value is very high, the search is more focused around the current best solutions. Values larger than 1 are allowed.

  • cache (bool): if True, memory is activated in the algorithm for multiple calls.

pygmo_bee_colony
"pygmo_bee_colony"

Minimize a scalar function using the artifical bee colony algorithm.

The Artificial Bee Colony Algorithm was originally proposed by [algo_20]. The implemented version of the algorithm is proposed in [algo_21]. The algorithm is only suited for bounded parameter spaces.

  • stopping.max_iterations (int): Number of generations to evolve.

  • seed (int): seed used by the internal random number generator.

  • discard_start_params (bool): If True, the start params are not guaranteed to be part of the initial population. This saves one criterion function evaluation that cannot be done in parallel with other evaluations. Default False.

  • max_n_trials (int): Maximum number of trials for abandoning a source. Default is 1.

  • population_size (int): Size of the population. If None, it’s twice the number of parameters but at least 20.

pygmo_de
"pygmo_de"

Minimize a scalar function using the differential evolution algorithm.

Differential Evolution is a heuristic optimizer originally presented in [algo_22]. The algorithm is only suited for bounded parameter spaces.

  • population_size (int): Size of the population. If None, it’s twice the number of parameters but at least 10.

  • seed (int): seed used by the internal random number generator.

  • discard_start_params (bool): If True, the start params are not guaranteed to be part of the initial population. This saves one criterion function evaluation that cannot be done in parallel with other evaluations. Default False.

  • stopping.max_iterations (int): Number of generations to evolve.

  • weight_coefficient (float): Weight coefficient. It is denoted by $F$ in the main paper and must lie in [0, 2]. It controls the amplification of the differential variation $(x_{r_2, G} - x_{r_3, G})$.

  • crossover_probability (float): Crossover probability.

  • mutation_variant (str or int): code for the mutation variant to create a new candidate individual. The default is . The following are available:

    • “best/1/exp” (1, when specified as int)

    • “rand/1/exp” (2, when specified as int)

    • “rand-to-best/1/exp” (3, when specified as int)

    • “best/2/exp” (4, when specified as int)

    • “rand/2/exp” (5, when specified as int)

    • “best/1/bin” (6, when specified as int)

    • “rand/1/bin” (7, when specified as int)

    • “rand-to-best/1/bin” (8, when specified as int)

    • “best/2/bin” (9, when specified as int)

    • “rand/2/bin” (10, when specified as int)

  • convergence.criterion_tolerance: stopping criteria on the criterion tolerance. Default is 1e-6. It is not clear whether this is the absolute or relative criterion tolerance.

  • convergence.relative_params_tolerance: stopping criteria on the x tolerance. In pygmo the default is 1e-6 but we use our default value of 1e-5.

pygmo_sea
"pygmo_sea"

Minimize a scalar function using the (N+1)-ES simple evolutionary algorithm.

This algorithm represents the simplest evolutionary strategy, where a population of $lambda$ individuals at each generation produces one offspring by mutating its best individual uniformly at random within the bounds. Should the offspring be better than the worst individual in the population it will substitute it.

See [algo_23].

The algorithm is only suited for bounded parameter spaces.

  • population_size (int): Size of the population. If None, it’s twice the number of parameters but at least 10.

  • seed (int): seed used by the internal random number generator.

  • discard_start_params (bool): If True, the start params are not guaranteed to be part of the initial population. This saves one criterion function evaluation that cannot be done in parallel with other evaluations. Default False.

  • stopping.max_iterations (int): number of generations to consider. Each generation will compute the objective function once.

pygmo_sga
"pygmo_sga"

Minimize a scalar function using a simple genetic algorithm.

A detailed description of the algorithm can be found in the pagmo2 documentation.

See also [algo_23].

  • population_size (int): Size of the population. If None, it’s twice the number of parameters but at least 64.

  • seed (int): seed used by the internal random number generator.

  • discard_start_params (bool): If True, the start params are not guaranteed to be part of the initial population. This saves one criterion function evaluation that cannot be done in parallel with other evaluations. Default False.

  • stopping.max_iterations (int): Number of generations to evolve.

  • crossover_probability (float): Crossover probability.

  • crossover_strategy (str): the crossover strategy. One of “exponential”,“binomial”, “single” or “sbx”. Default is “exponential”.

  • eta_c (float): distribution index for “sbx” crossover. This is an inactive parameter if other types of crossovers are selected. Can be in [1, 100].

  • mutation_probability (float): Mutation probability.

  • mutation_strategy (str): Mutation strategy. Must be “gaussian”, “polynomial” or “uniform”. Default is “polynomial”.

  • mutation_polynomial_distribution_index (float): Must be in [0, 1]. Default is 1.

  • mutation_gaussian_width (float): Must be in [0, 1]. Default is 1.

  • selection_strategy (str): Selection strategy. Must be “tournament” or “truncated”.

  • selection_truncated_n_best (int): number of best individuals to use in the “truncated” selection mechanism.

  • selection_tournament_size (int): size of the tournament in the “tournament” selection mechanism. Default is 1.

pygmo_sade
"pygmo_sade"

Minimize a scalar function using Self-adaptive Differential Evolution.

The original Differential Evolution algorithm (pygmo_de) can be significantly improved introducing the idea of parameter self-adaptation.

Many different proposals have been made to self-adapt both the crossover and the F parameters of the original differential evolution algorithm. pygmo’s implementation supports two different mechanisms. The first one, proposed by [algo_24], does not make use of the differential evolution operators to produce new values for the weight coefficient $F$ and the crossover probability $CR$ and, strictly speaking, is thus not self-adaptation, rather parameter control. The resulting differential evolution variant is often referred to as jDE. The second variant is inspired by the ideas introduced by [algo_25] and uses a variaton of the selected DE operator to produce new $CR$ anf $F$ parameters for each individual. This variant is referred to iDE.

  • population_size (int): Size of the population. If None, it’s twice the number of parameters but at least 64.

  • seed (int): seed used by the internal random number generator.

  • discard_start_params (bool): If True, the start params are not guaranteed to be part of the initial population. This saves one criterion function evaluation that cannot be done in parallel with other evaluations. Default False.

  • jde (bool): Whether to use the jDE self-adaptation variant to control the $F$ and $CR$ parameter. If True jDE is used, else iDE.

  • stopping.max_iterations (int): Number of generations to evolve.

  • mutation_variant (int or str): code for the mutation variant to create a new candidate individual. The default is “rand/1/exp”. The first ten are the classical mutation variants introduced in the orginal DE algorithm, the remaining ones are, instead, considered in the work by [algo_25]. The following are available:

    • “best/1/exp” or 1

    • “rand/1/exp” or 2

    • “rand-to-best/1/exp” or 3

    • “best/2/exp” or 4

    • “rand/2/exp” or 5

    • “best/1/bin” or 6

    • “rand/1/bin” or 7

    • “rand-to-best/1/bin” or 8

    • “best/2/bin” or 9

    • “rand/2/bin” or 10

    • “rand/3/exp” or 11

    • “rand/3/bin” or 12

    • “best/3/exp” or 13

    • “best/3/bin” or 14

    • “rand-to-current/2/exp” or 15

    • “rand-to-current/2/bin” or 16

    • “rand-to-best-and-current/2/exp” or 17

    • “rand-to-best-and-current/2/bin” or 18

  • keep_adapted_params (bool): when true the adapted parameters $CR$ anf $F$ are not reset between successive calls to the evolve method. Default is False.

  • ftol (float): stopping criteria on the x tolerance.

  • xtol (float): stopping criteria on the f tolerance.

pygmo_cmaes
"pygmo_cmaes"

Minimize a scalar function using the Covariance Matrix Evolutionary Strategy.

CMA-ES is one of the most successful algorithm, classified as an Evolutionary Strategy, for derivative-free global optimization. The version supported by estimagic is the version described in [algo_26].

In contrast to the pygmo version, estimagic always sets force_bounds to True. This avoids that ill defined parameter values are evaluated.

  • population_size (int): Size of the population. If None, it’s twice the number of parameters but at least 64.

  • seed (int): seed used by the internal random number generator.

  • discard_start_params (bool): If True, the start params are not guaranteed to be part of the initial population. This saves one criterion function evaluation that cannot be done in parallel with other evaluations. Default False.

  • stopping.max_iterations (int): Number of generations to evolve.

  • backward_horizon (float): backward time horizon for the evolution path. It must lie betwen 0 and 1.

  • variance_loss_compensation (float): makes partly up for the small variance loss in case the indicator is zero. cs in the MATLAB Code of [algo_26]. It must lie between 0 and 1.

  • learning_rate_rank_one_update (float): learning rate for the rank-one update of the covariance matrix. c1 in the pygmo and pagmo documentation. It must lie between 0 and 1.

  • learning_rate_rank_mu_update (float): learning rate for the rank-mu update of the covariance matrix. cmu in the pygmo and pagmo documentation. It must lie between 0 and 1.

  • initial_step_size (float): initial step size, \(\sigma^0\) in the original paper.

  • ftol (float): stopping criteria on the x tolerance.

  • xtol (float): stopping criteria on the f tolerance.

  • keep_adapted_params (bool): when true the adapted parameters are not reset between successive calls to the evolve method. Default is False.

pygmo_simulated_annealing
"pygmo_simulated_annealing"

Minimize a function with the simulated annealing algorithm.

This version of the simulated annealing algorithm is, essentially, an iterative random search procedure with adaptive moves along the coordinate directions. It permits uphill moves under the control of metropolis criterion, in the hope to avoid the first local minima encountered. This version is the one proposed in [algo_27].

  • population_size (int): Size of the population. If None, it’s twice the number of parameters but at least 64.

  • seed (int): seed used by the internal random number generator.

  • discard_start_params (bool): If True, the start params are not guaranteed to be part of the initial population. This saves one criterion function evaluation that cannot be done in parallel with other evaluations. Default False.

  • start_temperature (float): starting temperature. Must be > 0.

  • end_temperature (float): final temperature. Our default (0.01) is lower than in pygmo and pagmo. The final temperature must be positive.

  • n_temp_adjustments (int): number of temperature adjustments in the annealing schedule.

  • n_range_adjustments (int): number of adjustments of the search range performed at a constant temperature.

  • bin_size (int): number of mutations that are used to compute the acceptance rate.

  • start_range (float): starting range for mutating the decision vector. It must lie between 0 and 1.

pygmo_pso
"pygmo_pso"

Minimize a scalar function using Particle Swarm Optimization.

Particle swarm optimization (PSO) is a population based algorithm inspired by the foraging behaviour of swarms. In PSO each point has memory of the position where it achieved the best performance xli (local memory) and of the best decision vector \(x^g\) in a certain neighbourhood, and uses this information to update its position.

For a survey on particle swarm optimization algorithms, see [algo_28].

Each particle determines its future position \(x_{i+1} = x_i + v_i\) where

\[v_{i+1} = \omega (v_i + \eta_1 \cdot \mathbf{r}_1 \cdot (x_i - x^{l}_i) + \eta_2 \cdot \mathbf{r}_2 \cdot (x_i - x^g))\]
  • population_size (int): Size of the population. If None, it’s twice the number of parameters but at least 10.

  • seed (int): seed used by the internal random number generator.

  • discard_start_params (bool): If True, the start params are not guaranteed to be part of the initial population. This saves one criterion function evaluation that cannot be done in parallel with other evaluations. Default False.

  • stopping.max_iterations (int): Number of generations to evolve.

  • omega (float): depending on the variant chosen, \(\omega\) is the particles’ inertia weight or the construction coefficient. It must lie between 0 and 1.

  • force_of_previous_best (float): \(\eta_1\) in the equation above. It’s the magnitude of the force, applied to the particle’s velocity, in the direction of its previous best position. It must lie between 0 and 4.

  • force_of_best_in_neighborhood (float): \(\eta_2\) in the equation above. It’s the magnitude of the force, applied to the particle’s velocity, in the direction of the best position in its neighborhood. It must lie between 0 and 4.

  • max_velocity (float): maximum allowed particle velocity as fraction of the box bounds. It must lie between 0 and 1.

  • algo_variant (int or str): algorithm variant to be used:
    • 1 or “canonical_inertia”: Canonical (with inertia weight)

    • 2 or “social_and_cog_rand”: Same social and cognitive rand.

    • 3 or “all_components_rand”: Same rand. for all components

    • 4 or “one_rand”: Only one rand.

    • 5 or “canonical_constriction”: Canonical (with constriction fact.)

    • 6 or “fips”: Fully Informed (FIPS)

  • neighbor_definition (int or str): swarm topology that defines each particle’s neighbors that is to be used:

    • 1 or “gbest”

    • 2 or “lbest”

    • 3 or “Von Neumann”

    • 4 or “Adaptive random”

  • neighbor_param (int): the neighbourhood parameter. If the lbest topology is selected (neighbor_definition=2), it represents each particle’s indegree (also outdegree) in the swarm topology. Particles have neighbours up to a radius of k = neighbor_param / 2 in the ring. If the Randomly-varying neighbourhood topology is selected (neighbor_definition=4), it represents each particle’s maximum outdegree in the swarm topology. The minimum outdegree is 1 (the particle always connects back to itself). If neighbor_definition is 1 or 3 this parameter is ignored.

  • keep_velocities (bool): when true the particle velocities are not reset between successive calls to evolve.

pygmo_pso_gen
"pygmo_pso_gen"

Minimize a scalar function with generational Particle Swarm Optimization.

Particle Swarm Optimization (generational) is identical to pso, but does update the velocities of each particle before new particle positions are computed (taking into consideration all updated particle velocities). Each particle is thus evaluated on the same seed within a generation as opposed to the standard PSO which evaluates single particle at a time. Consequently, the generational PSO algorithm is suited for stochastic optimization problems.

For a survey on particle swarm optimization algorithms, see [algo_28].

Each particle determines its future position \(x_{i+1} = x_i + v_i\) where

\[v_{i+1} = \omega (v_i + \eta_1 \cdot \mathbf{r}_1 \cdot (x_i - x^{l}_i) + \eta_2 \cdot \mathbf{r}_2 \cdot (x_i - x^g))\]
  • population_size (int): Size of the population. If None, it’s twice the number of parameters but at least 10.

  • batch_evaluator (str or Callable): Name of a pre-implemented batch evaluator (currently ‘joblib’ and ‘pathos_mp’) or Callable with the same interface as the estimagic batch_evaluators. See Batch evaluators.

  • n_cores (int): Number of cores to use.

  • seed (int): seed used by the internal random number generator.

  • discard_start_params (bool): If True, the start params are not guaranteed to be part of the initial population. This saves one criterion function evaluation that cannot be done in parallel with other evaluations. Default False.

  • stopping.max_iterations (int): Number of generations to evolve.

  • omega (float): depending on the variant chosen, \(\omega\) is the particles’ inertia weight or the constructuion coefficient. It must lie between 0 and 1.

  • force_of_previous_best (float): \(\eta_1\) in the equation above. It’s the magnitude of the force, applied to the particle’s velocity, in the direction of its previous best position. It must lie between 0 and 4.

  • force_of_best_in_neighborhood (float): \(\eta_2\) in the equation above. It’s the magnitude of the force, applied to the particle’s velocity, in the direction of the best position in its neighborhood. It must lie between 0 and 4.

  • max_velocity (float): maximum allowed particle velocity as fraction of the box bounds. It must lie between 0 and 1.

  • algo_variant (int): code of the algorithm’s variant to be used:

    • 1 or “canonical_inertia”: Canonical (with inertia weight)

    • 2 or “social_and_cog_rand”: Same social and cognitive rand.

    • 3 or “all_components_rand”: Same rand. for all components

    • 4 or “one_rand”: Only one rand.

    • 5 or “canonical_constriction”: Canonical (with constriction fact.)

    • 6 or “fips”: Fully Informed (FIPS)

  • neighbor_definition (int): code for the swarm topology that defines each particle’s neighbors that is to be used:

    • 1 or “gbest”

    • 2 or “lbest”

    • 3 or “Von Neumann”

    • 4 or “Adaptive random”

  • neighbor_param (int): the neighbourhood parameter. If the lbest topology is selected (neighbor_definition=2), it represents each particle’s indegree (also outdegree) in the swarm topology. Particles have neighbours up to a radius of k = neighbor_param / 2 in the ring. If the Randomly-varying neighbourhood topology is selected (neighbor_definition=4), it represents each particle’s maximum outdegree in the swarm topology. The minimum outdegree is 1 (the particle always connects back to itself). If neighbor_definition is 1 or 3 this parameter is ignored.

  • keep_velocities (bool): when true the particle velocities are not reset between successive calls to evolve.

pygmo_mbh
"pygmo_mbh"

Minimize a scalar function using generalized Monotonic Basin Hopping.

Monotonic basin hopping, or simply, basin hopping, is an algorithm rooted in the idea of mapping the objective function $f(x_0)$ into the local minima found starting from $x_0$. This simple idea allows a substantial increase of efficiency in solving problems, such as the Lennard-Jones cluster or the MGA-1DSM interplanetary trajectory problem that are conjectured to have a so-called funnel structure.

See [algo_29] for the paper introducing the basin hopping idea for a Lennard-Jones cluster optimization.

pygmo provides an original generalization of this concept resulting in a meta-algorithm that operates on a population. When a population containing a single individual is used the original method is recovered.

  • population_size (int): Size of the population. If None, it’s twice the number of parameters but at least 250.

  • seed (int): seed used by the internal random number generator.

  • discard_start_params (bool): If True, the start params are not guaranteed to be part of the initial population. This saves one criterion function evaluation that cannot be done in parallel with other evaluations. Default False.

  • inner_algorithm (pygmo.algorithm): an pygmo algorithm or a user-defined algorithm, either C++ or Python. If None the pygmo.compass_search algorithm will be used.

  • stopping.max_inner_runs_without_improvement (int): consecutive runs of the inner algorithm that need to result in no improvement for mbh to stop.

  • perturbation (float): the perturbation to be applied to each component.

pygmo_xnes
"pygmo_xnes"

Minimize a scalar function using Exponential Evolution Strategies.

Exponential Natural Evolution Strategies is an algorithm closely related to CMAES and based on the adaptation of a gaussian sampling distribution via the so-called natural gradient. Like CMAES it is based on the idea of sampling new trial vectors from a multivariate distribution and using the new sampled points to update the distribution parameters. Naively this could be done following the gradient of the expected fitness as approximated by a finite number of sampled points. While this idea offers a powerful lead on algorithmic construction it has some major drawbacks that are solved in the so-called Natural Evolution Strategies class of algorithms by adopting, instead, the natural gradient. xNES is one of the most performing variants in this class.

See [algo_30] and the pagmo documentation on xNES for details.

  • population_size (int): Size of the population. If None, it’s twice the number of parameters but at least 64.

  • seed (int): seed used by the internal random number generator.

  • discard_start_params (bool): If True, the start params are not guaranteed to be part of the initial population. This saves one criterion function evaluation that cannot be done in parallel with other evaluations. Default False.

  • stopping.max_iterations (int): Number of generations to evolve.

  • learning_rate_mean_update (float): learning rate for the mean update (\(\eta_\mu\)). It must be between 0 and 1 or None.

  • learning_rate_step_size_update (float): learning rate for the step-size update. It must be between 0 and 1 or None.

  • learning_rate_cov_matrix_update (float): learning rate for the covariance matrix update. It must be between 0 and 1 or None.

  • initial_search_share (float): share of the given search space that will be initally searched. It must be between 0 and 1. Default is 1.

  • ftol (float): stopping criteria on the x tolerance.

  • xtol (float): stopping criteria on the f tolerance.

  • keep_adapted_params (bool): when true the adapted parameters are not reset between successive calls to the evolve method. Default is False.

pygmo_gwo
"pygmo_gwo"

Minimize a scalar function usinng the Grey Wolf Optimizer.

The grey wolf optimizer was proposed by [algo_31]. The pygmo implementation that is wrapped by estimagic is pased on the pseudo code provided in that paper.

This algorithm is a classic example of a highly criticizable line of search that led in the first decades of our millenia to the development of an entire zoo of metaphors inspiring optimzation heuristics. In our opinion they, as is the case for the grey wolf optimizer, are often but small variations of already existing heuristics rebranded with unnecessray and convoluted biological metaphors. In the case of GWO this is particularly evident as the position update rule is shokingly trivial and can also be easily seen as a product of an evolutionary metaphor or a particle swarm one. Such an update rule is also not particulary effective and results in a rather poor performance most of times.

  • population_size (int): Size of the population. If None, it’s twice the number of parameters but at least 64.

  • seed (int): seed used by the internal random number generator.

  • discard_start_params (bool): If True, the start params are not guaranteed to be part of the initial population. This saves one criterion function evaluation that cannot be done in parallel with other evaluations. Default False.

  • stopping.max_iterations (int): Number of generations to evolve.

pygmo_compass_search
"pygmo_compass_search"

Minimize a scalar function using compass search.

The algorithm is described in [algo_32].

It is considered slow but reliable. It should not be used for stochastic problems.

  • population_size (int): Size of the population. Even though the algorithm is not population based the population size does affect the results of the algorithm.

  • seed (int): seed used by the internal random number generator.

  • discard_start_params (bool): If True, the start params are not guaranteed to be part of the initial population. This saves one criterion function evaluation that cannot be done in parallel with other evaluations. Default False.

  • stopping.max_criterion_evaluations (int): maximum number of function evaluations.

  • start_range (float): the start range. Must be in (0, 1].

  • stop_range (float): the stop range. Must be in (0, start_range].

  • reduction_coeff (float): the range reduction coefficient. Must be in (0, 1).

pygmo_ihs
"pygmo_ihs"

Minimize a scalar function using the improved harmony search algorithm.

Improved harmony search (IHS) was introduced by [algo_33]. IHS supports stochastic problems.

  • population_size (int): Size of the population. If None, it’s twice the number of parameters.

  • seed (int): seed used by the internal random number generator.

  • discard_start_params (bool): If True, the start params are not guaranteed to be part of the initial population. This saves one criterion function evaluation that cannot be done in parallel with other evaluations. Default False.

  • stopping.max_iterations (int): Number of generations to evolve.

  • choose_from_memory_probability (float): probability of choosing from memory (similar to a crossover probability).

  • min_pitch_adjustment_rate (float): minimum pitch adjustment rate. (similar to a mutation rate). It must be between 0 and 1.

  • max_pitch_adjustment_rate (float): maximum pitch adjustment rate. (similar to a mutation rate). It must be between 0 and 1.

  • min_distance_bandwidth (float): minimum distance bandwidth. (similar to a mutation width). It must be positive.

  • max_distance_bandwidth (float): maximum distance bandwidth. (similar to a mutation width).

pygmo_de1220
"pygmo_de1220"

Minimize a scalar function using Self-adaptive Differential Evolution, pygmo flavor.

See the PAGMO documentation for details.

  • population_size (int): Size of the population. If None, it’s twice the number of parameters but at least 64.

  • seed (int): seed used by the internal random number generator.

  • discard_start_params (bool): If True, the start params are not guaranteed to be part of the initial population. This saves one criterion function evaluation that cannot be done in parallel with other evaluations. Default False.

  • jde (bool): Whether to use the jDE self-adaptation variant to control the $F$ and $CR$ parameter. If True jDE is used, else iDE.

  • stopping.max_iterations (int): Number of generations to evolve.

  • allowed_variants (array-like object): allowed mutation variants (can be codes or strings). Each code refers to one mutation variant to create a new candidate individual. The first ten refer to the classical mutation variants introduced in the original DE algorithm, the remaining ones are, instead, considered in the work by [algo_25]. The default is [“rand/1/exp”, “rand-to-best/1/exp”, “rand/1/bin”, “rand/2/bin”, “best/3/exp”, “best/3/bin”, “rand-to-current/2/exp”, “rand-to-current/2/bin”]. The following are available:

    • 1 or “best/1/exp”

    • 2 or “rand/1/exp”

    • 3 or “rand-to-best/1/exp”

    • 4 or “best/2/exp”

    • 5 or “rand/2/exp”

    • 6 or “best/1/bin”

    • 7 or “rand/1/bin”

    • 8 or “rand-to-best/1/bin”

    • 9 or “best/2/bin”

    • 10 or “rand/2/bin”

    • 11 or “rand/3/exp”

    • 12 or “rand/3/bin”

    • 13 or “best/3/exp”

    • 14 or “best/3/bin”

    • 15 or “rand-to-current/2/exp”

    • 16 or “rand-to-current/2/bin”

    • 17 or “rand-to-best-and-current/2/exp”

    • 18 or “rand-to-best-and-current/2/bin”

  • keep_adapted_params (bool): when true the adapted parameters $CR$ anf $F$ are not reset between successive calls to the evolve method. Default is False.

  • ftol (float): stopping criteria on the x tolerance.

  • xtol (float): stopping criteria on the f tolerance.

The Interior Point Optimizer (ipopt)#

estimagic’s support for the Interior Point Optimizer ([algo_34], [algo_35], [algo_36], [algo_37]) is built on cyipopt, a Python wrapper for the Ipopt optimization package.

To use ipopt, you need to have cyipopt installed (conda install cyipopt).

ipopt
"ipopt"

Minimize a scalar function using the Interior Point Optimizer.

This implementation of the Interior Point Optimizer ([algo_34], [algo_35], [algo_36], [algo_37]) relies on cyipopt, a Python wrapper for the Ipopt optimization package.

There are two levels of termination criteria. If the usual “desired” tolerances (see tol, dual_inf_tol etc) are satisfied at an iteration, the algorithm immediately terminates with a success message. On the other hand, if the algorithm encounters “acceptable_iter” many iterations in a row that are considered “acceptable”, it will terminate before the desired convergence tolerance is met. This is useful in cases where the algorithm might not be able to achieve the “desired” level of accuracy.

The options are analogous to the ones in the ipopt documentation with the exception of the linear solver options which are here bundled into a dictionary. Any argument that takes “yes” and “no” in the ipopt documentation can also be passed as a True and False, respectively. and any option that accepts “none” in ipopt accepts a Python None.

The following options are not supported:
  • num_linear_variables: since estimagic may reparametrize your problem and this changes the parameter problem, we do not support this option.

  • derivative checks

  • print options. Use estimagic’s dashboard to monitor your optimization.

  • convergence.relative_criterion_tolerance (float): The algorithm terminates successfully, if the (scaled) non linear programming error becomes smaller than this value.

  • mu_target (float): Desired value of complementarity. Usually, the barrier parameter is driven to zero and the termination test for complementarity is measured with respect to zero complementarity. However, in some cases it might be desired to have Ipopt solve barrier problem for strictly positive value of the barrier parameter. In this case, the value of “mu_target” specifies the final value of the barrier parameter, and the termination tests are then defined with respect to the barrier problem for this value of the barrier parameter. The valid range for this real option is 0 ≤ mu_target and its default value is 0.

  • s_max (float): Scaling threshold for the NLP error.

  • stopping.max_iterations (int): If the maximum number of iterations is reached, the optimization stops, but we do not count this as successful convergence. The difference to max_criterion_evaluations is that one iteration might need several criterion evaluations, for example in a line search or to determine if the trust region radius has to be shrunk.

  • stopping.max_wall_time_seconds (float): Maximum number of walltime clock seconds.

  • stopping.max_cpu_time (float): Maximum number of CPU seconds. A limit on CPU seconds that Ipopt can use to solve one problem. If during the convergence check this limit is exceeded, Ipopt will terminate with a corresponding message. The valid range for this real option is 0 < max_cpu_time and its default value is \(1e+20\) .

  • dual_inf_tol (float): Desired threshold for the dual infeasibility. Absolute tolerance on the dual infeasibility. Successful termination requires that the max-norm of the (unscaled) dual infeasibility is less than this threshold. The valid range for this real option is 0 < dual_inf_tol and its default value is 1.

  • constr_viol_tol (float): Desired threshold for the constraint and bound violation. Absolute tolerance on the constraint and variable bound violation. Successful termination requires that the max-norm of the (unscaled) constraint violation is less than this threshold. If option bound_relax_factor is not zero 0, then Ipopt relaxes given variable bounds. The value of constr_viol_tol is used to restrict the absolute amount of this bound relaxation. The valid range for this real option is 0 < constr_viol_tol and its default value is 0.0001.

  • compl_inf_tol (float): Desired threshold for the complementarity conditions. Absolute tolerance on the complementarity. Successful termination requires that the max-norm of the (unscaled) complementarity is less than this threshold. The valid range for this real option is 0 < text{compl_inf_tol and its default is 0.0001.

  • acceptable_iter (int): Number of “acceptable” iterates before termination. If the algorithm encounters this many successive “acceptable” iterates (see above on the acceptable heuristic), it terminates, assuming that the problem has been solved to best possible accuracy given round-off. If it is set to zero, this heuristic is disabled. The valid range for this integer option is 0 ≤ acceptable_iter.

  • acceptable_tol (float):”Acceptable” convergence tolerance (relative). Determines which (scaled) overall optimality error is considered to be “acceptable”. The valid range for this real option is 0 < acceptable_tol.

  • acceptable_dual_inf_tol (float): “Acceptance” threshold for the dual infeasibility. Absolute tolerance on the dual infeasibility. “Acceptable” termination requires that the (max-norm of the unscaled) dual infeasibility is less than this threshold; see also acceptable_tol . The valid range for this real option is 0 < acceptable_dual_inf_tol and its default value is \(1e+10.\)

  • acceptable_constr_viol_tol (float): “Acceptance” threshold for the constraint violation. Absolute tolerance on the constraint violation. “Acceptable” termination requires that the max-norm of the (unscaled) constraint violation is less than this threshold; see also acceptable_tol . The valid range for this real option is 0 < acceptable_constr_viol_tol and its default value is 0.01.

  • acceptable_compl_inf_tol (float): “Acceptance” threshold for the complementarity conditions. Absolute tolerance on the complementarity. “Acceptable” termination requires that the max-norm of the (unscaled) complementarity is less than this threshold; see also acceptable_tol . The valid range for this real option is 0 < text{acceptable_compl_inf_tol and its default value is 0.01.

  • acceptable_obj_change_tol (float): “Acceptance” stopping criterion based on objective function change. If the relative change of the objective function (scaled by \(max(1,|f(x)|)\) ) is less than this value, this part of the acceptable tolerance termination is satisfied; see also acceptable_tol . This is useful for the quasi-Newton option, which has trouble to bring down the dual infeasibility. The valid range for this real option is 0 ≤ acceptable_obj_change_tol and its default value is \(1e+20\) .

  • diverging_iterates_tol (float): Threshold for maximal value of primal iterates. If any component of the primal iterates exceeded this value (in absolute terms), the optimization is aborted with the exit message that the iterates seem to be diverging. The valid range for this real option is 0 < diverging_iterates_tol and its default value is \(1e+20\) .

  • nlp_lower_bound_inf (float): any bound less or equal this value will be considered -inf (i.e. not lwer bounded). The valid range for this real option is unrestricted and its default value is \(-1e+19\) .

  • nlp_upper_bound_inf (float): any bound greater or this value will be considered \(+\inf\) (i.e. not upper bunded). The valid range for this real option is unrestricted and its default value is \(1e+19\) .

  • fixed_variable_treatment (str): Determines how fixed variables should be handled. The main difference between those options is that the starting point in the “make_constraint” case still has the fixed variables at their given values, whereas in the case “make_parameter(_nodual)” the functions are always evaluated with the fixed values for those variables. Also, for “relax_bounds”, the fixing bound constraints are relaxed (according to bound_relax_factor ). For all but “make_parameter_nodual”, bound multipliers are computed for the fixed variables. The default value for this string option is “make_parameter”. Possible values:

    • “make_parameter”: Remove fixed variable from optimization variables

    • “make_parameter_nodual”: Remove fixed variable from optimization variables and do not compute bound multipliers for fixed variables

    • “make_constraint”: Add equality constraints fixing variables

    • “relax_bounds”: Relax fixing bound constraints

  • dependency_detector (str): Indicates which linear solver should be used to detect linearly dependent equality constraints. This is experimental and does not work well. The default value for this string option is “none”. Possible values:

    • “none” or None: don’t check; no extra work at beginning

    • “mumps”: use MUMPS

    • “wsmp”: use WSMP

    • “ma28”: use MA28

  • dependency_detection_with_rhs (str or bool): Indicates if the right hand sides of the constraints should be considered in addition to gradients during dependency detection. The default value for this string option is “no”. Possible values: ‘yes’, ‘no’, True, False.

  • kappa_d (float): Weight for linear damping term (to handle one-sided bounds). See Section 3.7 in implementation paper. The valid range for this real option is 0 ≤ kappa_d and its default value is \(1e-05\) .

  • bound_relax_factor (float): Factor for initial relaxation of the bounds. Before start of the optimization, the bounds given by the user are relaxed. This option sets the factor for this relaxation. Additional, the constraint violation tolerance constr_viol_tol is used to bound the relaxation by an absolute value. If it is set to zero, then then bounds relaxation is disabled. See Eqn.(35) in implementation paper. Note that the constraint violation reported by Ipopt at the end of the solution process does not include violations of the original (non-relaxed) variable bounds. See also option honor_original_bounds. The valid range for this real option is 0 ≤ bound_relax_factor and its default value is \(1e-08\) .

  • honor_original_bounds (str or bool): Indicates whether final points should be projected into original bunds. Ipopt might relax the bounds during the optimization (see, e.g., option bound_relax_factor ). This option determines whether the final point should be projected back into the user-provide original bounds after the optimization. Note that violations of constraints and complementarity reported by Ipopt at the end of the solution process are for the non-projected point. The default value for this string option is “no”. Possible values: ‘yes’, ‘no’, True, False

  • check_derivatives_for_naninf (str): whether to check for NaN / inf in the derivative matrices. Activating this option will cause an error if an invalid number is detected in the constraint Jacobians or the Lagrangian Hessian. If this is not activated, the test is skipped, and the algorithm might proceed with invalid numbers and fail. If test is activated and an invalid number is detected, the matrix is written to output with print_level corresponding to J_MORE_DETAILED; so beware of large output! The default value for this string option is “no”.

  • jac_c_constant (str or bool): Indicates whether to assume that all equality constraints are linear Activating this option will cause Ipopt to ask for the Jacobian of the equality constraints only once from the NLP and reuse this information later. The default value for this string option is “no”. Possible values: yes, no, True, False.

  • jac_d_constant (str or bool): Indicates whether to assume that all inequality constraints are linear Activating this option will cause Ipopt to ask for the Jacobian of the inequality constraints only once from the NLP and reuse this information later. The default value for this string option is “no”. Possible values: yes, no, True, False

  • hessian_constant (str or bool): Indicates whether to assume the problem is a QP (quadratic objective, linear constraints). Activating this option will cause Ipopt to ask for the Hessian of the Lagrangian function only once from the NLP and reuse this information later. The default value for this string option is “no”. Possible values: yes, no, True, False.

  • nlp_scaling_method (str): Select the technique used for scaling the NLP. Selects the technique used for scaling the problem internally before it is solved. For user-scaling, the parameters come from the NLP. If you are using AMPL, they can be specified through suffixes (“scaling_factor”) The default value for this string option is “gradient-based”. Possible values:

    • “none”: no problem scaling will be performed - “user-scaling”: scaling parameters will come from the user - “gradient-based”: scale the problem so the maximum gradient at the starting point is nlp_scaling_max_gradient .

    • “equilibration-based”: scale the problem so that first derivatives are of order 1 at random points (uses Harwell routine MC19)

  • obj_scaling_factor (float): Scaling factor for the objective function. This option sets a scaling factor for the objective function. The scaling is seen internally by Ipopt but the unscaled objective is reported in the console output. If additional scaling parameters are computed (e.g. user-scaling or gradient-based), both factors are multiplied. If this value is chosen to be negative, Ipopt will maximize the objective function instead of minimizing it. The valid range for this real option is unrestricted and its default value is 1.

  • nlp_scaling_max_gradient (float): Maximum gradient after NLP scaling. This is the gradient scaling cut-off. If the maximum gradient is above this value, then gradient based scaling will be performed. Scaling parameters are calculated to scale the maximum gradient back to this value. (This is g_max in Section 3.8 of the implementation paper.) Note: This option is only used if nlp_scaling_method is chosen as “gradient-based”. The valid range for this real option is \(0 < \text{nlp_scaling_max_gradient}\) and its default value is 100.

  • nlp_scaling_obj_target_gradient (float): advanced! Target value for objective function gradient size. If a positive number is chosen, the scaling factor for the objective function is computed so that the gradient has the max norm of the given size at the starting point. This overrides nlp_scaling_max_gradient for the objective function. The valid range for this real option is 0 ≤ nlp_scaling_obj_target_gradient and its default value is 0.

  • nlp_scaling_constr_target_gradient (float): Min value of gradient-based scaling values. This is the lower bound for the scaling factors computed by gradient-based scaling method. If some derivatives of some functions are huge, the scaling factors will otherwise become very small, and the (unscaled) final constraint violation, for example, might then be significant. Note: This option is only used if nlp_scaling_method is chosen as “gradient-based”. The valid range for this real option is 0 ≤ nlp_scaling_min_value and its default value is \(1e-08\).

  • nlp_scaling_min_value (float): Minimum value of gradient-based scaling values. This is the lower bound for the scaling factors computed by gradient-based scaling method. If some derivatives of some functions are huge, the scaling factors will otherwise become very small, and the (unscaled) final constraint violation, for example, might then be significant. Note: This option is only used if nlp_scaling_method is chosen as “gradient-based”. The valid range for this real option is 0 ≤ nlp_scaling_min_value and its default value is \(1e-08\).

  • bound_push (float): Desired minimum absolute distance from the initial point to bound. Determines how much the initial point might have to be modified in order to be sufficiently inside the bounds (together with bound_frac ). (This is kappa_1 in Section 3.6 of implementation paper.) The valid range for this real option is 0 < bound_push and its default value is 0.01.

  • bound_frac (float): Desired minimum relative distance from the initial point to bound. Determines how much the initial point might have to be modified in order to be sufficiently inside the bounds (together with “bound_push”). (This is kappa_2 in Section 3.6 of implementation paper.) The valid range for this real option is 0 < bound_frac ≤ 0.5 and its default value is 0.01.

  • slack_bound_push (float): Desired minimum absolute distance from the initial slack to bound. Determines how much the initial slack variables might have to be modified in order to be sufficiently inside the inequality bounds (together with slack_bound_frac ). (This is kappa_1 in Section 3.6 of implementation paper.) The valid range for this real option is 0 < slack_bound_push and its default value is 0.01.

  • slack_bound_frac (float): Desired minimum relative distance from the initial slack to bound. Determines how much the initial slack variables might have to be modified in order to be sufficiently inside the inequality bounds (together with slack_bound_push ). (This is kappa_2 in Section 3.6 of implementation paper.) The valid range for this real option is 0 < slack_bound_frac ≤ 0.5 and its default value is 0.01.

  • constr_mult_init_max (float): Maximum allowed least-square guess of constraint multipliers. Determines how large the initial least-square guesses of the constraint multipliers are allowed to be (in max-norm). If the guess is larger than this value, it is discarded and all constraint multipliers are set to zero. This options is also used when initializing the restoration phase. By default, “resto.constr_mult_init_max” (the one used in RestoIterateInitializer) is set to zero. The valid range for this real option is 0 ≤ constr_mult_init_max and its default value is 1000.

  • bound_mult_init_val (float): Initial value for the bound multipliers. All dual variables corresponding to bound constraints are initialized to this value. The valid range for this real option is 0 < bound_mult_init_val and its default value is 1.

  • bound_mult_init_method (str): Initialization method for bound multipliers This option defines how the iterates for the bound multipliers are initialized. If “constant” is chosen, then all bound multipliers are initialized to the value of bound_mult_init_val. If “mu-based” is chosen, the each value is initialized to the the value of “mu_init” divided by the corresponding slack variable. This latter option might be useful if the starting point is close to the optimal solution. The default value for this string option is “constant”. Possible values:

    • “constant”: set all bound multipliers to the value of bound_mult_init_val

    • “mu-based”: initialize to mu_init/x_slack

  • least_square_init_primal (str or bool): Least square initialization of the primal variables. If set to yes, Ipopt ignores the user provided point and solves a least square problem for the primal variables (x and s) to fit the linearized equality and inequality constraints.This might be useful if the user doesn’t know anything about the starting point, or for solving an LP or QP. The default value for this string option is “no”. Possible values:

    • “no”: take user-provided point

    • “yes”: overwrite user-provided point with least-square estimates

  • least_square_init_duals (str or bool): Least square initialization of all dual variables If set to yes, Ipopt tries to compute least-square multipliers (considering ALL dual variables). If successful, the bound multipliers are possibly corrected to be at least bound_mult_init_val . This might be useful if the user doesn’t know anything about the starting point, or for solving an LP or QP. This overwrites option bound_mult_init_method . The default value for this string option is “no”. Possible values:

    • “no”: use bound_mult_init_val and least-square equality constraint multipliers

    • “yes”: overwrite user-provided point with least-square estimates

  • warm_start_init_point (str or bool): Warm-start for initial point Indicates whether this optimization should use a warm start initialization, where values of primal and dual variables are given (e.g., from a previous optimization of a related problem.) The default value for this string option is “no”. Possible values:

    • “no” or False: do not use the warm start initialization

    • “yes” or True: use the warm start initialization

  • warm_start_same_structure (str or bool): Advanced feature! Indicates whether a problem with a structure identical t the previous one is to be solved. If enabled, then the algorithm assumes that an NLP is now to be solved whose structure is identical to one that already was considered (with the same NLP object). The default value for this string option is “no”. Possible values: yes, no, True, False.

  • warm_start_bound_push (float): same as bound_push for the regular initializer. The valid range for this real option is 0 < warm_start_bound_push and its default value is 0.001.

  • warm_start_bound_frac (float): same as bound_frac for the regular initializer The valid range for this real option is 0 < warm_start_bound_frac ≤ 0.5 and its default value is 0.001.

  • warm_start_slack_bound_push (float): same as slack_bound_push for the regular initializer The valid range for this real option is 0 < warm_start_slack_bound_push and its default value is 0.001.

  • warm_start_slack_bound_frac (float): same as slack_bound_frac for the regular initializer The valid range for this real option is 0 < warm_start_slack_bound_frac ≤ 0.5 and its default value is 0.001.

  • warm_start_mult_bound_push (float): same as mult_bound_push for the regular initializer The valid range for this real option is 0 < warm_start_mult_bound_push and its default value is 0.001.

  • warm_start_mult_init_max (float): Maximum initial value for the equality multipliers. The valid range for this real option is unrestricted and its default value is \(1e+06\) .

  • warm_start_entire_iterate (str or bool): Tells algorithm whether to use the GetWarmStartIterate method in the NLP. The default value for this string option is “no”. Possible values:

    • “no”: call GetStartingPoint in the NLP

    • “yes”: call GetWarmStartIterate in the NLP

  • warm_start_target_mu (float): Advanced and experimental! The valid range for this real option is unrestricted and its default value is 0.

  • option_file_name (str): File name of options file. By default, the name of the Ipopt options file is “ipopt.opt” - or something else if specified in the IpoptApplication::Initialize call. If this option is set by SetStringValue BEFORE the options file is read, it specifies the name of the options file. It does not make any sense to specify this option within the options file. Setting this option to an empty string disables reading of an options file.

  • replace_bounds (bool or str): Whether all variable bounds should be replaced by inequality constraints. This option must be set for the inexact algorithm. The default value for this string option is “no”. Possible values: “yes”, “no”, True, False.

  • skip_finalize_solution_call (str or bool): Whether a call to NLP::FinalizeSolution after optimization should be suppressed. In some Ipopt applications, the user might want to call the FinalizeSolution method separately. Setting this option to “yes” will cause the IpoptApplication object to suppress the default call to that method. The default value for this string option is “no”. Possible values: “yes”, “no”, True, False

  • timing_statistics (str or bool): Indicates whether to measure time spend in components of Ipopt and NLP evaluation. The overall algorithm time is unaffected by this option. The default value for this string option is “no”. Possible values: “yes”, “no”, True, False

  • mu_max_fact (float): Factor for initialization of maximum value for barrier parameter. This option determines the upper bound on the barrier parameter. This upper bound is computed as the average complementarity at the initial point times the value of this option. (Only used if option “mu_strategy” is chosen as “adaptive”.) The valid range for this real option is 0 < mu_max_fact and its default value is 1000.

  • mu_max (float): Maximum value for barrier parameter. This option specifies an upper bound on the barrier parameter in the adaptive mu selection mode. If this option is set, it overwrites the effect of mu_max_fact. (Only used if option “mu_strategy” is chosen as “adaptive”.) The valid range for this real option is 0 < mu_max and its default value is 100000.

  • mu_min (float): Minimum value for barrier parameter. This option specifies the lower bound on the barrier parameter in the adaptive mu selection mode. By default, it is set to the minimum of \(1e-11\) and min( tol , compl_inf_tol )/( barrier_tol_factor +1), which should be a reasonable value. (Only used if option mu_strategy is chosen as “adaptive”.) The valid range for this real option is 0 < mu_min and its default value is \(1e-11\) .

  • adaptive_mu_globalization (str): Globalization strategy for the adaptive mu selection mode. To achieve global convergence of the adaptive version, the algorithm has to switch to the monotone mode (Fiacco-McCormick approach) when convergence does not seem to appear. This option sets the criterion used to decide when to do this switch. (Only used if option “mu_strategy” is chosen as “adaptive”.) The default value for this string option is “obj-constr-filter”. Possible values:

    • “kkt-error”: nonmonotone decrease of kkt-error

    • “obj-constr-filter”: 2-dim filter for objective and constraint violation

    • “never-monotone-mode”: disables globalization.

  • adaptive_mu_kkterror_red_iters (float): advanced feature! Maximum number of iterations requiring sufficient progress. For the “kkt-error” based globalization strategy, sufficient progress must be made for “adaptive_mu_kkterror_red_iters” iterations. If this number of iterations is exceeded, the globalization strategy switches to the monotone mode. The valid range for this integer option is 0 ≤ adaptive_mu_kkterror_red_iters and its default value is 4.

  • adaptive_mu_kkterror_red_fact (float): advanced feature! Sufficient decrease factor for “kkt-error” globalization strategy. For the “kkt-error” based globalization strategy, the error must decrease by this factor to be deemed sufficient decrease. The valid range for this real option is 0 < adaptive_mu_kkterror_red_fact < 1 and its default value is 0.9999.

  • filter_margin_fact (float): advanced feature! Factor determining width of margin for obj-constr-filter adaptive globalization strategy. When using the adaptive globalization strategy, “obj-constr-filter”, sufficient progress for a filter entry is defined as follows: (new obj) < (filter obj) - filter_margin_fact*(new constr-viol) OR (new constr-viol) < (filter constr-viol) - filter_margin_fact*(new constr-viol). For the description of the “kkt-error-filter” option see filter_max_margin . The valid range for this real option is 0 < filter_margin_fact < 1 and its default value is \(10-05\) .

  • filter_max_margin (float): advanced feature! Maximum width of margin in obj-constr-filter adaptive globalization strategy. The valid range for this real option is 0 < filter_max_margin and its default value is 1.

  • adaptive_mu_restore_previous_iterate (str or bool): advanced feature! Indicates if the previous accepted iterate should be restored if the monotone mode is entered. When the globalization strategy for the adaptive barrier algorithm switches to the monotone mode, it can either start from the most recent iterate (no), or from the last iterate that was accepted (yes). The default value for this string option is “no”. Possible values: “yes”, “no”, True, False

  • adaptive_mu_monotone_init_factor (float): advanced feature! Determines the initial value of the barrier parameter when switching to the monotone mode. When the globalization strategy for the adaptive barrier algorithm switches to the monotone mode and fixed_mu_oracle is chosen as “average_compl”, the barrier parameter is set to the current average complementarity times the value of “adaptive_mu_monotone_init_factor”. The valid range for this real option is 0 < adaptive_mu_monotone_init_factor and its default value is 0.8.

  • adaptive_mu_kkt_norm_type (str): advanced! Norm used for the KKT error in the adaptive mu globalization strategies. When computing the KKT error for the globalization strategies, the norm to be used is specified with this option. Note, this option is also used in the QualityFunctionMuOracle. The default value for this string option is “2-norm-squared”. Possible values:

    • “1-norm”: use the 1-norm (abs sum)

    • “2-norm-squared”: use the 2-norm squared (sum of squares)

    • “max-norm”: use the infinity norm (max)

    • “2-norm”: use 2-norm

  • mu_strategy (str): Update strategy for barrier parameter. Determines which barrier parameter update strategy is to be used. The default value for this string option is “monotone”. Possible values:

    • “monotone”: use the monotone (Fiacco-McCormick) strategy

    • “adaptive”: use the adaptive update strategy

  • mu_oracle (str): Oracle for a new barrier parameter in the adaptive strategy. Determines how a new barrier parameter is computed in each “free-mode” iteration of the adaptive barrier parameter strategy. (Only considered if “adaptive” is selected for option “mu_strategy”). The default value for this string option is “quality-function”. Possible values:

    • “probing”: Mehrotra’s probing heuristic

    • “loqo”: LOQO’s centrality rule

    • “quality-function”: minimize a quality function

  • fixed_mu_oracle (str): Oracle for the barrier parameter when switching to fixed mode. Determines how the first value of the barrier parameter should be computed when switching to the “monotone mode” in the adaptive strategy. (Only considered if “adaptive” is selected for option “mu_strategy”.) The default value for this string option is “average_compl”. Possible values:

    • “probing”: Mehrotra’s probing heuristic

    • “loqo”: LOQO’s centrality rule

    • “quality-function”: minimize a quality function

    • “average_compl”: base on current average complementarity

  • mu_init (float): Initial value for the barrier parameter. This option determines the initial value for the barrier parameter (mu). It is only relevant in the monotone, Fiacco-McCormick version of the algorithm. (i.e., if “mu_strategy” is chosen as “monotone”) The valid range for this real option is 0 < mu_init and its default value is 0.1.

  • barrier_tol_factor (float): Factor for mu in barrier stop test. The convergence tolerance for each barrier problem in the monotone mode is the value of the barrier parameter times “barrier_tol_factor”. This option is also used in the adaptive mu strategy during the monotone mode. This is kappa_epsilon in implementation paper. The valid range for this real option is 0 < barrier_tol_factor and its default value is 10.

  • mu_linear_decrease_factor (float): Determines linear decrease rate of barrier parameter. For the Fiacco-McCormick update procedure the new barrier parameter mu is obtained by taking the minimum of mu*”mu_linear_decrease_factor” and mu^”superlinear_decrease_power”. This is kappa_mu in implementation paper. This option is also used in the adaptive mu strategy during the monotone mode. The valid range for this real option is 0 < mu_linear_decrease_factor < 1 and its default value is 0.2.

  • mu_superlinear_decrease_power (float): Determines superlinear decrease rate of barrier parameter. For the Fiacco-McCormick update procedure the new barrier parameter mu is obtained by taking the minimum of mu*”mu_linear_decrease_factor” and mu^”superlinear_decrease_power”. This is theta_mu in implementation paper. This option is also used in the adaptive mu strategy during the monotone mode. The valid range for this real option is 1 < mu_superlinear_decrease_power < 2 and its default value is 1.5.

  • mu_allow_fast_monotone_decrease (str or bool): Advanced feature! Allow skipping of barrier problem if barrier test i already met. The default value for this string option is “yes”. Possible values:

    • “no”: Take at least one iteration per barrier problem even if the barrier test is already met for the updated barrier parameter

    • “yes”: Allow fast decrease of mu if barrier test it met

  • tau_min (float): Advanced feature! Lower bound on fraction-to-the-boundary parameter tau. This is tau_min in the implementation paper. This option is also used in the adaptive mu strategy during the monotone mode. The valid range for this real option is 0 < tau_min < 1 and its default value is 0.99.

  • sigma_max (float): Advanced feature! Maximum value of the centering parameter. This is the upper bound for the centering parameter chosen by the quality function based barrier parameter update. Only used if option “mu_oracle” is set to “quality-function”. The valid range for this real option is 0 < sigma_max and its default value is 100.

  • sigma_min (float): Advanced feature! Minimum value of the centering parameter. This is the lower bound for the centering parameter chosen by the quality function based barrier parameter update. Only used if option “mu_oracle” is set to “quality-function”. The valid range for this real option is 0 ≤ sigma_min and its default value is \(10-06\) .

  • quality_function_norm_type (str): Advanced feature. Norm used for components of the quality function. Only used if option “mu_oracle” is set to “quality-function”. The default value for this string option is “2-norm-squared”. Possible values:

    • “1-norm”: use the 1-norm (abs sum)

    • “2-norm-squared”: use the 2-norm squared (sum of squares)

    • “max-norm”: use the infinity norm (max)

    • “2-norm”: use 2-norm

  • quality_function_centrality (str): Advanced feature. The penalty term for centrality that is included in quality function. This determines whether a term is added to the quality function to penalize deviation from centrality with respect to complementarity. The complementarity measure here is the xi in the Loqo update rule. Only used if option “mu_oracle” is set to “quality-function”. The default value for this string option is “none”. Possible values:

    • “none”: no penalty term is added

    • “log”: complementarity * the log of the centrality measure

    • “reciprocal”: complementarity * the reciprocal of the centrality measure

    • “cubed-reciprocal”: complementarity * the reciprocal of the centrality measure cubed

  • quality_function_balancing_term (str): Advanced feature. The balancing term included in the quality function for centrality. This determines whether a term is added to the quality function that penalizes situations where the complementarity is much smaller than dual and primal infeasibilities. Only used if option “mu_oracle” is set to “quality-function”. The default value for this string option is “none”. Possible values:

    • “none”: no balancing term is adde

    • “cubic”: \(max(0,\max(\text{dual_inf},\text{primal_inf})-\text{compl})^3\)

  • quality_function_max_section_steps (int): Maximum number of search steps during direct search procedure determining the optimal centering parameter. The golden section search is performed for the quality function based mu oracle. Only used if option “mu_oracle” is set to “quality-function”. The valid range for this integer option is 0 ≤ quality_function_max_section_steps and its default value is 8.

  • quality_function_section_sigma_tol (float): advanced feature! Tolerance for the section search procedure determining the optimal centering parameter (in sigma space). The golden section search is performed for the quality function based mu oracle. Only used if option “mu_oracle” is set to “quality-function”. The valid range for this real option is 0 ≤ quality_function_section_sigma_tol < 1 and its default value is 0.01.

  • quality_function_section_qf_tol (float): advanced feature! Tolerance for the golden section search procedure determining the optimal centering parameter (in the function value space). The golden section search is performed for the quality function based mu oracle. Only used if option “mu_oracle” is set to “quality-function”. The valid range for this real option is 0 ≤ quality_function_section_qf_tol < 1 and its default value is 0.

  • line_search_method (str): Advanced feature. Globalization method used in backtracking line search. Only the “filter” choice is officially supported. But sometimes, good results might be obtained with the other choices. The default value for this string option is “filter”. Possible values:

    • “filter”: Filter method

    • “cg-penalty”: Chen-Goldfarb penalty function

    • “penalty”: Standard penalty function

  • alpha_red_factor (float): Advanced feature. Fractional reduction of the trial step size in the backtracking lne search. At every step of the backtracking line search, the trial step size is reduced by this factor. The valid range for this real option is 0 < alpha_red_factor < 1 and its default value is 0.5.

  • accept_every_trial_step (str or bool): Always accept the first trial step. Setting this option to “yes” essentially disables the line search and makes the algorithm take aggressive steps, without global convergence guarantees. The default value for this string option is “no”. Possible values: “yes”, “no”, True, False.

  • accept_after_max_steps (float): advanced feature. Accept a trial point after maximal this number of steps een if it does not satisfy line search conditions. Setting this to -1 disables this option. The valid range for this integer option is -1 ≤ accept_after_max_steps and its default value is -1.

  • alpha_for_y (str): Method to determine the step size for constraint multipliers (alpha_y) . The default value for this string option is “primal”. Possible values:

    • “primal”: use primal step size

    • “bound-mult”: use step size for the bound multipliers (good for LPs)

    • “min”: use the min of primal and bound multipliers

    • “max”: use the max of primal and bound multipliers

    • “full”: take a full step of size one

    • “min-dual-infeas”: choose step size minimizing new dual infeasibility

    • “safer-min-dual-infeas”: like “min_dual_infeas”, but safeguarded by “min” and “max”

    • “primal-and-full”: use the primal step size, and full step if delta_x <= alpha_for_y_tol

    • “dual-and-full”: use the dual step size, and full step if delta_x <= alpha_for_y_tol

    • “acceptor”: Call LSAcceptor to get step size for y

  • alpha_for_y_tol (float): Tolerance for switching to full equality multiplier steps. This is only relevant if “alpha_for_y” is chosen “primal-and-full” or “dual-and-full”. The step size for the equality constraint multipliers is taken to be one if the max-norm of the primal step is less than this tolerance. The valid range for this real option is 0 ≤ alpha_for_y_tol and its default value is 10.

  • tiny_step_tol (float): Advanced feature. Tolerance for detecting numerically insignificant steps. If the search direction in the primal variables (x and s) is, in relative terms for each component, less than this value, the algorithm accepts the full step without line search. If this happens repeatedly, the algorithm will terminate with a corresponding exit message. The default value is 10 times machine precision. The valid range for this real option is 0 ≤ tiny_step_tol and its default value is 2.22045 · \(1e-15\).

  • tiny_step_y_tol (float): Advanced feature. Tolerance for quitting because of numerically insignificant steps. If the search direction in the primal variables (x and s) is, in relative terms for each component, repeatedly less than tiny_step_tol, and the step in the y variables is smaller than this threshold, the algorithm will terminate. The valid range for this real option is 0 ≤ tiny_step_y_tol and its default value is 0.01.

  • watchdog_shortened_iter_trigger (int): Number of shortened iterations that trigger the watchdog. If the number of successive iterations in which the backtracking line search did not accept the first trial point exceeds this number, the watchdog procedure is activated. Choosing “0” here disables the watchdog procedure. The valid range for this integer option is 0 ≤ watchdog_shortened_iter_trigger and its default value is 10.

  • watchdog_trial_iter_max (int): Maximum number of watchdog iterations. This option determines the number of trial iterations allowed before the watchdog procedure is aborted and the algorithm returns to the stored point. The valid range for this integer option is 1 ≤ watchdog_trial_iter_max and its default value is 3. theta_max_fact (float): Advanced feature. Determines upper bound for constraint violation in the filter. The algorithmic parameter theta_max is determined as theta_max_fact times the maximum of 1 and the constraint violation at initial point. Any point with a constraint violation larger than theta_max is unacceptable to the filter (see Eqn. (21) in the implementation paper). The valid range for this real option is 0 < theta_max_fact and its default value is 10000.

  • theta_min_fact (float): advanced feature. Determines constraint violation threshold in the switching rule. The algorithmic parameter theta_min is determined as theta_min_fact times the maximum of 1 and the constraint violation at initial point. The switching rules treats an iteration as an h-type iteration whenever the current constraint violation is larger than theta_min (see paragraph before Eqn. (19) in the implementation paper). The valid range for this real option is 0 < theta_min_fact and its default value is 0.0001.

  • eta_phi (float): advanced! Relaxation factor in the Armijo condition. See Eqn. (20) in the implementation paper. The valid range for this real option is 0 < eta_phi < 0.5 and its default value is \(1e-08\).

  • delta (float): advanced! Multiplier for constraint violation in the switching rule. See Eqn. (19) in the implementation paper. The valid range for this real option is 0 < delta and its default value is 1.

  • s_phi (float): advanced! Exponent for linear barrier function model in the switching rule. See Eqn. (19) in the implementation paper. The valid range for this real option is 1 < s_phi and its default value is 2.3.

  • s_theta (float): advanced! Exponent for current constraint violation in the switching rule. See Eqn. (19) in the implementation paper. The valid range for this real option is 1 < s_theta and its default value is 1.1.

  • gamma_phi (float): advanced! Relaxation factor in the filter margin for the barrier function. See Eqn. (18a) in the implementation paper. The valid range for this real option is 0 < gamma_phi < 1 and its default value is \(1e-08\).

  • gamma_theta (float): advanced! Relaxation factor in the filter margin for the constraint violation. See Eqn. (18b) in the implementation paper. The valid range for this real option is 0 < gamma_theta < 1 and its default value is \(1e-05\).

  • alpha_min_frac (float): advanced! Safety factor for the minimal step size (before switching to restoration phase). This is gamma_alpha in Eqn. (20) in the implementation paper. The valid range for this real option is 0 < alpha_min_frac < 1 and its default value is 0.05.

  • max_soc (int): Maximum number of second order correction trial steps at each iteration. Choosing 0 disables the second order corrections. This is p^{max} of Step A-5.9 of Algorithm A in the implementation paper. The valid range for this integer option is 0 ≤ max_soc and its default value is 4.

  • kappa_soc (float): advanced! Factor in the sufficient reduction rule for second order correction. This option determines how much a second order correction step must reduce the constraint violation so that further correction steps are attempted. See Step A-5.9 of Algorithm A in the implementation paper. The valid range for this real option is 0 < kappa_soc and its default value is 0.99.

  • obj_max_inc (float): advanced! Determines the upper bound on the acceptable increase of barrier objective function. Trial points are rejected if they lead to an increase in the barrier objective function by more than obj_max_inc orders of magnitude. The valid range for this real option is 1 < obj_max_inc and its default value is 5.

  • max_filter_resets (int): advanced! Maximal allowed number of filter resets. A positive number enables a heuristic that resets the filter, whenever in more than “filter_reset_trigger” successive iterations the last rejected trial steps size was rejected because of the filter. This option determine the maximal number of resets that are allowed to take place. The valid range for this integer option is 0 ≤ max_filter_resets and its default value is 5.

  • filter_reset_trigger (int): Advanced! Number of iterations that trigger the filter reset. If the filter reset heuristic is active and the number of successive iterations in which the last rejected trial step size was rejected because of the filter, the filter is reset. The valid range for this integer option is 1 ≤ filter_reset_trigger and its default value is 5.

  • corrector_type (str): advanced! The type of corrector steps that should be taken. If “mu_strategy” is “adaptive”, this option determines what kind of corrector steps should be tried. Changing this option is experimental. The default value for this string option is “none”. Possible values:

    • “none” or None: no corrector

    • “affine”: corrector step towards mu=0

    • “primal-dual”: corrector step towards current mu

  • skip_corr_if_neg_curv (str or bool): advanced! Whether to skip the corrector step in negative curvature iteration. The corrector step is not tried if negative curvature has been encountered during the computation of the search direction in the current iteration. This option is only used if “mu_strategy” is “adaptive”. Changing this option is experimental. The default value for this string option is “yes”. Possible values: “yes”, “no”, True, False.

  • skip_corr_in_monotone_mode (str or bool): Advanced! Whether to skip the corrector step during monotone brrier parameter mode. The corrector step is not tried if the algorithm is currently in the monotone mode (see also option “barrier_strategy”). This option is only used if “mu_strategy” is “adaptive”. Changing this option is experimental. The default value for this string option is “yes”. Possible values: “yes”, “no”, True, False

  • corrector_compl_avrg_red_fact (int): advanced! Complementarity tolerance factor for accepting corrector step. This option determines the factor by which complementarity is allowed to increase for a corrector step to be accepted. Changing this option is experimental. The valid range for this real option is 0 < corrector_compl_avrg_red_fact and its default value is 1.

  • soc_method (int): Ways to apply second order correction. This option determines the way to apply second order correction, 0 is the method described in the implementation paper. 1 is the modified way which adds alpha on the rhs of x and s rows. Officially, the valid range for this integer option is 0 ≤ soc_method ≤ 1 and its default value is 0 but only 0 and 1 are allowed.

  • nu_init (float): advanced! Initial value of the penalty parameter. The valid range for this real option is 0 < nu_init and its default value is \(1e-06\).

  • nu_inc (float): advanced! Increment of the penalty parameter. The valid range for this real option is 0 < nu_inc and its default value is 0.0001.

  • rho (float): advanced! Value in penalty parameter update formula. The valid range for this real option is 0 < rho < 1 and its default value is 0.1.

  • kappa_sigma (float): advanced! Factor limiting the deviation of dual variables from primal estimates. If the dual variables deviate from their primal estimates, a correction is performed. See Eqn. (16) in the implementation paper. Setting the value to less than 1 disables the correction. The valid range for this real option is 0 < kappa_sigma and its default value is \(1e+10\).

  • recalc_y (str or bool): Tells the algorithm to recalculate the equality and inequality multipliers as least square estimates. This asks the algorithm to recompute the multipliers, whenever the current infeasibility is less than recalc_y_feas_tol. Choosing yes might be helpful in the quasi-Newton option. However, each recalculation requires an extra factorization of the linear system. If a limited memory quasi-Newton option is chosen, this is used by default. The default value for this string option is “no”. Possible values:

    • “no” or False: use the Newton step to update the multipliers

    • “yes” or True: use least-square multiplier

  • estimates recalc_y_feas_tol (float): Feasibility threshold for recomputation of multipliers. If recalc_y is chosen and the current infeasibility is less than this value, then the multipliers are recomputed. The valid range for this real option is 0 < recalc_y_feas_tol and its default value is \(1e-06\).

  • slack_move (float): advanced! Correction size for very small slacks. Due to numerical issues or the lack of an interior, the slack variables might become very small. If a slack becomes very small compared to machine precision, the corresponding bound is moved slightly. This parameter determines how large the move should be. Its default value is mach_eps^{3/4}. See also end of Section 3.5 in implementation paper - but actual implementation might be somewhat different. The valid range for this real option is 0 ≤ slack_move and its default value is 1.81899 · \(1e-12\).

  • constraint_violation_norm_type (str): advanced! Norm to be used for the constraint violation in te line search. Determines which norm should be used when the algorithm computes the constraint violation in the line search. The default value for this string option is “1-norm”. Possible values:

    • “1-norm”: use the 1-norm

    • “2-norm”: use the 2-norm

    • “max-norm”: use the infinity norm

  • mehrotra_algorithm (str or bool): Indicates whether to do Mehrotra’s predictor-corrector algorithm. If enabled, line search is disabled and the (unglobalized) adaptive mu strategy is chosen with the “probing” oracle, and “corrector_type=affine” is used without any safeguards; you should not set any of those options explicitly in addition. Also, unless otherwise specified, the values of bound_push , bound_frac , and bound_mult_init_val are set more aggressive, and sets “alpha_for_y=bound_mult”. The Mehrotra’s predictor-corrector algorithm works usually very well for LPs and convex QPs. The default value for this string option is “no”. Possible values: “yes”, “no”, True, False.

  • fast_step_computation (str or bool): Indicates if the linear system should be solved quickly. If enabled, the algorithm assumes that the linear system that is solved to obtain the search direction is solved sufficiently well. In that case, no residuals are computed to verify the solution and the computation of the search direction is a little faster. The default value for this string option is “no”. Possible values: “yes”, “no”, True, False.

  • min_refinement_steps (int): Minimum number of iterative refinement steps per linear system solve. Iterative refinement (on the full asymmetric system) is performed for each right hand side. This option determines the minimum number of iterative refinements (i.e. at least “min_refinement_steps” iterative refinement steps are enforced per right hand side.) The valid range for this integer option is 0 ≤ min_refinement_steps and its default value is 1.

  • max_refinement_steps (int): Maximum number of iterative refinement steps per linear system solve. Iterative refinement (on the full unsymmetric system) is performed for each right hand side. This option determines the maximum number of iterative refinement steps. The valid range for this integer option is 0 ≤ max_refinement_steps and its default value is 10.

  • residual_ratio_max (float): advanced! Iterative refinement tolerance. Iterative refinement is performed until the residual test ratio is less than this tolerance (or until “max_refinement_steps” refinement steps are performed). The valid range for this real option is 0 < residual_ratio_max and its default value is \(1e-10\).

  • residual_ratio_singular (float): advanced! Threshold for declaring linear system singular after filed iterative refinement. If the residual test ratio is larger than this value after failed iterative refinement, the algorithm pretends that the linear system is singular. The valid range for this real option is 0 < residual_ratio_singular and its default value is \(1e-05\).

  • residual_improvement_factor (float): advanced! Minimal required reduction of residual test ratio in iterative refinement. If the improvement of the residual test ratio made by one iterative refinement step is not better than this factor, iterative refinement is aborted. The valid range for this real option is 0 < residual_improvement_factor and its default value is 1.

  • neg_curv_test_tol (float): Tolerance for heuristic to ignore wrong inertia. If nonzero, incorrect inertia in the augmented system is ignored, and Ipopt tests if the direction is a direction of positive curvature. This tolerance is alpha_n in the paper by [algo_38] and it determines when the direction is considered to be sufficiently positive. A value in the range of [1e-12, 1e-11] is recommended. The valid range for this real option is 0 ≤ neg_curv_test_tol and its default value is 0.

  • neg_curv_test_reg (str or bool): Whether to do the curvature test with the primal regularization (see [algo_38]). The default value for this string option is “yes”. Possible values:

    • “yes” or True: use primal regularization with the inertia-free curvature test

    • “no” or False: use original IPOPT approach, in which the primal regularization is ignored

  • max_hessian_perturbation (float): Maximum value of regularization parameter for handling negative curvature. In order to guarantee that the search directions are indeed proper descent directions, Ipopt requires that the inertia of the (augmented) linear system for the step computation has the correct number of negative and positive eigenvalues. The idea is that this guides the algorithm away from maximizers and makes Ipopt more likely converge to first order optimal points that are minimizers. If the inertia is not correct, a multiple of the identity matrix is added to the Hessian of the Lagrangian in the augmented system. This parameter gives the maximum value of the regularization parameter. If a regularization of that size is not enough, the algorithm skips this iteration and goes to the restoration phase. This is delta_w^max in the implementation paper. The valid range for this real option is 0 < max_hessian_perturbation and its default value is \(1e+20\).

  • min_hessian_perturbation (float): Smallest perturbation of the Hessian block. The size of the perturbation of the Hessian block is never selected smaller than this value, unless no perturbation is necessary. This is delta_w^min in implementation paper. The valid range for this real option is 0 ≤ min_hessian_perturbation and its default value is \(1e-20\).

  • perturb_inc_fact_first (float): Increase factor for x-s perturbation for very first perturbation. The factor by which the perturbation is increased when a trial value was not sufficient - this value is used for the computation of the very first perturbation and allows a different value for the first perturbation than that used for the remaining perturbations. This is bar_kappa_w^+ in the implementation paper. The valid range for this real option is 1 < perturb_inc_fact_first and its default value is 100.

  • perturb_inc_fact (float): Increase factor for x-s perturbation. The factor by which the perturbation is increased when a trial value was not sufficient - this value is used for the computation of all perturbations except for the first. This is kappa_w^+ in the implementation paper. The valid range for this real option is 1 < perturb_inc_fact and its default value is 8.

  • perturb_dec_fact (float): Decrease factor for x-s perturbation. The factor by which the perturbation is decreased when a trial value is deduced from the size of the most recent successful perturbation. This is kappa_w^- in the implementation paper. The valid range for this real option is 0 < perturb_dec_fact < 1 and its default value is 0.333333.

  • first_hessian_perturbation (float): Size of first x-s perturbation tried. The first value tried for the x-s perturbation in the inertia correction scheme. This is delta_0 in the implementation paper. The valid range for this real option is 0 < first_hessian_perturbation and its default value is 0.0001.

  • jacobian_regularization_value (float): Size of the regularization for rank-deficient constraint Jacobians. This is bar delta_c in the implementation paper. The valid range for this real option is 0 ≤ jacobian_regularization_value and its default value is \(1e-08\).

  • jacobian_regularization_exponent (float): advanced! Exponent for mu in the regularization for rnk-deficient constraint Jacobians. This is kappa_c in the implementation paper. The valid range for this real option is 0 ≤ jacobian_regularization_exponent and its default value is 0.25.

  • perturb_always_cd (str or bool): advanced! Active permanent perturbation of constraint linearization. Enabling this option leads to using the delta_c and delta_d perturbation for the computation of every search direction. Usually, it is only used when the iteration matrix is singular. The default value for this string option is “no”. Possible values: “yes”, “no”, True, False.

  • expect_infeasible_problem (str or bool): Enable heuristics to quickly detect an infeasible problem. This options is meant to activate heuristics that may speed up the infeasibility determination if you expect that there is a good chance for the problem to be infeasible. In the filter line search procedure, the restoration phase is called more quickly than usually, and more reduction in the constraint violation is enforced before the restoration phase is left. If the problem is square, this option is enabled automatically. The default value for this string option is “no”. Possible values: “yes”, “no”, True, False.

  • expect_infeasible_problem_ctol (float): Threshold for disabling “expect_infeasible_problem” option. If the constraint violation becomes smaller than this threshold, the “expect_infeasible_problem” heuristics in the filter line search are disabled. If the problem is square, this options is set to 0. The valid range for this real option is 0 ≤ expect_infeasible_problem_ctol and its default value is 0.001.

  • expect_infeasible_problem_ytol (float): Multiplier threshold for activating “xpect_infeasible_problem” option. If the max norm of the constraint multipliers becomes larger than this value and “expect_infeasible_problem” is chosen, then the restoration phase is entered. The valid range for this real option is 0 < expect_infeasible_problem_ytol and its default value is \(1e+08\).

  • start_with_resto (str or bool): Whether to switch to restoration phase in first iteration.Setting this option to “yes” forces the algorithm to switch to the feasibility restoration phase in the first iteration. If the initial point is feasible, the algorithm will abort with a failure. The default value for this string option is “no”. Possible values: “yes”, “no”, True, False

  • soft_resto_pderror_reduction_factor (float): Required reduction in primal-dual error in the soft restoration phase. The soft restoration phase attempts to reduce the primal-dual error with regular steps. If the damped primal-dual step (damped only to satisfy the fraction-to-the-boundary rule) is not decreasing the primal-dual error by at least this factor, then the regular restoration phase is called. Choosing “0” here disables the soft restoration phase. The valid range for this real option is 0 ≤ soft_resto_pderror_reduction_factor and its default value is 0.9999.

  • max_soft_resto_iters (int): advanced! Maximum number of iterations performed successively in soft rstoration phase. If the soft restoration phase is performed for more than so many iterations in a row, the regular restoration phase is called. The valid range for this integer option is 0 ≤ max_soft_resto_iters and its default value is 10.

  • required_infeasibility_reduction (float): Required reduction of infeasibility before leaving restoration phase. The restoration phase algorithm is performed, until a point is found that is acceptable to the filter and the infeasibility has been reduced by at least the fraction given by this option. The valid range for this real option is 0 ≤ required_infeasibility_reduction < 1 and its default value is 0.9.

  • max_resto_iter (int): advanced! Maximum number of successive iterations in restoration phase.The algorithm terminates with an error message if the number of iterations successively taken in the restoration phase exceeds this number. The valid range for this integer option is 0 ≤ max_resto_iter and its default value is 3000000.

  • evaluate_orig_obj_at_resto_trial (str or bool): Determines if the original objective function should be evaluated at restoration phase trial points. Enabling this option makes the restoration phase algorithm evaluate the objective function of the original problem at every trial point encountered during the restoration phase, even if this value is not required. In this way, it is guaranteed that the original objective function can be evaluated without error at all accepted iterates; otherwise the algorithm might fail at a point where the restoration phase accepts an iterate that is good for the restoration phase problem, but not the original problem. On the other hand, if the evaluation of the original objective is expensive, this might be costly. The default value for this string option is “yes”. Possible values: “yes”, “no”, True, False

  • resto_penalty_parameter (float): advanced! Penalty parameter in the restoration phase objective function. This is the parameter rho in equation (31a) in the Ipopt implementation paper. The valid range for this real option is 0 < resto_penalty_parameter and its default value is 1000.

  • resto_proximity_weight (float): advanced! Weighting factor for the proximity term in restoration pase objective. This determines how the parameter zeta in equation (29a) in the implementation paper is computed. zeta here is resto_proximity_weight*sqrt(mu), where mu is the current barrier parameter. The valid range for this real option is 0 ≤ resto_proximity_weight and its default value is 1.

  • bound_mult_reset_threshold (float): Threshold for resetting bound multipliers after the restoration pase. After returning from the restoration phase, the bound multipliers are updated with a Newton step for complementarity. Here, the change in the primal variables during the entire restoration phase is taken to be the corresponding primal Newton step. However, if after the update the largest bound multiplier exceeds the threshold specified by this option, the multipliers are all reset to 1. The valid range for this real option is 0 ≤ bound_mult_reset_threshold and its default value is 1000.

  • constr_mult_reset_threshold (float): Threshold for resetting equality and inequality multipliers ater restoration phase. After returning from the restoration phase, the constraint multipliers are recomputed by a least square estimate. This option triggers when those least-square estimates should be ignored. The valid range for this real option is 0 ≤ constr_mult_reset_threshold and its default value is 0.

  • resto_failure_feasibility_threshold (float): advanced! Threshold for primal infeasibility to declare failure of restoration phase. If the restoration phase is terminated because of the “acceptable” termination criteria and the primal infeasibility is smaller than this value, the restoration phase is declared to have failed. The default value is actually 1e2*tol, where tol is the general termination tolerance. The valid range for this real option is 0 ≤ resto_failure_feasibility_threshold and its default value is 0.

  • limited_memory_aug_solver (str): advanced! Strategy for solving the augmented system for low-rank Hessian. The default value for this string option is “sherman-morrison”. Possible values:

    • “sherman-morrison”: use Sherman-Morrison formula

    • “extended”: use an extended augmented system

  • limited_memory_max_history (int): Maximum size of the history for the limited quasi-Newton Hessian approximation. This option determines the number of most recent iterations that are taken into account for the limited-memory quasi-Newton approximation. The valid range for this integer option is 0 ≤ limited_memory_max_history and its default value is 6.

  • limited_memory_update_type (str): Quasi-Newton update formula for the limited memory quasi-Newton approximation. The default value for this string option is “bfgs”. Possible values:

    • “bfgs”: BFGS update (with skipping)

    • “sr1”: SR1 (not working well)

  • limited_memory_initialization (str): Initialization strategy for the limited memory quasi-Newton aproximation. Determines how the diagonal Matrix B_0 as the first term in the limited memory approximation should be computed. The default value for this string option is “scalar1”. Possible values:

    • “scalar1”: sigma = s^Ty/s^Ts

    • “scalar2”: sigma = y^Ty/s^Ty

    • “scalar3”: arithmetic average of scalar1 and scalar2

    • “scalar4”: geometric average of scalar1 and scalar2

    • “constant”: sigma = limited_memory_init_val

  • limited_memory_init_val (float): Value for B0 in low-rank update. The starting matrix in the low rank update, B0, is chosen to be this multiple of the identity in the first iteration (when no updates have been performed yet), and is constantly chosen as this value, if “limited_memory_initialization” is “constant”. The valid range for this real option is 0 < limited_memory_init_val and its default value is 1.

  • limited_memory_init_val_max (float): Upper bound on value for B0 in low-rank update. The starting matrix in the low rank update, B0, is chosen to be this multiple of the identity in the first iteration (when no updates have been performed yet), and is constantly chosen as this value, if “limited_memory_initialization” is “constant”. The valid range for this real option is 0 < limited_memory_init_val_max and its default value is \(1e+08\).

  • limited_memory_init_val_min (float): Lower bound on value for B0 in low-rank update. The starting matrix in the low rank update, B0, is chosen to be this multiple of the identity in the first iteration (when no updates have been performed yet), and is constantly chosen as this value, if “limited_memory_initialization” is “constant”. The valid range for this real option is 0 < limited_memory_init_val_min and its default value is \(1e-08\).

  • limited_memory_max_skipping (int): Threshold for successive iterations where update is skipped. If the update is skipped more than this number of successive iterations, the quasi-Newton approximation is reset. The valid range for this integer option is 1 ≤ limited_memory_max_skipping and its default value is 2.

  • limited_memory_special_for_resto (str or bool): Determines if the quasi-Newton updates should be special dring the restoration phase. Until Nov 2010, Ipopt used a special update during the restoration phase, but it turned out that this does not work well. The new default uses the regular update procedure and it improves results. If for some reason you want to get back to the original update, set this option to “yes”. The default value for this string option is “no”. Possible values: “yes”, “no”, True, False.

  • hessian_approximation (str): Indicates what Hessian information is to be used. This determines which kind of information for the Hessian of the Lagrangian function is used by the algorithm. The default value for this string option is “limited-memory”. Possible values: - “exact”: Use second derivatives provided by the NLP. - “limited-memory”: Perform a limited-memory quasi-Newton approximation

  • hessian_approximation_space (str): advanced! Indicates in which subspace the Hessian information is to be approximated. The default value for this string option is “nonlinear-variables”. Possible values: - “nonlinear-variables”: only in space of nonlinear variables. - “all-variables”: in space of all variables (without slacks)

  • linear_solver (str): Linear solver used for step computations. Determines which linear algebra package is to be used for the solution of the augmented linear system (for obtaining the search directions). The default value for this string option is “ma27”. Possible values:

    • “mumps” (use the Mumps package, default)

    • “ma27” (load the Harwell routine MA27 from library at runtime)

    • “ma57” (load the Harwell routine MA57 from library at runtime)

    • “ma77” (load the Harwell routine HSL_MA77 from library at runtime)

    • “ma86” (load the Harwell routine MA86 from library at runtime)

    • “ma97” (load the Harwell routine MA97 from library at runtime)

    • “pardiso” (load the Pardiso package from pardiso-project.org from user-provided library at runtime)

    • “custom” (use custom linear solver (expert use))

  • linear_solver_options (dict or None): dictionary with the linear solver options, possibly including linear_system_scaling, hsllib and pardisolib. See the ipopt documentation for details. The linear solver options are not automatically converted to float at the moment.]

The Fides Optimizer#

estimagic supports the Fides Optimizer. To use Fides, you need to have the fides package installed (pip install fides>=0.7.4, make sure you have at least 0.7.1).

fides
"fides"

Fides implements an Interior Trust Region Reflective for boundary costrained optimization problems based on the papers [algo_39] and [algo_40]. Accordingly, Fides is named after the Roman goddess of trust and reliability. In contrast to other optimizers, Fides solves the full trust-region subproblem exactly, which can yields higher quality proposal steps, but is computationally more expensive. This makes Fides particularly attractive for optimization problems with objective functions that are computationally expensive to evaluate and the computational cost of solving the trust-region subproblem is negligible.

  • hessian_update_strategy (str): Hessian Update Strategy to employ. You can provide a lowercase or uppercase string or a fides.hession_approximation.HessianApproximation class instance. FX, SSM, TSSM and GNSBFGS are not supported by estimagic. The available update strategies are:

    • bb: Broydens “bad” method as introduced [algo_41].

    • bfgs: Broyden-Fletcher-Goldfarb-Shanno update strategy.

    • bg: Broydens “good” method as introduced in [algo_41].

    • You can use a general BroydenClass Update scheme using the Broyden class from fides.hessian_approximation. This is a generalization of BFGS/DFP methods where the parameter \(phi\) controls the convex combination between the two. This is a rank 2 update strategy that preserves positive-semidefiniteness and symmetry (if \(\phi \in [0,1]\)). It is described in [algo_42], Chapter 6.3.

    • dfp: Davidon-Fletcher-Powell update strategy.

    • sr1: Symmetric Rank 1 update strategy as described in [algo_42], Chapter 6.2.

  • convergence.absolute_criterion_tolerance (float): absolute convergence criterion tolerance. This is only the interpretation of this parameter if the relative criterion tolerance is set to 0. Denoting the absolute criterion tolerance by \(\alpha\) and the relative criterion tolerance by \(\beta\), the convergence condition on the criterion improvement is \(|f(x_k) - f(x_{k-1})| < \alpha + \beta \cdot |f(x_{k-1})|\)

  • convergence.relative_criterion_tolerance (float): relative convergence criterion tolerance. This is only the interpretation of this parameter if the absolute criterion tolerance is set to 0 (as is the default). Denoting the absolute criterion tolerance by \(\alpha\) and the relative criterion tolerance by \(\beta\), the convergence condition on the criterion improvement is \(|f(x_k) - f(x_{k-1})| < \alpha + \beta \cdot |f(x_{k-1})|\)

  • convergence.absolute_params_tolerance (float): The optimization terminates successfully when the step size falls below this number, i.e. when \(||x_{k+1} - x_k||\) is smaller than this tolerance.

  • convergence.absolute_gradient_tolerance (float): The optimization terminates successfully when the gradient norm is less or equal than this tolerance.

  • convergence.relative_gradient_tolerance (float): The optimization terminates successfully when the norm of the gradient divided by the absolute function value is less or equal to this tolerance.

  • stopping.max_iterations (int): maximum number of allowed iterations.

  • stopping.max_seconds (int): maximum number of walltime seconds, deactivated by default.

  • trustregion.initial_radius (float): Initial trust region radius. Default is 1.

  • trustregion.stepback_strategy (str): search refinement strategy if proposed step reaches a parameter bound. The default is “truncate”. The available options are:

    • “reflect”: recursive reflections at boundary.

    • “reflect_single”: single reflection at boundary.

    • “truncate”: truncate step at boundary and re-solve the restricted subproblem

    • “mixed”: mix reflections and truncations

  • trustregion.subspace_dimension (str): Subspace dimension in which the subproblem will be solved. The default is “2D”. The following values are available:

    • “2D”: Two dimensional Newton/Gradient subspace

    • “full”: full dimensionality

    • “scg”: Conjugated Gradient subspace via Steihaug’s method

  • trustregion.max_stepback_fraction (float): Stepback parameter that controls how close steps are allowed to get to the boundary. It is the maximal fraction of a step to take if full step would reach breakpoint.

  • trustregion.decrease_threshold (float): Acceptance threshold for trust region ratio. The default is 0.25 ([algo_3]). The radius is decreased if the trust region ratio is below this value. This is denoted by \(\\mu\) in algorithm 4.1 in [algo_3].

  • trustregion.increase_threshold (float): Threshold for the trust region radius ratio above which the trust region radius can be increased. This is denoted by \(\eta\) in algorithm 4.1 in [algo_3]. The default is 0.75 ([algo_3]).

  • trustregion.decrease_factor (float): factor by which trust region radius will be decreased in case it is decreased. This is denoted by \(\gamma_1\) in algorithm 4.1 in [algo_3] and its default is 0.25.

  • trustregion.increase_factor (float): factor by which trust region radius will be increase in case it is increase. This is denoted by \(\gamma_2\) in algorithm 4.1 in [algo_3] and its default is 2.0.

  • trustregion.refine_stepback (bool): whether to refine stepbacks via optimization. Default is False.

  • trustregion.scaled_gradient_as_possible_stepback (bool): whether the scaled gradient should be added to the set of possible stepback proposals. Default is False.

The NLOPT Optimizers (nlopt)#

estimagic supports the following NLOPT algorithms. Please add the appropriate citations in addition to estimagic when using an NLOPT algorithm. To install nlopt run conda install nlopt.

nlopt_bobyqa
"nlopt_bobyqa"

Minimize a scalar function using the BOBYQA algorithm.

The implementation is derived from the BOBYQA subroutine of M. J. D. Powell.

The algorithm performs derivative free bound-constrained optimization using an iteratively constructed quadratic approximation for the objective function. Due to its use of quadratic appoximation, the algorithm may perform poorly for objective functions that are not twice-differentiable.

For details see [algo_15].

  • convergence.relative_params_tolerance (float): Stop when the relative movement between parameter vectors is smaller than this.

  • convergence.absolute_params_tolerance (float): Stop when the absolute movement between parameter vectors is smaller than this.

  • convergence.relative_criterion_tolerance (float): Stop when the relative improvement between two iterations is smaller than this.

  • convergence.absolute_criterion_tolerance (float): Stop when the change of the criterion function between two iterations is smaller than this.

  • stopping.max_criterion_evaluations (int): If the maximum number of function evaluation is reached, the optimization stops but we do not count this as convergence.

nlopt_neldermead
"nlopt_neldermead"

Minimize a scalar function using the Nelder-Mead simplex algorithm.

The basic algorithm is described in [algo_43].

The difference between the nlopt implementation an the original implementation is that the nlopt version supports bounds. This is done by moving all new points that would lie outside the bounds exactly on the bounds.

  • convergence.relative_params_tolerance (float): Stop when the relative movement between parameter vectors is smaller than this.

  • convergence.absolute_params_tolerance (float): Stop when the absolute movement between parameter vectors is smaller than this.

  • convergence.relative_criterion_tolerance (float): Stop when the relative improvement between two iterations is smaller than this.

  • convergence.absolute_criterion_tolerance (float): Stop when the change of the criterion function between two iterations is smaller than this.

  • stopping.max_criterion_evaluations (int): If the maximum number of function evaluation is reached, the optimization stops but we do not count this as convergence.

nlopt_praxis
"nlopt_praxis"

Minimize a scalar function using principal-axis method.

This is a gradient-free local optimizer originally described in [algo_44]. It assumes quadratic form of the optimized function and repeatedly updates a set of conjugate search directions.

The algorithm is not invariant to scaling of the objective function and may fail under its certain rank-preserving transformations (e.g., will lead to a non-quadratic shape of the objective function).

The algorithm is not determenistic and it is not possible to achieve detereminancy via seed setting.

The algorithm failed on a simple benchmark function with finite parameter bounds. Passing arguments lower_bounds and upper_bounds has been disabled for this algorithm.

The difference between the nlopt implementation an the original implementation is that the nlopt version supports bounds. This is done by returning infinity (Inf) when the constraints are violated. The implementation of bound constraints is achieved at the const of significantly reduced speed of convergence. In case of bounded constraints, this method is dominated by nlopt_bobyqa and nlopt_cobyla.

  • convergence.relative_params_tolerance (float): Stop when the relative movement between parameter vectors is smaller than this.

  • convergence.absolute_params_tolerance (float): Stop when the absolute movement between parameter vectors is smaller than this.

  • convergence.relative_criterion_tolerance (float): Stop when the relative improvement between two iterations is smaller than this.

  • convergence.absolute_criterion_tolerance (float): Stop when the change of the criterion function between two iterations is smaller than this.

  • stopping.max_criterion_evaluations (int): If the maximum number of function evaluation is reached, the optimization stops but we do not count this as convergence.

nlopt_cobyla
"nlopt_cobyla"

Minimize a scalar function using the cobyla method.

The alggorithm is derived from Powell’s Constrained Optimization BY Linear Approximations (COBYLA) algorithm. It is a derivative-free optimizer with nonlinear inequality and equality constrains, described in :cite`Powell1994`.

It constructs successive linear approximations of the objective function and constraints via a simplex of n+1 points (in n dimensions), and optimizes these approximations in a trust region at each step.

The the nlopt implementation differs from the original implementation in a a few ways: - Incorporates all of the NLopt termination criteria. - Adds explicit support for bound constraints. - Allows the algorithm to increase the trust-reion radius if the predicted imptoovement was approximately right and the simplex is satisfactory. - Pseudo-randomizes simplex steps in the algorithm, aimproving robustness by avoiding accidentally taking steps that don’t improve conditioning, preserving the deterministic nature of the algorithm. - Supports unequal initial-step sizes in the different parameters.

  • convergence.relative_params_tolerance (float): Stop when the relative movement between parameter vectors is smaller than this.

  • convergence.absolute_params_tolerance (float): Stop when the absolute movement between parameter vectors is smaller than this.

  • convergence.relative_criterion_tolerance (float): Stop when the relative improvement between two iterations is smaller than this.

  • convergence.absolute_criterion_tolerance (float): Stop when the change of the criterion function between two iterations is smaller than this.

  • stopping.max_criterion_evaluations (int): If the maximum number of function evaluation is reached, the optimization stops but we do not count this as convergence.

nlopt_sbplx
"nlopt_sbplx"

Minimize a scalar function using the “Subplex” algorithm.

The alggorithm is a reimplementation of Tom Rowan’s “Subplex” algorithm. See [algo_45]. Subplex is a variant of Nedler-Mead that uses Nedler-Mead on a sequence of subspaces. It is climed to be more efficient and robust than the original Nedler-Mead algorithm.

The difference between this re-implementation and the original algorithm of Rowan, is that it explicitly supports bound constraints providing big improvement in the case where the optimum lies against one of the constraints.

  • convergence.relative_params_tolerance (float): Stop when the relative movement between parameter vectors is smaller than this.

  • convergence.absolute_params_tolerance (float): Stop when the absolute movement between parameter vectors is smaller than this.

  • convergence.relative_criterion_tolerance (float): Stop when the relative improvement between two iterations is smaller than this.

  • convergence.absolute_criterion_tolerance (float): Stop when the change of the criterion function between two iterations is smaller than this.

  • stopping.max_criterion_evaluations (int): If the maximum number of function evaluation is reached, the optimization stops but we do not count this as convergence.

nlopt_newuoa
"nlopt_newuoa"

Minimize a scalar function using the NEWUOA algorithm.

The algorithm is derived from the NEWUOA subroutine of M.J.D Powell which uses iteratively constructed quadratic approximation of the objctive fucntion to perform derivative-free unconstrained optimization. Fore more details see: [algo_46].

The algorithm in nlopt has been modified to support bound constraints. If all of the bound constraints are infinite, this function calls the nlopt.LN_NEWUOA optimizers for uncsonstrained optimization. Otherwise, the nlopt.LN_NEWUOA_BOUND optimizer for constrained problems.

NEWUOA requires the dimension n of the parameter space to be ≥ 2, i.e. the implementation does not handle one-dimensional optimization problems.

  • convergence.relative_params_tolerance (float): Stop when the relative movement between parameter vectors is smaller than this.

  • convergence.absolute_params_tolerance (float): Stop when the absolute movement between parameter vectors is smaller than this.

  • convergence.relative_criterion_tolerance (float): Stop when the relative improvement between two iterations is smaller than this.

  • convergence.absolute_criterion_tolerance (float): Stop when the change of the criterion function between two iterations is smaller than this.

  • stopping.max_criterion_evaluations (int): If the maximum number of function evaluation is reached, the optimization stops but we do not count this as convergence.

nlopt_tnewton
"nlopt_tnewton"

Minimize a scalar function using the “TNEWTON” algorithm.

The alggorithm is based on a Fortran implementation of a preconditioned inexact truncated Newton algorithm written by Prof. Ladislav Luksan.

Truncated Newton methods are a set of algorithms designed to solve large scale optimization problems. The algorithms use (inaccurate) approximations of the solutions to Newton equations, using conjugate gradient methodds, to handle the expensive calculations of derivatives during each iteration.

Detailed description of algorithms is given in [algo_47].

  • convergence.relative_params_tolerance (float): Stop when the relative movement between parameter vectors is smaller than this.

  • convergence.absolute_params_tolerance (float): Stop when the absolute movement between parameter vectors is smaller than this.

  • convergence.relative_criterion_tolerance (float): Stop when the relative improvement between two iterations is smaller than this.

  • convergence.absolute_criterion_tolerance (float): Stop when the change of the criterion function between two iterations is smaller than this.

  • stopping.max_criterion_evaluations (int): If the maximum number of function evaluation is reached, the optimization stops but we do not count this as convergence.

nlopt_lbfgs
"nlopt_lbfgs"

Minimize a scalar function using the “LBFGS” algorithm.

The alggorithm is based on a Fortran implementation of low storage BFGS algorithm written by Prof. Ladislav Luksan.

LFBGS is an approximation of the original Broyden–Fletcher–Goldfarb–Shanno algorithm based on limited use of memory. Memory efficiency is obtained by preserving a limi- ted number (<10) of past updates of candidate points and gradient values and using them to approximate the hessian matrix.

Detailed description of algorithms is given in [algo_48], [algo_49].

  • convergence.relative_params_tolerance (float): Stop when the relative movement between parameter vectors is smaller than this.

  • convergence.absolute_params_tolerance (float): Stop when the absolute movement between parameter vectors is smaller than this.

  • convergence.relative_criterion_tolerance (float): Stop when the relative improvement between two iterations is smaller than this.

  • convergence.absolute_criterion_tolerance (float): Stop when the change of the criterion function between two iterations is smaller than this.

  • stopping.max_criterion_evaluations (int): If the maximum number of function evaluation is reached, the optimization stops but we do not count this as convergence.

nlopt_ccsaq
"nlopt_ccsaq"

Minimize a scalar function using CCSAQ algorithm.

CCSAQ uses the quadratic variant of the conservative convex separable approximation. The algorithm performs gradient based local optimization with equality (but not inequality) constraints. At each candidate point x, a quadratic approximation to the criterion faunction is computed using the value of gradient at point x. A penalty term is incorporated to render optimizaion convex and conservative. The algorithm is “globally convergent” in the sense that it is guaranteed to con- verge to a local optimum from any feasible starting point.

The implementation is based on CCSA algorithm described in [algo_50].

  • convergence.relative_params_tolerance (float): Stop when the relative movement between parameter vectors is smaller than this.

  • convergence.absolute_params_tolerance (float): Stop when the absolute movement between parameter vectors is smaller than this.

  • convergence.relative_criterion_tolerance (float): Stop when the relative improvement between two iterations is smaller than this.

  • convergence.absolute_criterion_tolerance (float): Stop when the change of the criterion function between two iterations is smaller than this.

  • stopping.max_criterion_evaluations (int): If the maximum number of function evaluation is reached, the optimization stops but we do not count this as convergence.

nlopt_mma
"nlopt_mma"

Minimize a scalar function using the method of moving asymptotes (MMA).

The implementation is based on an algorithm described in [algo_50].

The algorithm performs gradient based local optimization with equality (but not inequality) constraints. At each candidate point x, an approximation to the criterion faunction is computed using the value of gradient at point x. A quadratic penalty term is incorporated to render optimizaion convex and conservative. The algorithm is “globally convergent” in the sense that it is guaranteed to con- verge to a local optimum from any feasible starting point.

  • convergence.relative_params_tolerance (float): Stop when the relative movement between parameter vectors is smaller than this.

  • convergence.absolute_params_tolerance (float): Stop when the absolute movement between parameter vectors is smaller than this.

  • convergence.relative_criterion_tolerance (float): Stop when the relative improvement between two iterations is smaller than this.

  • convergence.absolute_criterion_tolerance (float): Stop when the change of the criterion function between two iterations is smaller than this.

  • stopping.max_criterion_evaluations (int): If the maximum number of function evaluation is reached, the optimization stops but we do not count this as convergence.

nlopt_var
"nlopt_var"

Minimize a scalar function limited memory switching variable-metric method.

The algorithm relies on saving only limited number M of past updates of the gradient to approximate the inverse hessian. The large is M, the more memory is consumed

Detailed explanation of the algorithm, including its two variations of rank-2 and rank-1 methods can be found in the following paper [algo_51] .

  • convergence.relative_params_tolerance (float): Stop when the relative movement between parameter vectors is smaller than this.

  • convergence.absolute_params_tolerance (float): Stop when the absolute movement between parameter vectors is smaller than this.

  • convergence.relative_criterion_tolerance (float): Stop when the relative improvement between two iterations is smaller than this.

  • convergence.absolute_criterion_tolerance (float): Stop when the change of the criterion function between two iterations is smaller than this.

  • stopping.max_criterion_evaluations (int): If the maximum number of function evaluation is reached, the optimization stops but we do not count this as convergence.

  • rank_1_update (bool): Whether I rank-1 or rank-2 update is used.

nlopt_slsqp
"nlopt_slsqp"

Optimize a scalar function based on SLSQP method.

SLSQP solves gradient based nonlinearly constrained optimization problems. The algorithm treats the optimization problem as a sequence of constrained least-squares problems.

The implementation is based on the procedure described in [algo_1] and [algo_52] .

  • convergence.relative_params_tolerance (float): Stop when the relative movement between parameter vectors is smaller than this.

  • convergence.absolute_params_tolerance (float): Stop when the absolute movement between parameter vectors is smaller than this.

  • convergence.relative_criterion_tolerance (float): Stop when the relative improvement between two iterations is smaller than this.

  • convergence.absolute_criterion_tolerance (float): Stop when the change of the criterion function between two iterations is smaller than this.

  • stopping.max_criterion_evaluations (int): If the maximum number of function evaluation is reached, the optimization stops but we do not count this as convergence.

nlopt_direct
"nlopt_direct"

Optimize a scalar function based on DIRECT method.

DIRECT is the DIviding RECTangles algorithm for global optimization, described in [algo_53] .

Variations of the algorithm include locally biased routines (distinguished by _L suffix) that prove to be more efficients for functions that have few local minima. See the following for the DIRECT_L variant [algo_54] .

Locally biased algorithms can be implmented both with deterministic and random (distinguished by _RAND suffix) search algorithm.

Finally, both original and locally biased variants can be implemented with and without the rescaling of the bound constraints.

Boolean arguments locally_biased, ‘random_search’, and ‘unscaled_bouds’ can be set to True or False to determine which method is run. The comprehensive list of available methods are: - “DIRECT” - “DIRECT_L” - “DIRECT_L_NOSCAL” - “DIRECT_L_RAND” - “DIRECT_L_RAND_NOSCAL” - “DIRECT_RAND”

  • convergence.relative_params_tolerance (float): Stop when the relative movement between parameter vectors is smaller than this.

  • convergence.absolute_params_tolerance (float): Stop when the absolute movement between parameter vectors is smaller than this.

  • convergence.relative_criterion_tolerance (float): Stop when the relative improvement between two iterations is smaller than this.

  • convergence.absolute_criterion_tolerance (float): Stop when the change of the criterion function between two iterations is smaller than this.

  • stopping.max_criterion_evaluations (int): If the maximum number of function evaluation is reached, the optimization stops but we do not count this as convergence.

  • locally_biased (bool): Whether the “L” version of the algorithm is selected.

  • random_search (bool): Whether the randomized version of the algorithm is selected.

  • unscaled_bounds (bool): Whether the “NOSCAL” version of the algorithm is selected.

nlopt_esch
"nlopt_esch"

Optimize a scalar function using the ESCH algorithm.

ESCH is an evolutionary algorithm that supports bound constraints only. Specifi cally, it does not support nonlinear constraints.

More information on this method can be found in [algo_55] , [algo_56] , [algo_57] and [algo_58] .

  • convergence.relative_params_tolerance (float): Stop when the relative movement between parameter vectors is smaller than this.

  • convergence.absolute_params_tolerance (float): Stop when the absolute movement between parameter vectors is smaller than this.

  • convergence.relative_criterion_tolerance (float): Stop when the relative improvement between two iterations is smaller than this.

  • convergence.absolute_criterion_tolerance (float): Stop when the change of the criterion function between two iterations is smaller than this.

  • stopping.max_criterion_evaluations (int): If the maximum number of function evaluation is reached, the optimization stops but we do not count this as convergence.

nlopt_isres
"nlopt_isres"

Optimize a scalar function using the ISRES algorithm.

ISRES is an implementation of “Improved Stochastic Evolution Strategy” written for solving optimization problems with non-linear constraints. The algorithm is supposed to be a global method, in that it has heuristics to avoid local minima. However, no convergence proof is available.

The original method and a refined version can be found, respecively, in [algo_59] and [algo_60] .

  • convergence.relative_params_tolerance (float): Stop when the relative movement between parameter vectors is smaller than this.

  • convergence.absolute_params_tolerance (float): Stop when the absolute movement between parameter vectors is smaller than this.

  • convergence.relative_criterion_tolerance (float): Stop when the relative improvement between two iterations is smaller than this.

  • convergence.absolute_criterion_tolerance (float): Stop when the change of the criterion function between two iterations is smaller than this.

  • stopping.max_criterion_evaluations (int): If the maximum number of function evaluation is reached, the optimization stops but we do not count this as convergence.

nlopt_crs2_lm
"nlopt_crs2_lm"

Optimize a scalar function using the CRS2_LM algorithm.

This implementation of controlled random search method with local mutation is based on [algo_61] .

The original CRS method is described in [algo_62] and [algo_63] .

CRS class of algorithms starts with random population of points and evolves the points “randomly”. The size of the initial population can be set via the param- meter population_size. If the user doesn’t specify a value, it is set to the nlopt default of 10*(n+1).

  • convergence.relative_params_tolerance (float): Stop when the relative movement between parameter vectors is smaller than this.

  • convergence.absolute_params_tolerance (float): Stop when the absolute movement between parameter vectors is smaller than this.

  • convergence.relative_criterion_tolerance (float): Stop when the relative improvement between two iterations is smaller than this.

  • convergence.absolute_criterion_tolerance (float): Stop when the change of the criterion function between two iterations is smaller than this.

  • stopping.max_criterion_evaluations (int): If the maximum number of function evaluation is reached, the optimization stops but we do not count this as convergence.

  • population_size (int): Size of the population. If None, it’s set to be 10 * (number of parameters + 1).

The SimOpt Optimizers (simopt)#

estimagic supports the following SimOpt algorithms. Please add the appropriate citations in addition to estimagic when using a SimOpt algorithm. To install simopt run pip install simoptlib==1.0.1.

simopt_adam
"simopt_adam"

Minimize a scalar function using the ADAM algorithm from SimOpt.

  • stopping_max_iterations (int): If the maximum number of iterations is reached, the optimization stops, but we do not count this as convergence.

  • crn_across_solns (bool): Use CRN across solutions? Default True.

  • r (int): Number of replications taken at each solution. Default 1.

  • beta_1 (float): Exponential decay of the rate for the first moment estimates. Default 0.9.

  • beta_2 (float): Exponential decay rate for the second-moment estimates. Default 0.999.

  • alpha (float): Step size. Default 1.0.

  • epsilon (float): A small value to prevent zero-division. Default 10e-8.

  • sensitivity (float): Shrinking scale for variable bounds. Default 10e-7.

simopt_astrodf
"simopt_astrodf"

Minimize a scalar function using the ASTRODF algorithm from SimOpt.

  • stopping_max_iterations (int): If the maximum number of iterations is reached, the optimization stops, but we do not count this as convergence.

  • bounds_padding (float): Subtract (add) this value of the bounds which will be used by ASTRODF internally. Default 1e-8.

  • crn_across_solns (bool): Use CRN across solutions? Default True.

  • delta_max (float): Maximum value of the trust-region radius. Default 50.0

  • eta_1 (float): Threshhold for a successful iteration. Default 0.1.

  • eta_2 (float): Threshhold for a very successful iteration. Default 0.5.

  • gamma_1 (float): Very successful step trust-region radius increase. Default 2.0.

  • gamma_2 (float): Unsuccessful step trust-region radius decrease. Default 0.5.

  • w (float): Trust-region radius rate of shrinkage in contracation loop. Default 0.85.

  • mu (int): Trust-region radius ratio upper bound in contraction loop. Default 1000.

  • beta (int): Trust-region radius ratio lower bound in contraction loop. Default 10.

  • lambda_min (int): Minimum sample size value. Default 8.

  • simple_solve (bool): Solve subproblem with Cauchy point (rough approximate)? Default False.

  • criticality_select (bool): Skip contraction loop if not near critical region? Default True.

  • criticality_threshold (float): Threshold on gradient norm indicating near-critical region. Default 0.1.

Note

To get more accurate results in the case of bounds we revert the subtraction of a large value from the bounds that is done internally in simopt. Since the algorithm is numerically instable in the case of binding bounds without this substraction, we subtract a (small) value defined by bounds_padding. See the ASTRODF source code for details.

simopt_spsa
"simopt_spsa"

Minimize a scalar function using the SPSA algorithm from SimOpt.

  • stopping_max_iterations (int): If the maximum number of iterations is reached, the optimization stops, but we do not count this as convergence.

  • crn_across_solns (bool): Use CRN across solutions? Default True.

  • alpha (float): Non-negative coefficient in the SPSA gain sequecence ak. Default 0.602.

  • gamma (float): Non-negative coefficient in the SPSA gain sequence ck. Default 0.101.

  • step (float): Initial desired magnitude of change in the theta elements. Default 0.5.

  • gavg (int): Averaged SP gradients used per iteration. Default 1.

  • n_reps (int): Number of replications takes at each solution. Default 2.

  • n_loss (int): Number of loss function evaluations used in this gain calculation. Default 2.

  • eval_pct (float): Percentage of the expected number of loss evaluations per run. Default 2/3.

  • iter_pct (float): Percentage of the maximum expected number of iterations. Default 0.1.

simopt_strong
"simopt_strong"

Minimize a scalar function using the STRONG algorithm from SimOpt.

  • stopping_max_iterations (int): If the maximum number of iterations is reached, the optimization stops, but we do not count this as convergence.

  • crn_across_solns (bool): Use CRN across solutions? Default True.

  • n0 (int): Initial sample size Default 10.

  • n_r (int): Number of replications taken at each solution. Default 1.

  • sensitivity (float): Shrinking scale for VarBds. Default 10e-7.

  • delta_threshold (float): Maximum value of the radius. Default 1.2.

  • delta_T (float): Initial size of trust region. Default 2.0.

  • eta_0 (float): Constant for accepting. Default 0.01.

  • eta_1 (float): Constant for more confident accepting. Default 0.3.

  • gamma_1 (float): Constant for shrinking the trust region. Default 0.9.

  • gamma_2 (float): Constant for expanding the trust region. Default 1.11.

  • lambda (int): Magnifying factor for n_r inside the finite difference function. Default 2.

  • lambda_2 (float): Magnifying factor for n_r in stage I and stage II. Default 1.01.

References#

algo_1(1,2)

Dieter Kraft. A software package for sequential quadratic programming. Technical Report, DLR German Aerospace Center – Institute for Flight Mechanics, Köln, Germany, 1988. URL: http://degenerateconic.com/wp-content/uploads/2018/03/DFVLR_FB_88_28.pdf.

algo_2

Fuchang Gao and Lixing Han. Implementing the nelder-mead simplex algorithm with adaptive parameters. Computational Optimization and Applications, 51(1):259–277, 2012.

algo_3(1,2,3,4,5,6,7,8)

Jorge Nocedal and Stephen Wright. Numerical optimization. Springer Science & Business Media, 2006.

algo_4

M Powell. A direct search optimization method that models the objective and constraint functions by linear interpolation, pages 51–67. Kluwer Academic, Dordrecht, 1994.

algo_5

Michael JD Powell. Direct search algorithms for optimization calculations. Acta numerica, pages 287–336, 1998.

algo_6

Michael JD Powell. A view of algorithms for optimization without derivatives. Mathematics Today-Bulletin of the Institute of Mathematics and its Applications, 43(5):170–174, 2007.

algo_7

Marucha Lalee, Jorge Nocedal, and Todd Plantenga. On the implementation of an algorithm for large-scale equality constrained optimization. SIAM Journal on Optimization, 8(3):682–706, 1998.

algo_8

AR Conn, NI Gould, and PL Toint. Nonlinear equations and nonlinear fitting. In Trust region methods, volume 1, pages 749–774. Siam, 2000.

algo_9

Richard H Byrd, Mary E Hribar, and Jorge Nocedal. An interior point algorithm for large-scale nonlinear programming. SIAM Journal on Optimization, 9(4):877–900, 1999.

algo_10

Ernst R. Berndt, Bronwyn Hall, Robert Hall, and Jerry Hausman. Estimation and inference in nonlinear structural models. Annals of Economic and Social Measurement, pages 653–665, 1974.

algo_11

Halbert White. Maximum likelihood estimation of misspecified models. Econometrica, 50(1):1–25, 1982.

algo_12(1,2)

S Benson, LC McInnes, JJ Moré, T Munson, and J Sarich. Tao user manual (revision 3.7). Technical Report, Technical Report ANL/MCS-TM-322, Argonne National Laboratory, 2017. URL: http://web.mit.edu/tao-petsc_v3.7/tao_manual.pdf.

algo_13(1,2)

Stefan M. Wild. Solving derivative-free nonlinear least squares problems with pounders. Technical Report, Argonne National Laboratory, 2015. URL: https://doi.org/10.1137/1.9781611974683.ch40.

algo_14(1,2,3)

Coralia Cartis, Jan Fiala, Benjamin Marteau, and Lindon Roberts. Improving the flexibility and robustness of model-based derivative-free optimization solvers. 2018. arXiv:1804.00154.

algo_15(1,2,3)

Michael JD Powell. The bobyqa algorithm for bound constrained optimization without derivatives. Cambridge NA Report NA2009/06, University of Cambridge, Cambridge, pages 26–46, 2009.

algo_16(1,2)

Coralia Cartis, Jan Fiala, Benjamin Marteau, and Lindon Roberts. Improving the flexibility and robustness of model-based derivative-free optimization solvers. 2018. arXiv:1804.00154.

algo_17(1,2,3)

Coralia Cartis, Lindon Roberts, and Oliver Sheridan-Methven. Escaping local minima with derivative-free methods: a numerical investigation. 2018. arXiv:1812.11343.

algo_18

Francesco Biscani and Dario Izzo. A parallel global multiobjective framework for optimization: pagmo. Journal of Open Source Software, 5(53):2338, 2020. URL: https://doi.org/10.21105/joss.02338, doi:10.21105/joss.02338.

algo_19

Martin Schlüter, Jose A. Egea, and Julio R. Banga. Extended ant colony optimization for non-convex mixed integer nonlinear programming. Computers & Operations Research, 36(7):2217–2229, jul 2009. doi:10.1016/j.cor.2008.08.015.

algo_20

Dervis Karaboga and Bahriye Basturk. A powerful and efficient algorithm for numerical function optimization: artificial bee colony (ABC) algorithm. Journal of Global Optimization, 39(3):459–471, apr 2007. doi:10.1007/s10898-007-9149-x.

algo_21

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