# How to specify constraints#

## Constraints vs bounds#

optimagic distinguishes between bounds and constraints. Bounds are lower and upper
bounds for parameters. In the literature, they are sometimes called box constraints.
Bounds are specified as `lower_bounds`

and `upper_bounds`

argument to `maximize`

and
`minimize`

.

Examples with bounds can be found in this tutorial.

To specify more general constraints on your parameters, you can use the argument
`constraints`

. The variety of constraints you can impose ranges from rather simple ones
(e.g. parameters are fixed to a value, a group of parameters is required to be equal) to
more complex ones (like general linear constraints, or even nonlinear constraints).

## Can you use constraints with all optimizers?#

With the exception of general nonlinear constraints, we implement constraints via reparametrizations. Details are explained here. This means that you can use all of the constraints with any optimizer that supports bounds. Some constraints (e.g. fixing parameters) can even be used with optimizers that do not support bounds.

## Example criterion function#

Let’s look at a variation of the sphere function to illustrate what kinds of constraints you can impose and how you specify them in optimagic:

```
>>> import numpy as np
>>> import optimagic as om
>>> def fun(params):
... offset = np.linspace(1, 0, len(params))
... x = params - offset
... return x @ x
```

The unconstrained optimum of a six-dimensional version of this problem is:

```
>>> res = om.minimize(
... fun=fun,
... params=np.array([2.5, 1, 1, 1, 1, -2.5]),
... algorithm="scipy_lbfgsb",
... )
>>> res.params.round(3)
array([1. , 0.8, 0.6, 0.4, 0.2, 0. ])
```

The unconstrained optimum is usually easy to see because all parameters enter the criterion function in a additively separable way.

## Types of constraints#

Below, we show a very simple example of each type of constraint implemented in
optimagic. For each constraint, we will select a subset of the parameters on which the
constraint is imposed via the `selector`

argument, which is a function that takes in the
full parameter vector and returns the subset of parameters that should be constrained.

##
fixed

The simplest (but very useful) constraint fixes parameters at their start values.

Let’s take the above example and fix the first and last parameter to 2.5 and -2.5, respectively.

```
>>> res = om.minimize(
... fun=fun,
... params=np.array([2.5, 1, 1, 1, 1, -2.5]),
... algorithm="scipy_lbfgsb",
... constraints=om.FixedConstraint(
... selector=lambda params: params[[0, 5]]
... ),
... )
```

Looking at the optimization result, we get:

```
>>> res.params.round(3)
array([ 2.5, 0.8, 0.6, 0.4, 0.2, -2.5])
```

Which is indeed the correct constrained optimum. Fixes are compatible with all optimizers.

##
increasing

In our unconstrained example, the optimal parameters are decreasing from left to right. Let’s impose the constraint that the second, third and fourth parameter increase (weakly):

```
>>> res = om.minimize(
... fun=fun,
... params=np.array([1, 1, 1, 1, 1, 1]),
... algorithm="scipy_lbfgsb",
... constraints=om.IncreasingConstraint(
... selector=lambda params: params[[1, 2, 3]]
... ),
... )
```

Imposing the constraint on positions `params[[1, 2, 3]]`

means that the parameter value
at index position `2`

has to be (weakly) greater than the value at position `1`

.
Likewise, the parameter value at index position `3`

has to be (weakly) greater than the
value at position `2`

. Hence, imposing an increasing constraint with
only one selected parameter has no effect. We need to specify at least two parameters to make
a meaningful *relative* comparison.
Note that the increasing constraint affect all three parameters, i.e. `params[1]`

,
`params[2]`

, and `params[3]`

because the optimal parameters in the unconstrained case
are decreasing from left to right.

Looking at the optimization result, we get:

```
>>> res.params.round(3)
array([1. , 0.6, 0.6, 0.6, 0.2, 0. ])
```

Which is indeed the correct constrained optimum. Increasing constraints are only compatible with optimizers that support bounds.

##
decreasing

In our unconstrained example, the optimal parameters are decreasing from left to right already - without imposing any constraints. If we imposed an decreasing constraint without changing the order, it would simply have no effect.

So let’s impose one in a different order:

```
>>> res = om.minimize(
... fun=fun,
... params=np.array([1, 1, 1, 1, 1, 1]),
... algorithm="scipy_lbfgsb",
... constraints=om.DecreasingConstraint(
... selector=lambda params: params[[3, 0, 4]]
... ),
... )
```

Imposing the constraint on positions `params[[3, 0, 4]]`

means that the parameter value
at index position `0`

has to be (weakly) smaller than the value at position `3`

.
Likewise, the parameter value at index position `4`

has to be (weakly) smaller than the
value at position `0`

. Hence, imposing a decreasing constraint with
only one selected parameter has no effect. We need to specify at least two parameters to make
a meaningful *relative* comparison.
Note that the decreasing constraint should have no effect on `params[4]`

because it is
smaller than the other two anyways in the unconstrained optimum, but it will change
the optimal values of `params[3]`

and `params[0]`

. Indeed we get:

```
>>> res.params.round(3)
array([ 0.7, 0.8, 0.6, 0.7, 0.2, -0. ])
```

Which is the correct optimum. Decreasing constraints are only compatible with optimizers that support bounds.

##
equality

In our example, all optimal parameters are different. Let’s constrain the first and last to be equal to each other:

```
>>> res = om.minimize(
... fun=fun,
... params=np.array([1, 1, 1, 1, 1, 1]),
... algorithm="scipy_lbfgsb",
... constraints=om.EqualityConstraint(
... selector=lambda params: params[[0, 5]]
... ),
... )
```

This yields:

```
>>> res.params.round(3)
array([0.5, 0.8, 0.6, 0.4, 0.2, 0.5])
```

Which is the correct solution. Equality constraints are compatible with all optimizers.

##
pairwise_equality

Pairwise equality constraints are similar to equality constraints but impose that two or more groups of parameters are pairwise equal. Let’s look at an example:

```
>>> res = om.minimize(
... fun=fun,
... params=np.array([1, 1, 1, 1, 1, 1]),
... algorithm="scipy_lbfgsb",
... constraints=om.PairwiseEqualityConstraint(
... selectors=[
... lambda params: params[[0, 1]],
... lambda params: params[[2, 3]]
... ],
... ),
... )
```

This constraint imposes that `params[0] == params[2]`

and
`params[1] == params[3]`

. The optimal parameters with this constraint are:

```
>>> res.params.round(3)
array([ 0.8, 0.6, 0.8, 0.6, 0.2, -0. ])
```

##
probability

Let’s impose the constraint that the first four parameters form valid probabilities, i.e. they should add up to one and be between zero and one.

```
>>> res = om.minimize(
... fun=fun,
... params=np.array([0.3, 0.2, 0.25, 0.25, 1, 1]),
... algorithm="scipy_lbfgsb",
... constraints=om.ProbabilityConstraint(
... selector=lambda params: params[:4]
... ),
... )
```

This yields again the correct result:

```
>>> res.params.round(2)
array([0.53, 0.33, 0.13, 0. , 0.2 , 0. ])
```

##
covariance

In many estimation problems, particularly when doing a maximum likelihood estimation,
one has to estimate the covariance matrix of a random variable. The
`covariance`

costraint ensures that such a covariance matrix is always valid,
i.e. positive semi-definite and symmetric. Due to its symmetry, only the lower
triangle of a covariance matrix actually has to be estimated.

Let’s look at an example. We want to impose that the first three elements form the lower triangle of a valid covariance matrix.

```
>>> res = om.minimize(
... fun=fun,
... params=np.ones(6),
... algorithm="scipy_lbfgsb",
... constraints=om.FlatCovConstraint(
... selector=lambda params: params[:3]
... ),
... )
```

This yields the same solution as an unconstrained estimation because the constraint is not binding:

```
>>> res.params.round(3)
array([ 1.006, 0.784, 0.61 , 0.4 , 0.2 , -0. ])
```

We can now use one of optimagic’s utility functions to actually build the covariance matrix out of the first three parameters:

```
>>> from optimagic.utilities import cov_params_to_matrix
>>> cov_params_to_matrix(res.params[:3]).round(2)
array([[1.01, 0.78],
[0.78, 0.61]])
```

##
sdcorr

`sdcorr`

constraints are very similar to `covariance`

constraints. The only
difference is that instead of estimating a covariance matrix, we estimate
standard deviations and the correlation matrix of random variables.

Let’s look at an example. We want to impose that the first three elements form valid standard deviations and a correlation matrix.

```
>>> res = om.minimize(
... fun=fun,
... params=np.ones(6),
... algorithm="scipy_lbfgsb",
... constraints=om.FlatSDCorrConstraint(
... selector=lambda params: params[:3]
... ),
... )
```

This yields the same solution as an unconstrained estimation because the constraint is not binding:

```
>>> res.params.round(3)
array([ 1. , 0.8, 0.6, 0.4, 0.2, -0. ])
```

We can now use one of optimagic’s utility functions to actually build the standard deviations and the correlation matrix:

```
>>> from optimagic.utilities import sdcorr_params_to_sds_and_corr
>>> sd, corr = sdcorr_params_to_sds_and_corr(res.params[:3])
>>> sd.round(2)
array([1. , 0.8])
>>> corr.round(2)
array([[1. , 0.6],
[0.6, 1. ]])
```

##
linear

Linear constraints are the most difficult but also the most powerful constraints
in your toolkit. They can be used to express constraints of the form
`lower_bound <= weights.dot(x) <= upper_bound`

or
`weights.dot(x) = value`

where `x`

are the selected parameters.

Linear constraints have many of the other constraint types as special cases, but typically it is more convenient to use the special cases instead of expressing them as a linear constraint. Internally, it will make no difference.

Let’s impose the constraint that the average of the first four parameters is at least 0.95.

```
>>> res = om.minimize(
... fun=fun,
... params=np.ones(6),
... algorithm="scipy_lbfgsb",
... constraints=om.LinearConstraint(
... selector=lambda params: params[:4],
... lower_bound=0.95,
... weights=0.25,
... ),
... )
```

This yields:

```
>>> res.params.round(2)
array([ 1.25, 1.05, 0.85, 0.65, 0.2 , -0. ])
```

Where the first four parameters have an average of 0.95.

In the above example, `lower_bound`

and `weights`

are scalars. They may, however,
also be arrays (or even pytrees) with bounds and weights for each selected
parameter.

##
nonlinear

Warning

General nonlinear constraints that are specified via a black-box constraint function can only be used if you choose an optimizer that supports it. This feature is currently supported by the algorithms:

`ipopt`

`nlopt`

:`cobyla`

,`slsqp`

,`isres`

,`mma`

`scipy`

:`cobyla`

,`slsqp`

,`trust_constr`

You can use nonlinear constraints to express restrictions of the form
`lower_bound <= func(x) <= upper_bound`

or
`func(x) = value`

where `x`

are the selected parameters and `func`

is the
constraint function.

Let’s impose the constraint that the product of all but the last parameter is 1.

```
>>> res = om.minimize(
... fun=fun,
... params=np.ones(6),
... algorithm="scipy_slsqp",
... constraints=om.NonlinearConstraint(
... selector=lambda params: params[:-1],
... func=lambda x: np.prod(x),
... value=1.0,
... ),
... )
```

This yields:

```
>>> res.params.round(2)
array([ 1.31, 1.16, 1.01, 0.87, 0.75, -0. ])
```

Where the product of all but the last parameters is equal to 1.

If you have a function that calculates the derivative of your constraint, you can add this under the key “derivative” to the constraint dictionary. Otherwise, numerical derivatives are calculated for you if needed.

## Imposing multiple constraints at once#

The above examples all just impose one constraint at a time. To impose multiple constraints simultaneously, simple pass in a list of constraints. For example:

```
>>> res = om.minimize(
... fun=fun,
... params=np.ones(6),
... algorithm="scipy_lbfgsb",
... constraints=[
... om.EqualityConstraint(selector=lambda params: params[:2]),
... om.LinearConstraint(
... selector=lambda params: params[2:5],
... weights=1,
... value=3,
... ),
... ],
... )
This yields:
>>> res.params.round(2)
array([0.9, 0.9, 1.2, 1. , 0.8, 0. ])
```

There are limits regarding the compatibility of overlapping constraints. You will get a descriptive error message if your constraints are not compatible.

## How to select the parameters?#

The parameters can be selected via a `selector`

function. This function takes in the
full parameter vector and returns the subset of parameters that should be constrained.

Let’s assume we have defined parameters in a nested dictionary:

```
params = {"a": np.ones(2), "b": {"c": 3, "d": pd.Series([4, 5])}}
```

It is probably not a good idea to use a nested dictionary for so few parameters, but let’s ignore that.

Now assume we want to fix the parameters in the pandas Series at their start values. We can do so as follows:

```
res = om.minimize(
fun=some_fun,
params=params,
algorithm="scipy_lbfgsb",
constraints=om.FixedConstraint(selector=lambda params: params["b"]["d"]),
)
```

I.e. the value corresponding to `selector`

is a python function that takes the full
`params`

and returns a subset. The selected subset does not have to be a numpy array, it
can be an arbitrary pytree.

Using lambda functions if often convenient, but we could have just as well defined the selector function using def.

```
def my_selector(params):
return params["b"]["d"]
res = om.minimize(
fun=some_fun,
params=params,
algorithm="scipy_lbfgsb",
constraints=om.FixedConstraint(selector=my_selector),
)
```