# How to write objective functions#

optimagic is very flexible when it comes to the objective function and its derivatives. In this how-to guide we start with simple examples, that would also work with scipy.optimize before we show advanced options and their advantages.

## The simplest case#

In the simplest case, `fun`

maps a numpy array into a scalar objective value. The name
of first argument of `fun`

is arbitrary.

```
import numpy as np
import optimagic as om
def sphere(x):
return x @ x
res = om.minimize(
fun=sphere,
params=np.arange(3),
algorithm="scipy_lbfgsb",
)
res.params.round(6)
```

```
array([ 0., -0., 0.])
```

## More flexible `params`

#

In all but the most simple problems, a flat numpy array is not ideal to keep track of all the different parameters one wants to optimize over. Therefore, optimagic accepts objective functions that work with other parameter formats. Below we show a simple example. More examples can be found here.

```
def dict_fun(x):
return x["a"] ** 2 + x["b"] ** 4
res = om.minimize(
fun=dict_fun,
params={"a": 1, "b": 2},
algorithm="scipy_lbfgsb",
)
res.params
```

```
{'a': np.float64(2.2389496983474372e-07),
'b': np.float64(0.005724249755832909)}
```

The important thing is that the `params`

provided to `minimize`

need to have the format
that is expected by the objective function.

## Functions with additional arguments#

In many applications, the objective function takes more than `params`

as argument.
This can be achieved via `fun_kwargs`

. Take the following simplified example:

```
def shifted_sphere(x, offset):
return (x - offset) @ (x - offset)
res = om.minimize(
fun=shifted_sphere,
params=np.arange(3),
algorithm="scipy_lbfgsb",
fun_kwargs={"offset": np.ones(3)},
)
res.params
```

```
array([0.99999999, 0.99999999, 0.99999999])
```

`fun_kwargs`

is a dictionary with keyword arguments for `fun`

. There is no constraint
on the number or names of those arguments.

## Least-Squares problems#

Many estimation problems have a least-squares structure. If so, specialized optimizers that exploit this structure can be much faster than standard optimizers. The `sphere`

function from above is the simplest possible least-squarse problem you could imagine: the least-squares residuals are just the params.

To use least-squares optimizers in optimagic, you need to mark your function with a decorator and return the least-squares residuals instead of the aggregated function value.

```
@om.mark.least_squares
def ls_sphere(params):
return params
res = om.minimize(
fun=ls_sphere,
params=np.arange(3),
algorithm="pounders",
)
res.params.round(5)
```

```
array([ 0., 0., -0.])
```

Any least-squares optimization problem is also a standard optimization problem. You can therefore optimize least-squares functions with scalar optimizers as well:

```
res = om.minimize(
fun=ls_sphere,
params=np.arange(3),
algorithm="scipy_lbfgsb",
)
res.params.round(5)
```

```
array([ 0., -0., 0.])
```

## Returning additional information#

You can return additional information such as intermediate results, debugging information, etc. in your objective function. This information will be stored in a database if you use logging.

To do so, you need to return a `FunctionValue`

object.

```
def sphere_with_info(x):
return om.FunctionValue(value=x @ x, info={"avg": x.mean()})
res = om.minimize(
fun=sphere_with_info,
params=np.arange(3),
algorithm="scipy_lbfgsb",
)
res.params.round(6)
```

```
array([ 0., -0., 0.])
```

The `info`

can be an arbitrary dictionary. In the oversimplified example we returned the
mean of the parameters, which could have been recovered from the params history that
is collected anyways but in real applications this feature can be helpful.