# How to specify constraints#

## Constraints vs bounds#

Estimagic 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 estimagic:

>>> import numpy as np
>>> import estimagic as em
>>> def criterion(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 = em.minimize(
...    criterion=criterion,
...    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 estimagic. For each constraint, we will select a subset of the parameters on which the constraint is imposed via the “loc” key. Generalizations for selecting subsets of params that are not a flat numpy array are explained in the next section.

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 = em.minimize(
...    criterion=criterion,
...    params=np.array([2.5, 1, 1, 1, 1, -2.5]),
...    algorithm="scipy_lbfgsb",
...    constraints={"loc": [0, 5], "type": "fixed"},
...    )


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.

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 = em.minimize(
...    criterion=criterion,
...    params=np.array([1, 1, 1, 1, 1, 1]),
...    algorithm="scipy_lbfgsb",
...    constraints={"loc": [1, 2, 3], "type": "increasing"},
...    )


Imposing the constraint on positions "loc": [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 entry in “loc” 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.

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 = em.minimize(
...    criterion=criterion,
...    params=np.array([1, 1, 1, 1, 1, 1]),
...    algorithm="scipy_lbfgsb",
...    constraints={"loc": [3, 0, 4], "type": "decreasing"},
...    )


Imposing the constraint on positions "loc": [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 entry in “loc” 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.

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

>>> res = em.minimize(
...    criterion=criterion,
...    params=np.array([1, 1, 1, 1, 1, 1]),
...    algorithm="scipy_lbfgsb",
...    constraints={"loc": [0, 5], "type": "equality"},
...    )


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 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 = em.minimize(
...    criterion=criterion,
...    params=np.array([1, 1, 1, 1, 1, 1]),
...    algorithm="scipy_lbfgsb",
...    constraints={"locs": [[0, 1], [2, 3]], "type": "pairwise_equality"},
...    )


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. ])


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 = em.minimize(
...    criterion=criterion,
...    params=np.array([0.3, 0.2, 0.25, 0.25, 1, 1]),
...    algorithm="scipy_lbfgsb",
...    constraints={"loc": [0, 1, 2, 3], "type": "probability"},
...    )


This yields again the correct result:

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


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 = em.minimize(
...    criterion=criterion,
...    params=np.ones(6),
...    algorithm="scipy_lbfgsb",
...    constraints={"loc": [0, 1, 2], "type": "covariance"},
...    )


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 estimagic’s utility functions to actually build the covariance matrix out of the first three parameters:

>>> from estimagic.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 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 = em.minimize(
...    criterion=criterion,
...    params=np.ones(6),
...    algorithm="scipy_lbfgsb",
...    constraints={"loc": [0, 1, 2], "type": "sdcorr"},
...    )


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 estimagic’s utility functions to actually build the standard deviations and the correlation matrix:

>>> from estimagic.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 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 = em.minimize(
...    criterion=criterion,
...    params=np.ones(6),
...    algorithm="scipy_lbfgsb",
...    constraints={
...    "loc": [0, 1, 2, 3],
...    "type": "linear",
...    "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.

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 = em.minimize(
...    criterion=criterion,
...    params=np.ones(6),
...    algorithm="scipy_slsqp",
...    constraints={
...    "type": "nonlinear",
...    "selector": lambda x: x[:-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 = em.minimize(
...    criterion=criterion,
...    params=np.ones(6),
...    algorithm="scipy_lbfgsb",
...    constraints=[
...    {"loc": [0, 1], "type": "equality"},
...    {"loc": [2, 3, 4], "type": "linear", "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?#

All the above examples use a "loc" entry in the constraint dictionary to select the subset of params on which the constraint is imposed. This is just one out of several ways to do it. Which methods are available also depends on whether your parameters are a numpy array, DataFrame, or general pytree.

 loc query selector 1d-array ✅ (positions) ❌ ✅ DataFrame ✅ (labels) ✅ ✅ Pytree ❌ ❌ ✅

Below we show how to use each of these selection methods in simple examples

In all the examples above, we imposed constraints where our params are a numpy array and the loc method is used to select the constraint parameters. So now, we turn to DataFrame params.

Let’s assume our params are a DataFrame with a two level index. The names of the index levels are category and name. Something like this could, for example, be the params of an Ordered Logit model.

 value category name betas a 0.95 betas b 0.9 cutoffs a 0 cutoffs b 0.4

Now, let;s impose the constraint that the cutoffs (i.e. the last two parameters) are increasing.

res = em.minimize(
criterion=some_criterion,
params=params,
algorithm="scipy_lbfgsb",
constraints={"loc": "cutoffs", "type": "increasing"},
)


The value corresponding to "loc" can be anything you would pass to pandas’ DataFrame.loc method, too. So, if you know pandas, imposing constraints in estimagic via "loc" should feel already familiar. Imposing constraints this way can be extremely powerful if you have a well designed MultiIndex, as you can easily select groups of parameters or single paramaters.

Let’s assume our params are a DataFrame with a two level index. The names of the index levels are category and name. Something like this could for example be the params of an Ordered Logit model.

 value category name betas a 0.95 betas b 0.9 cutoffs a 0 cutoffs b 0.4

This time, we want to fix all betas as well as all parameters where the second index level is equal to "a". If we wanted to do that using loc, we would have to type out three index tuples. So let’s do that with a query instead:

res = em.minimize(
criterion=some_criterion,
params=params,
algorithm="scipy_lbfgsb",
constraints={"query": "category == 'betas' | name == 'a'", "type": "fixed"},
)


The value corresponding to "query" can be anything you would pass to pandas’ DataFrame.query method, too. So, if you know pandas, imposing constraints in estimagic via "query" should feel just the same.

Using selector to select the parameters is the most general way and works for all params. 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 = em.minimize(
criterion=some_criterion,
params=params,
algorithm="scipy_lbfgsb",
constraints={"selector": lambda params: params["b"]["d"], "type": "fixed"},
)


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 = em.minimize(
criterion=some_criterion,
params=params,
algorithm="scipy_lbfgsb",
constraints={"selector": my_selector, "type": "fixed"},
)