Matrix Inversion¶

In this case study, the optimization of a matrix shall be illustrated. Of course, we all know that there are very efficient algorithms for calculating an inverse of a matrix. However, for the sake of illustration, a small example shall show that pymoo can also be used to optimize matrices or even tensors.

Assuming matrix A has a size of n x n, the problem can be defined by optimizing a vector consisting of n**2 variables. During evaluation the vector x, is reshaped to inversion of the matrix to be found (and also stored as the attribute A_inv to be retrieved later).

[1]:

from pymoo.core.problem import ElementwiseProblem


class MatrixInversionProblem(ElementwiseProblem):

    def __init__(self, A, **kwargs):
        self.A = A
        self.n = len(A)
        super().__init__(n_var=self.n**2, n_obj=1, xl=-100.0, xu=+100.0, **kwargs)


    def _evaluate(self, x, out, *args, **kwargs):
        A_inv = x.reshape((self.n, self.n))
        out["A_inv"] = A_inv

        I = np.eye(self.n)
        out["F"] = ((I - (A @ A_inv)) ** 2).sum()

Now let us see what solution is found to be optimal

[2]:

import numpy as np
from pymoo.algorithms.soo.nonconvex.de import DE
from pymoo.optimize import minimize

np.random.seed(1)
A = np.random.random((2, 2))

problem = MatrixInversionProblem(A)

algorithm = DE()

res = minimize(problem,
               algorithm,
               seed=1,
               verbose=False)

opt = res.opt[0]

In this case the true optimum is actually known. It is

[3]:

np.linalg.inv(A)

[3]:

array([[ 2.39952297e+00, -5.71699951e+00],
       [-9.07758630e-04,  3.30977861e+00]])

Let us see if the black-box optimization algorithm has found something similar

[4]:

opt.get("A_inv")

[4]:

array([[ 2.39916052e+00, -5.71656622e+00],
       [-8.41267527e-04,  3.30978797e+00]])

This small example shall have illustrated how a matrix can be optimized. In fact, this is implemented by optimizing a vector of variables that are reshaped during evaluation.