On the Design of a Matrix Adaptation Evolution Strategy for Optimization on General Quadratic Manifolds

Abstract

An evolution strategy design is presented that allows for an evolution on general quadratic manifolds. That is, it covers elliptic, parabolic, and hyperbolic equality constraints. The peculiarity of the presented algorithm design is that it is an interior point method. It evaluates the objective function only for feasible search parameter vectors and it evolves itself on the nonlinear constraint manifold. Such a characteristic is particularly important in situations where it is not possible to evaluate infeasible parameter vectors, e.g., in simulation-based optimization. This is achieved by a closed form transformation of an individual’s parameter vector, which is in contrast to iterative repair mechanisms. This constraint handling approach is incorporated into a matrix adaptation evolution strategy making such algorithms capable of handling problems containing the constraints considered. Results of different experiments are presented. A test problem consisting of a spherical objective function and a single hyperbolic/parabolic equality constraint is used. It is designed to be scalable in the dimension. As a further benchmark, the Thomson problem is used. Both problems are used to compare the performance of the developed algorithm with other optimization methods supporting constraints. The experiments show the effectiveness of the proposed algorithm on the considered problems. Additionally, an idea for handling multiple constraints is discussed. And for a better understanding of the dynamical behavior of the proposed algorithm, single run dynamics are presented.