CMA-ES for Discrete and Mixed-Variable Optimization on Sets of Points

TL;DR

This paper introduces CMA-ES-SoP, an extension of CMA-ES for optimizing over sets of points in discrete and mixed-variable problems, effectively preventing premature convergence through adaptive margin correction.

Contribution

It proposes CMA-ES-SoP, a novel extension of CMA-ES that handles set-based optimization with adaptive margin correction to improve performance on discrete and mixed-variable problems.

Findings

CMA-ES-SoP successfully optimized set-based problems where naive CMA-ES failed.

The adaptive margin correction prevents premature convergence.

Numerical simulations validate the effectiveness of CMA-ES-SoP.

Abstract

Discrete and mixed-variable optimization problems have appeared in several real-world applications. Most of the research on mixed-variable optimization considers a mixture of integer and continuous variables, and several integer handlings have been developed to inherit the optimization performance of the continuous optimization methods to mixed-integer optimization. In some applications, acceptable solutions are given by selecting possible points in the disjoint subspaces. This paper focuses on the optimization on sets of points and proposes an optimization method by extending the covariance matrix adaptation evolution strategy (CMA-ES), termed the CMA-ES on sets of points (CMA-ES-SoP). The CMA-ES-SoP incorporates margin correction that maintains the generation probability of neighboring points to prevent premature convergence to a specific non-optimal point, which is an effective integer-handling technique for CMA-ES. In addition, because margin correction with a fixed margin value tends to increase the marginal probabilities for a portion of neighboring points more than necessary, the CMA-ES-SoP updates the target margin value adaptively to make the average of the marginal probabilities close to a predefined target probability. Numerical simulations demonstrated that the CMA-ES-SoP successfully optimized the optimization problems on sets of points, whereas the naive CMA-ES failed to optimize them due to premature convergence.