Marginal Probability-Based Integer Handling for CMA-ES Tackling Single-and Multi-Objective Mixed-Integer Black-Box Optimization

TL;DR

This paper introduces a novel marginal probability-based integer handling method for CMA-ES to effectively optimize mixed-integer black-box problems, addressing stagnation issues caused by discretization and demonstrating improved performance in single- and multi-objective scenarios.

Contribution

The study proposes a simple yet effective integer handling technique for CMA-ES based on marginal probabilities, enhancing optimization in mixed-integer black-box problems.

Findings

Demonstrates improved efficiency and robustness on benchmark problems.

Addresses stagnation caused by discretization in integer variables.

Validates the method's effectiveness in both single- and multi-objective optimization.

Abstract

This study targets the mixed-integer black-box optimization (MI-BBO) problem where continuous and integer variables should be optimized simultaneously. The CMA-ES, our focus in this study, is a population-based stochastic search method that samples solution candidates from a multivariate Gaussian distribution (MGD), which shows excellent performance in continuous BBO. The parameters of MGD, mean and (co)variance, are updated based on the evaluation value of candidate solutions in the CMA-ES. If the CMA-ES is applied to the MI-BBO with straightforward discretization, however, the variance corresponding to the integer variables becomes much smaller than the granularity of the discretization before reaching the optimal solution, which leads to the stagnation of the optimization. In particular, when binary variables are included in the problem, this stagnation more likely occurs because the granularity of the discretization becomes wider, and the existing integer handling for the CMA-ES does not address this stagnation. To overcome these limitations, we propose a simple integer handling for the CMA-ES based on lower-bounding the marginal probabilities associated with the generation of integer variables in the MGD. The numerical experiments on the MI-BBO benchmark problems demonstrate the efficiency and robustness of the proposed method. Furthermore, in order to demonstrate the generality of the idea of the proposed method, in addition to the single-objective optimization case, we incorporate it into multi-objective CMA-ES and verify its performance on bi-objective mixed-integer benchmark problems.