Static and Dynamic Multimodal Optimization by Improved Covariance Matrix Self-Adaptation Evolution Strategy With Repelling Subpopulations

The covariance matrix self-adaptation evolution strategy with repelling subpopulations (RS-CMSA-ES) is one of the most successful multimodal optimization (MMO) methods currently available. However, some of its components may become inefficient in certain situations. This study introduces the second variant of this method, called RS-CMSA-ESII. It improves the adaptation schemes for the normalized taboo distances of the archived solutions and the covariance matrix of the subpopulation, the termination criteria for the subpopulations, and the way in which the infeasible solutions are treated. It also improves the time complexity of RS-CMSA-ES by updating the initialization procedure of a subpopulation and developing a more accurate metric for determining critical taboo regions. The effects of these modifications are illustrated by designing controlled numerical simulations. RS-CMSA-ESII is then compared with the most successful and recent niching methods for MMO on a widely adopted test suite. The results obtained reveal the superiority of RS-CMSA-ESII over these methods, including the winners of the competition on niching methods for MMO in previous years. Besides, this study extends RS-CMSA-ESII to dynamic MMO and compares it with a few recently proposed methods on the modified moving peak benchmark functions.

I. Introduction

An optimization problem has been traditionally perceived as that of finding a single best solution (the global optimum) given an objective function, decision parameters, and possibly, problem constraints. There are real-world problems with features that are not easy to formulate, e.g., a product’s aesthetics, or a company’s preferences toward specific suppliers. In these situations, the decision maker may be interested or even need a set of diverse optimal solutions instead of one. The best solution to the actual problem may then be selected from this diverse set of near-optimal solutions considering hard-to-formulate aspects of the actual problem. Additionally, knowing all near-optimal solutions of a problem might be crucial in certain problems. For example, a design engineer may need to know all the resonance frequencies of a mechanical system [1].