Matrix Adaptation Evolution Strategy with Multi-Objective Optimization for Multimodal Optimization

Wei Li

doi:10.3390/a12030056

Abstract

The standard covariance matrix adaptation evolution strategy (CMA-ES) is highly effective at locating a single global optimum. However, it shows unsatisfactory performance for solving multimodal optimization problems (MMOPs). In this paper, an improved algorithm based on the MA-ES, which is called the matrix adaptation evolution strategy with multi-objective optimization algorithm, is proposed to solve multimodal optimization problems (MA-ESN-MO). Taking advantage of the multi-objective optimization in maintaining population diversity, MA-ESN-MO transforms an MMOP into a bi-objective optimization problem. The archive is employed to save better solutions for improving the convergence of the algorithm. Moreover, the peaks found by the algorithm can be maintained until the end of the run. Multiple subpopulations are used to explore and exploit in parallel to find multiple optimal solutions for the given problem. Experimental results on CEC2013 test problems show that the covariance matrix adaptation with Niching and the multi-objective optimization algorithm (CMA-NMO), CMA Niching with the Mahalanobis Metric and the multi-objective optimization algorithm (CMA-NMM-MO), and matrix adaptation evolution strategy Niching with the multi-objective optimization algorithm (MA-ESN-MO) have overall better performance compared with the covariance matrix adaptation evolution strategy (CMA-ES), matrix adaptation evolution strategy (MA-ES), CMA Niching (CMA-N), CMA-ES Niching with Mahalanobis Metric (CMA-NMM), and MA-ES-Niching (MA-ESN).

Highlights

Many problems from the real world are classified as optimization problems
In order to overcome the weakness, niching techniques are incorporated into evolutionary algorithms (EAs), such as differential evolution [1,2], particle swarm optimization [3], the covariance matrix adaptation evolution strategy (CMA-ES) [4], self-adaptive niching CMA-ES [4], and genetic algorithm [5], to solve multimodal optimization problems
This paper proposes an improved algorithm based on the MA-ES called the matrix adaptation evolution strategy with multi-objective optimization algorithm (MA-ESN-MO)

Summary

Introduction

Many problems from the real world are classified as optimization problems. Some optimization problems that have one global solution are called single modal optimization problems, while others that have multiple global and local optima are known as multimodal optimization problems. Traditional evolutionary algorithms (EAs) are effective at converging to a single global optimum because of the global selection strategy used. It is inappropriate for EAs to solve multimodal optimization problems. In order to overcome the weakness, niching techniques are incorporated into EAs, such as differential evolution [1,2], particle swarm optimization [3], the covariance matrix adaptation evolution strategy (CMA-ES) [4], self-adaptive niching CMA-ES [4], and genetic algorithm [5], to solve multimodal optimization problems. The representative niching strategies include crowing [6], restricted tournament selection [7], fitness sharing [8], clearing [9], and speciation [5]

Methods

Results

Conclusion