Maintaining Diversity Provably Helps in Evolutionary Multimodal Optimization

TL;DR

This paper demonstrates that maintaining diversity in the solution space of evolutionary algorithms significantly improves their ability to solve multimodal optimization problems efficiently, with theoretical and experimental validation.

Contribution

It provides the first rigorous analysis showing that diversity maintenance in solution space accelerates convergence in evolutionary algorithms for multimodal problems.

Findings

Diversity consideration can lead to polynomial or exponential speedups.

Theoretical analysis applies to both single-objective and multi-objective problems.

Experimental results support the theoretical claims.

Abstract

In the real world, there exist a class of optimization problems that multiple (local) optimal solutions in the solution space correspond to a single point in the objective space. In this paper, we theoretically show that for such multimodal problems, a simple method that considers the diversity of solutions in the solution space can benefit the search in evolutionary algorithms (EAs). Specifically, we prove that the proposed method, working with crossover, can help enhance the exploration, leading to polynomial or even exponential acceleration on the expected running time. This result is derived by rigorous running time analysis in both single-objective and multi-objective scenarios, including $(\mu+1)$-GA solving the widely studied single-objective problem, Jump, and NSGA-II and SMS-EMOA (two well-established multi-objective EAs) solving the widely studied bi-objective problem, OneJumpZeroJump. Experiments are also conducted to validate the theoretical results. We hope that our results may encourage the exploration of diversity maintenance in the solution space for multi-objective optimization, where existing EAs usually only consider the diversity in the objective space and can easily be trapped in local optima.