Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Maintaining Diversity Provably Helps in Evolutionary Multimodal Optimization

Shengjie Ren, Zhijia Qiu, Chao Bian, Miqing Li, Chao Qian

TL;DR

This work studies multimodal optimization problems (MMOPs) where multiple solution-space optima map to a single objective, and shows that maintaining diversity in the solution space can boost evolutionary search when used with crossover. The authors introduce a simple diversity-maintenance mechanism that favors distant solutions during population updates for $(\mu+1)$-GA, NSGA-II, and SMS-EMOA, and provide rigorous running-time analyses proving polynomial or exponential speedups on Jump (single-objective) and OneJumpZeroJump (bi-objective), respectively, supported by experiments. For Jump, the bound is $O(\mu^2 4^k+ \mu n\log n+n\sqrt{k}(\mu\log\mu+\log n))$, while for OneJumpZeroJump it is $O(\mu^2 4^{k} + \mu n\log n)$ across NSGA-II and SMS-EMOA; the speedups grow with $k$ and are especially pronounced when $k$ is large. Overall, the paper highlights the practical and theoretical value of preserving solution-space diversity to enhance exploration in MOEAs and MMOPs.

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 $(μ+1)$-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.

Maintaining Diversity Provably Helps in Evolutionary Multimodal Optimization

TL;DR

This work studies multimodal optimization problems (MMOPs) where multiple solution-space optima map to a single objective, and shows that maintaining diversity in the solution space can boost evolutionary search when used with crossover. The authors introduce a simple diversity-maintenance mechanism that favors distant solutions during population updates for -GA, NSGA-II, and SMS-EMOA, and provide rigorous running-time analyses proving polynomial or exponential speedups on Jump (single-objective) and OneJumpZeroJump (bi-objective), respectively, supported by experiments. For Jump, the bound is , while for OneJumpZeroJump it is across NSGA-II and SMS-EMOA; the speedups grow with and are especially pronounced when is large. Overall, the paper highlights the practical and theoretical value of preserving solution-space diversity to enhance exploration in MOEAs and MMOPs.

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 -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.
Paper Structure (10 sections, 4 theorems, 10 equations, 2 figures, 2 algorithms)

This paper contains 10 sections, 4 theorems, 10 equations, 2 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Multimodal Optimization
  4. Evolutionary Algorithms
  5. Proposed Diversity Maintenance Method
  6. $(\mu+1)$-GA on Jump
  7. NSGA-II on OneJumpZeroJump
  8. SMS-EMOA on OneJumpZeroJump
  9. Experiments
  10. Conclusion

Key Result

Theorem 1

For $(\mu+1)$-GA solving Jump with $k\le n/4$, if using the diversity maintenance method, and a population size $\mu$ such that $\mu\ge 2$, then the expected number of fitness evaluations for finding the global optimal solution $1^n$ is $O(\mu^2 4^k+\mu n\log n + n\sqrt{k}(\mu\log\mu+\log n))$.

Figures (2)

  • Figure 1: The function value of the Jump and OneJumpZeroJump problems vs. the number of 1-bits of a solution when $n=20$ and $k=5$. Left subfigure: Jump; right subfigure: OneJumpZeroJump.
  • Figure 2: Average number of fitness evaluations of $(\mu+1)$-GA for solving Jump, and NSGA-II/SMS-EMOA for solving OneJumpZeroJump, when the diversity maintenance method is used or not.

Theorems & Definitions (11)

  • Definition 1: Droste02
  • Definition 2
  • Definition 3: doerr2021ojzj
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • ...and 1 more