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Not Just for Archiving: Provable Benefits of Reusing the Archive in Evolutionary Multi-objective Optimization

Shengjie Ren, Zimin Liang, Miqing Li, Chao Qian

TL;DR

This work analyzes the use of archives in SMS-EMOA for multi-objective optimization, showing that reusing archived non-dominated solutions yields provable, polynomial-time speedups on OneJumpZeroJump problems and their variant, beyond the benefits of archiving alone. The authors prove that archive reuse enhances Pareto front coverage and exploration, enabling faster convergence with small populations and competing effectively with large-population approaches. They validate the theory with experiments on both synthetic and real-world problems (knapsack, TSP, QAP, NK landscapes), demonstrating improved hypervolume performance. The results inform practical MOEA design by advocating archive reuse as a concrete, beneficial mechanism beyond mere storage of good solutions.

Abstract

Evolutionary Algorithms (EAs) have become the most popular tool for solving widely-existed multi-objective optimization problems. In Multi-Objective EAs (MOEAs), there is increasing interest in using an archive to store non-dominated solutions generated during the search. This approach can 1) mitigate the effects of population oscillation, a common issue in many MOEAs, and 2) allow for the use of smaller, more practical population sizes. In this paper, we analytically show that the archive can even further help MOEAs through reusing its solutions during the process of new solution generation. We first prove that using a small population size alongside an archive (without incorporating archived solutions in the generation process) may fail on certain problems, as the population may remove previously discovered but promising solutions. We then prove that reusing archive solutions can overcome this limitation, resulting in at least a polynomial speedup on the expected running time. Our analysis focuses on the well-established SMS-EMOA algorithm applied to the commonly studied OneJumpZeroJump problem as well as one of its variants. We also show that reusing archive solutions can be better than using a large population size directly. Finally, we show that our theoretical findings can generally hold in practice by experiments on well-known practical optimization problems -- multi-objective 0-1 Knapsack, TSP, QAP and NK-landscape problems -- with realistic settings.

Not Just for Archiving: Provable Benefits of Reusing the Archive in Evolutionary Multi-objective Optimization

TL;DR

This work analyzes the use of archives in SMS-EMOA for multi-objective optimization, showing that reusing archived non-dominated solutions yields provable, polynomial-time speedups on OneJumpZeroJump problems and their variant, beyond the benefits of archiving alone. The authors prove that archive reuse enhances Pareto front coverage and exploration, enabling faster convergence with small populations and competing effectively with large-population approaches. They validate the theory with experiments on both synthetic and real-world problems (knapsack, TSP, QAP, NK landscapes), demonstrating improved hypervolume performance. The results inform practical MOEA design by advocating archive reuse as a concrete, beneficial mechanism beyond mere storage of good solutions.

Abstract

Evolutionary Algorithms (EAs) have become the most popular tool for solving widely-existed multi-objective optimization problems. In Multi-Objective EAs (MOEAs), there is increasing interest in using an archive to store non-dominated solutions generated during the search. This approach can 1) mitigate the effects of population oscillation, a common issue in many MOEAs, and 2) allow for the use of smaller, more practical population sizes. In this paper, we analytically show that the archive can even further help MOEAs through reusing its solutions during the process of new solution generation. We first prove that using a small population size alongside an archive (without incorporating archived solutions in the generation process) may fail on certain problems, as the population may remove previously discovered but promising solutions. We then prove that reusing archive solutions can overcome this limitation, resulting in at least a polynomial speedup on the expected running time. Our analysis focuses on the well-established SMS-EMOA algorithm applied to the commonly studied OneJumpZeroJump problem as well as one of its variants. We also show that reusing archive solutions can be better than using a large population size directly. Finally, we show that our theoretical findings can generally hold in practice by experiments on well-known practical optimization problems -- multi-objective 0-1 Knapsack, TSP, QAP and NK-landscape problems -- with realistic settings.

Paper Structure

This paper contains 14 sections, 11 theorems, 24 equations, 2 figures, 4 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Multi-objective Optimization
  4. SMS-EMOA and Archiving Mechianism
  5. Reusing the Archive
  6. Benchmark Problems
  7. Provable Benefits of Reusing the Archive
  8. Archive Reuse Enhances Pareto Front Coverage
  9. Archive Reuse Helps Exploration
  10. Reusing Archive vs. Using A Large Population
  11. Experiments
  12. Conclusion
  13. Proof of Theorem \ref{['thm:sms-arc-omm']} and Theorem \ref{['thm:sms-arc-lotz']}
  14. OneMinMax and LeadingOnesTrailingZeroes

Key Result

Lemma 1

For a positive integer $n$, we have Stirling's formula as $\sqrt{2\pi n}(n/e)^n \le n! \le e^{1/12}\sqrt{2\pi n}(n/e)^n$

Figures (2)

  • Figure 1: Illustration of the OneJumpZeroJump$_{SS}$ problem when $n=20$, $k=5$ and $a=2$. The left subfigure: the function values $f_1$ and $f_2$ vs. the number of 1-bits of a solution; the right subfigure: $f_2$ vs. $f_1$.
  • Figure 2: Non-dominated solutions obtained by SMS-EMOA-A (archive only to store non-dominated solutions), SMS-EMOA-AR (archive with reuse), and SMS-EMOA-L (a large population size) on the multi-objective 0/1 Knapsack (KP), NK-Landscape ,(NK), TSP and QAP problems, with 100 (top panel), 200 (middle panel), and 500 (bottom panel) decision variables respectively. The Knapsack and NK-Landscape are maximization problems, and the TSP and QAP are minimization problems.

Theorems & Definitions (24)

  • Definition 1: Multi-objective Optimization
  • Definition 2
  • Definition 3: Doerr and Zheng doerr2021ojzj
  • Definition 4
  • Lemma 1: Robbins robbins1955remark
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • ...and 14 more