A Theoretical Perspective on Why Stochastic Population Update Needs an Archive in Evolutionary Multi-objective Optimization

TL;DR

This paper provides a theoretical analysis showing that incorporating an archive in stochastic population update methods can significantly improve the efficiency of multi-objective evolutionary algorithms, especially with small populations.

Contribution

It introduces a theoretical framework demonstrating how an archive enhances stochastic population updates in MOEAs, reducing expected runtime and improving search performance.

Findings

Using an archive can exponentially reduce expected runtime.

Small populations with an archive can outperform larger populations without one.

The $(+)$ update mode is more suitable for SPU than the $(+1)$ mode.

Abstract

Evolutionary algorithms (EAs) have been widely applied to multi-objective optimization due to their population-based nature. Population update, a key component in multi-objective EAs (MOEAs), is usually performed in a greedy, deterministic manner. However, recent studies have questioned this practice and shown that stochastic population update (SPU), which allows inferior solutions have a chance to be preserved, can help MOEAs jump out of local optima more easily. Nevertheless, SPU risks losing high-quality solutions, potentially requiring a large population. Intuitively, a possible solution to this issue is to introduce an archive that stores the best solutions ever found. In this paper, we theoretically show that using an archive allows a small population and may enhance the search performance of SPU-based MOEAs. We examine two classic algorithms, SMS-EMOA and NSGA-II, on the bi-objective problem OneJumpZeroJump, and prove that using an archive can reduce the expected running time upper bound (even exponentially). The comparison between SMS-EMOA and NSGA-II also suggests that the $(\mu+\mu)$ update mode may be more suitable for SPU than the $(\mu+1)$ update mode. We also validate our findings empirically. We hope this work may provide theoretical support to explore different ideas of designing algorithms in evolutionary multi-objective optimization.