A First Running Time Analysis of the Strength Pareto Evolutionary Algorithm 2 (SPEA2)

TL;DR

This paper provides the first theoretical running time analysis of SPEA2, a practical multi-objective evolutionary algorithm, on three benchmark problems, establishing bounds and conditions for its expected performance.

Contribution

It introduces the first running time bounds for SPEA2 on common multi-objective problems, using general theorems applicable to other MOEAs, advancing theoretical understanding.

Findings

Expected running time bounds for SPEA2 on three problems.

Population size requirements for theoretical guarantees.

General theorems applicable to other MOEAs.

Abstract

Evolutionary algorithms (EAs) have emerged as a predominant approach for addressing multi-objective optimization problems. However, the theoretical foundation of multi-objective EAs (MOEAs), particularly the fundamental aspects like running time analysis, remains largely underexplored. Existing theoretical studies mainly focus on basic MOEAs, with little attention given to practical MOEAs. In this paper, we present a running time analysis of strength Pareto evolutionary algorithm 2 (SPEA2) for the first time. Specifically, we prove that the expected running time of SPEA2 for solving three commonly used multi-objective problems, i.e., $m$OneMinMax, $m$LeadingOnesTrailingZeroes, and $m$-OneJumpZeroJump, is $O(\mu n\cdot \min\{m\log n, n\})$, $O(\mu n^2)$, and $O(\mu n^k \cdot \min\{mn, 3^{m/2}\})$, respectively. Here $m$ denotes the number of objectives, and the population size $\mu$ is required to be at least $(2n/m+1)^{m/2}$, $(2n/m+1)^{m-1}$ and $(2n/m-2k+3)^{m/2}$, respectively. The proofs are accomplished through general theorems which are also applicable for analyzing the expected running time of other MOEAs on these problems, and thus can be helpful for future theoretical analysis of MOEAs.