A First Running Time Analysis of the Strength Pareto Evolutionary Algorithm 2 (SPEA2)
Shengjie Ren, Chao Bian, Miqing Li, Chao Qian
TL;DR
The paper addresses the theoretical running-time analysis of multi-objective evolutionary algorithms by deriving general bounds for SPEA2 on three classic benchmark problems: $m$OneMinMax, $m$LeadingOnesTrailingZeroes, and $m$OneJumpZeroJump. It introduces general theorems that hinge on preserving the non-dominated set and on archive-size constraints, then applies them to SPEA2 to obtain explicit expected fitness-evaluation bounds: $O(\bar{\mu} n\cdot \min\{m\log n, n\})$, $O(\bar{\mu} n^2)$, and $O(\bar{\mu} n^k \cdot \min\{mn, 3^{m/2}\})$ for the three problems, respectively, given suitable $\bar{\mu}$. The theorems are framed to accommodate other MOEAs, offering a unified analytical approach and aligning with existing results in the literature. The work thus provides a rigorous, general toolset for running-time analysis of MOEAs and highlights the role of archive design in shaping theoretical guarantees.
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 $O(μn\cdot \min\{m\log n, n\})$, $O(μn^2)$, and $O(μn^k \cdot \min\{mn, 3^{m/2}\})$, respectively. Here $m$ denotes the number of objectives, and the population size $μ$ 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.