Improved Runtime Guarantees for the SPEA2 Multi-Objective Optimizer

TL;DR

This paper analyzes SPEA2, a domination-based MOEA, and shows its runtime bounds can be surprisingly independent of population size over broad parameter ranges. By dissecting SPEA2’s σ-criterion-driven selection, the authors establish balanced preservation of objective values and multiplicative growth of value multiplicities, enabling tighter runtime guarantees on three classic benchmarks: OneMinMax, OneJumpZeroJump, and LeadingOnes-TrailingZeros. They derive concrete bounds: for OneMinMax, $O((\mu+\lambda)n + n^2 \log n)$ function evaluations (when $\mu \ge n+1$); for LOTZ, $O((\mu+\lambda)n \log(\mu/(n+1)) + n^3 + \lambda n)$; and for OJZJ$_k$, $O((\mu+\lambda)n + n^{k+1})$ (when $\mu \ge n-2k+3$). Collectively, these results imply SPEA2 is more robust to parameter settings than NSGA-II, offering practical guidance that effective performance can be achieved with a wide range of $\mu$ and $\lambda$, particularly when scaled as $O(n^k)$.

Abstract

Together with the NSGA-II, the SPEA2 is one of the most widely used domination-based multi-objective evolutionary algorithms. For both algorithms, the known runtime guarantees are linear in the population size; for the NSGA-II, matching lower bounds exist. With a careful study of the more complex selection mechanism of the SPEA2, we show that it has very different population dynamics. From these, we prove runtime guarantees for the OneMinMax, LeadingOnesTrailingZeros, and OneJumpZeroJump benchmarks that depend less on the population size. For example, we show that the SPEA2 with parent population size $μ\ge n - 2k + 3$ and offspring population size $λ$ computes the Pareto front of the OneJumpZeroJump benchmark with gap size $k$ in an expected number of $O( (λ+μ)n + n^{k+1})$ function evaluations. This shows that the best runtime guarantee of $O(n^{k+1})$ is not only achieved for $μ= Θ(n)$ and $λ= O(n)$ but for arbitrary $μ, λ= O(n^k)$. Thus, choosing suitable parameters -- a key challenge in using heuristic algorithms -- is much easier for the SPEA2 than the NSGA-II.

Theorems & Definitions (26)

• Lemma 1
• proof
• Lemma 2
• Theorem 3
• proof : Proof of \ref{['lem:mult-growth']}
• Lemma 4
• proof
• Theorem 5
• Lemma 6
• proof