Proven Approximation Guarantees in Multi-Objective Optimization: SPEA2 Beats NSGA-II

TL;DR

This paper provides the first mathematical proof that SPEA2 can efficiently compute optimal Pareto front approximations in polynomial time, outperforming NSGA-II in multi-objective optimization.

Contribution

It introduces a formal proof demonstrating SPEA2's superior approximation guarantees compared to NSGA-II, specifically for the OneMinMax benchmark.

Findings

SPEA2 can compute optimal Pareto front approximations in polynomial time.

NSGA-II's approximation guarantees are roughly a factor of two.

Experiments support the theoretical advantage of SPEA2 over NSGA-II.

Abstract

Together with the NSGA-II and SMS-EMOA, the strength Pareto evolutionary algorithm 2 (SPEA2) is one of the most prominent dominance-based multi-objective evolutionary algorithms (MOEAs). Different from the NSGA-II, it does not employ the crowding distance (essentially the distance to neighboring solutions) to compare pairwise non-dominating solutions but a complex system of $\sigma$-distances that builds on the distances to all other solutions. In this work, we give a first mathematical proof showing that this more complex system of distances can be superior. More specifically, we prove that a simple steady-state SPEA2 can compute optimal approximations of the Pareto front of the OneMinMax benchmark in polynomial time. The best proven guarantee for a comparable variant of the NSGA-II only assures approximation ratios of roughly a factor of two, and both mathematical analyses and experiments indicate that optimal approximations are not found efficiently.