A Generalized Scalarization Method for Evolutionary Multi-Objective Optimization

TL;DR

This paper introduces a generalized scalarization method for MOEA/D that ensures better alignment between subproblems and solutions across different scalarization types, improving performance on complex multi-objective problems.

Contribution

It proposes a G$L_p$ scalarization that guarantees avoidance of mismatches in MOEA/D for any $L_p$ scalarization, extending applicability to non-convex Pareto fronts.

Findings

G$L_p$ scalarization ensures no mismatches with global replacement.

Theoretical proof of mismatch avoidance for all $p \\geq 1$.

Experimental results confirm the theoretical advantages.

Abstract

The decomposition-based multi-objective evolutionary algorithm (MOEA/D) transforms a multi-objective optimization problem (MOP) into a set of single-objective subproblems for collaborative optimization. Mismatches between subproblems and solutions can lead to severe performance degradation of MOEA/D. Most existing mismatch coping strategies only work when the $L_{\infty}$ scalarization is used. A mismatch coping strategy that can use any $L_{p}$ scalarization, even when facing MOPs with non-convex Pareto fronts, is of great significance for MOEA/D. This paper uses the global replacement (GR) as the backbone. We analyze how GR can no longer avoid mismatches when $L_{\infty}$ is replaced by another $L_{p}$ with $p\in [1,\infty)$, and find that the $L_p$-based ($1\leq p<\infty$) subproblems having inconsistently large preference regions. When $p$ is set to a small value, some middle subproblems have very small preference regions so that their direction vectors cannot pass through their corresponding preference regions. Therefore, we propose a generalized $L_p$ (G$L_p$) scalarization to ensure that the subproblem's direction vector passes through its preference region. Our theoretical analysis shows that GR can always avoid mismatches when using the G$L_p$ scalarization for any $p\geq 1$. The experimental studies on various MOPs conform to the theoretical analysis.