A Generalized Scalarization Method for Evolutionary Multi-objective Optimization
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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_∞ 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_∞ is replaced by another L_p with p∈ [1,∞), and find that the L_p-based (1≤ p<∞) 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 (GL_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 GL_p scalarization for any p≥ 1. The experimental studies on various MOPs conform to the theoretical analysis.
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