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.

翻译：基于分解的多目标进化算法（MOEA/D）将多目标优化问题转化为一组单目标子问题以进行协同优化。子问题与解之间的不匹配可能导致MOEA/D性能严重下降。现有的大多数不匹配应对策略仅适用于使用$L_{\infty}$标量化的情况。一种能够使用任意$L_{p}$标量化（即使面对具有非凸帕累托前沿的多目标优化问题）的不匹配应对策略，对MOEA/D具有重要意义。本文以全局替换（GR）为核心方法。我们分析了当$L_{\infty}$被另一种$L_{p}$（$p\in [1,\infty)$）取代时，GR为何无法再避免不匹配，并发现基于$L_p$（$1\leq p<\infty$）的子问题存在不一致的大偏好区域。当$p$设为较小值时，部分中间子问题的偏好区域非常小，导致其方向向量无法穿过对应的偏好区域。因此，我们提出了一种广义$L_p$（G$L_p$）标量化方法，以确保子问题的方向向量能穿过其偏好区域。理论分析表明，当使用G$L_p$标量化时，对于任意$p\geq 1$，GR总能避免不匹配。针对各类多目标优化问题的实验研究验证了理论分析结果。