  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.
