The decomposition-based multi-objective evolutionary algorithm (MOEA/D) does not directly optimize a given multi-objective function $f$, but instead optimizes $N + 1$ single-objective subproblems of $f$ in a co-evolutionary manner. It maintains an archive of all non-dominated solutions found and outputs it as approximation to the Pareto front. Once the MOEA/D found all optima of the subproblems (the $g$-optima), it may still miss Pareto optima of $f$. The algorithm is then tasked to find the remaining Pareto optima directly by mutating the $g$-optima. In this work, we analyze for the first time how the MOEA/D with only standard mutation operators computes the whole Pareto front of the OneMinMax benchmark when the $g$-optima are a strict subset of the Pareto front. For standard bit mutation, we prove an expected runtime of $O(n N \log n + n^{n/(2N)} N \log n)$ function evaluations. Especially for the second, more interesting phase when the algorithm start with all $g$-optima, we prove an $\Omega(n^{(1/2)(n/N + 1)} \sqrt{N} 2^{-n/N})$ expected runtime. This runtime is super-polynomial if $N = o(n)$, since this leaves large gaps between the $g$-optima, which require costly mutations to cover. For power-law mutation with exponent $\beta \in (1, 2)$, we prove an expected runtime of $O\left(n N \log n + n^{\beta} \log n\right)$ function evaluations. The $O\left(n^{\beta} \log n\right)$ term stems from the second phase of starting with all $g$-optima, and it is independent of the number of subproblems $N$. This leads to a huge speedup compared to the lower bound for standard bit mutation. In general, our overall bound for power-law suggests that the MOEA/D performs best for $N = O(n^{\beta - 1})$, resulting in an $O(n^\beta \log n)$ bound. In contrast to standard bit mutation, smaller values of $N$ are better for power-law mutation, as it is capable of easily creating missing solutions.

翻译：基于分解的多目标进化算法(MOEA/D)并非直接优化给定的多目标函数$f$，而是以协同进化方式优化$f$的$N+1$个单目标子问题。该算法维护一个包含所有已发现非支配解的存档，并将其作为帕累托前沿的近似解输出。当MOEA/D找到所有子问题的最优解（$g$-最优解）时，仍可能遗漏$f$的帕累托最优解。此时算法需通过变异$g$-最优解直接寻找剩余帕累托最优解。本研究首次分析了仅使用标准变异算子的MOEA/D在$g$-最优解为帕累托前沿严格子集时，如何计算OneMinMax基准函数的完整帕累托前沿。对于标准位变异，我们证明了期望运行时为$O(n N \log n + n^{n/(2N)} N \log n)$次函数评估。特别针对更具研究价值的第二阶段（当算法初始即拥有所有$g$-最优解时），我们证明了期望运行时为$\Omega(n^{(1/2)(n/N + 1)} \sqrt{N} 2^{-n/N})$。当$N = o(n)$时，该运行时超多项式级，因为$g$-最优解之间的大间隔需通过高代价变异才能覆盖。对于指数为$\beta \in (1, 2)$的幂律变异，我们证明了期望运行时为$O\left(n N \log n + n^{\beta} \log n\right)$次函数评估。其中$O\left(n^{\beta} \log n\right)$项源自初始即拥有所有$g$-最优解的第二阶段，且与子问题数$N$无关。相较于标准位变异的下界，这带来了显著加速。总体而言，我们的幂律变异全局界表明：当$N = O(n^{\beta - 1})$时MOEA/D性能最优，此时运行界为$O(n^\beta \log n)$。与标准位变异不同，较小$N$值更利于幂律变异，因其能轻松生成缺失解。