基于分解的多目标进化算法（MOEA/D）并不直接优化给定的多目标函数$f$，而是以协同进化的方式优化$f$的$N + 1$个单目标子问题。它维护一个包含所有已找到的非支配解的存档，并将其作为对Pareto前沿的近似输出。一旦MOEA/D找到了所有子问题的最优解（即$g$-最优解），它仍可能遗漏$f$的Pareto最优解。此时，算法被要求通过变异$g$-最优解来直接寻找剩余的Pareto最优解。\n\n在本工作中，我们首次分析了当$g$-最优解是Pareto前沿的严格子集时，仅使用标准变异算子的MOEA/D如何计算OneMinMax基准问题的整个Pareto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。特别是对于第二个更有趣的阶段，即算法从所有$g$-最优解开始时，我们证明了$\Omega(n^{(1/2)(n/N + 1)} \sqrt{N} 2^{-n/N})$的期望运行时。如果$N = o(n)$，该运行时是超多项式的，因为这会在$g$-最优解之间留下较大的间隙，需要代价高昂的变异来覆盖。\n\n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。$O\left(n^\beta \log n\right)$项来自从所有$g$-最优解开始的第二阶段，并且它与子问题数量$N$无关。与标准位变异的下界相比，这带来了巨大的加速。总体而言，我们对幂律变异的整体上界表明，MOEA/D在$N = O(n^{\beta - 1})$时表现最佳，从而得到$O(n^\beta \log n)$的上界。与标准位变异相反，对于幂律变异而言，较小的$N$值更好，因为它能够轻松地创建缺失的解。
英文摘要：
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 $Ω(n^{(1/2)(n/N + 1)} \sqrt{N} 2^{-n/N})$ 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 $β\in (1, 2)$, we prove an expected runtime of $O\left(n N \log n + n^β \log n\right)$ function evaluations. The $O\left(n^β \log n\right)$ 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 $N = O(n^{β- 1})$, resulting in an $O(n^β\log n)$ 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.
Proven Advantage of Multiobjective Evolutionary Algorithms for Problems with Different Degrees of Conflict
School of Computer Science and Technology, International Research Institute for Artificial Intelligence, Harbin Institute of Technology, Shenzhen, China（哈尔滨工业大学深圳，计算机科学与技术学院，人工智能国际研究院）
First Mathematical Runtime Analyses of Multi-Objective Evolutionary Algorithms for Multi-Valued Decision Variables
School of Computer Science and Technology, National Key Laboratory of Smart Farm Technologies and Systems, International Research Institute for Artificial Intelligence（计算机科学与技术学院，智慧农业技术与系统国家重点实验室，人工智能国际研究院）；Pengcheng Laboratory, Shenzhen, China（鹏城实验室）；Harbin Institute of Technology, Shenzhen（哈尔滨工业大学（深圳））；Institut Polytechnique de Paris, Palaiseau, France（巴黎理工学院）；Laboratoire d’Informatique (LIX), CNRS, École Polytechnique（计算机实验室（LIX），法国国家科学研究中心，巴黎综合理工学院）
Mathematical Runtime Analysis for the Non-Dominated Sorting Genetic Algorithm II (NSGA-II)
Southern University of Science and Technology（南方科技大学）；International Research Institute for Artificial Intelligence（国际人工智能研究院）；School of Computer Science and Technology（计算机科学与技术学院）；Harbin Institute of Technology（哈尔滨工业大学）；Laboratoire d’Informatique (LIX)（计算机科学实验室（LIX））；École Polytechnique, CNRS（法国国家科学研究中心、巴黎综合理工学院）；Institute Polytechnique de Paris（巴黎理工学院）
Neural Multi-Objective Combinatorial Optimization with Diversity Enhancement
Sun Yat-sen University（中山大学）；Singapore Management University（新加坡管理大学）；Eindhoven University of Technology（埃因霍温理工大学）；National University of Singapore（新加坡国立大学）
Theoretical Analyses of Multiobjective Evolutionary Algorithms on Multimodal Objectives
School of Computer Science and Technology（计算机科学与技术学院）；International Research Institute for Artificial Intelligence（国际人工智能研究院）；Harbin Institute of Technology（哈尔滨工业大学）；Southern University of Science and Technology（南方科技大学）；Laboratoire d’Informatique (LIX)（信息学实验室）；École Polytechnique（巴黎综合理工学院）；CNRS（法国国家科学研究中心）；Institut Polytechnique de Paris（巴黎理工学院）
