Multiobjective evolutionary algorithms (MOEAs) have progressed significantly in recent decades, but most of them are designed to solve unconstrained multiobjective optimization problems. In fact, many real-world multiobjective problems contain a number of constraints. To promote research on constrained multiobjective optimization, we first propose a problem classification scheme with three primary types of difficulty, which reflect various types of challenges presented by real-world optimization problems, in order to characterize the constraint functions in constrained multiobjective optimization problems (CMOPs). These are feasibility-hardness, convergence-hardness, and diversity-hardness. We then develop a general toolkit to construct difficulty adjustable and scalable CMOPs (DAS-CMOPs, or DAS-CMaOPs when the number of objectives is greater than three) with three types of parameterized constraint functions developed to capture the three proposed types of difficulty. In fact, the combination of the three primary constraint functions with different parameters allows the construction of a large variety of CMOPs, with difficulty that can be defined by a triplet, with each of its parameters specifying the level of one of the types of primary difficulty. Furthermore, the number of objectives in this toolkit can be scaled beyond three. Based on this toolkit, we suggest nine difficulty adjustable and scalable CMOPs and nine CMaOPs, to be called DAS-CMOP1-9 and DAS-CMaOP1-9, respectively. To evaluate the proposed test problems, two popular CMOEAs—MOEA/D-CDP (MOEA/D with constraint dominance principle) and NSGA-II-CDP (NSGA-II with constraint dominance principle) and two popular constrained many-objective evolutionary algorithms (CMaOEAs)—C-MOEA/DD and C-NSGA-III—are used to compare performance on DAS-CMOP1-9 and DAS-CMaOP1-9 with a variety of difficulty triplets, respectively. The experimental results reveal that mechanisms in MOEA/D-CDP may be more effective in solving convergence-hard DAS-CMOPs, while mechanisms of NSGA-II-CDP may be more effective in solving DAS-CMOPs with simultaneous diversity-, feasibility-, and convergence-hardness. Mechanisms in C-NSGA-III may be more effective in solving feasibility-hard CMaOPs, while mechanisms of C-MOEA/DD may be more effective in solving CMaOPs with convergence-hardness. In addition, none of them can solve these problems efficiently, which stimulates us to continue to develop new CMOEAs and CMaOEAs to solve the suggested DAS-CMOPs and DAS-CMaOPs.

中文翻译：

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难度可调整和可扩展的约束多目标测试问题工具包

近几十年来，多目标进化算法 (MOEA) 取得了显着进展，但其中大多数旨在解决无约束的多目标优化问题。事实上，许多现实世界的多目标问题包含许多约束。为了促进约束多目标优化的研究，我们首先提出了一个具有三种主要难度类型的问题分类方案，反映了现实世界优化问题提出的各种类型的挑战，以表征约束多目标优化问题（CMOPs）中的约束函数）。它们是可行性-硬度、收敛-硬度和多样性-硬度。然后我们开发了一个通用工具包来构建难度可调和可扩展的 CMOP（DAS-CMOP，或 DAS-CMaOP，当目标数量大于三个时），开发了三种类型的参数化约束函数来捕获三种建议的难度类型。事实上，三个具有不同参数的主要约束函数的组合允许构建种类繁多的 CMOP，其难度可以由三元组定义，其每个参数指定一种主要难度类型的级别. 此外，该工具包中的目标数量可以扩展到三个以上。基于这个工具包，我们建议九个难度可调和可扩展的 CMOP 和九个 CMaOP，分别称为 DAS-CMOP1-9 和 DAS-CMaOP1-9。为了评估建议的测试问题，两种流行的 CMOEA——MOEA/D-CDP（MOEA/D with constraint Dominance Principle）和 NSGA-II-CDP（NSGA-II with Constraint Dominance Principle）和两种流行的约束多目标进化算法（CMaOEAs）——C-MOEA/ DD 和 C-NSGA-III——分别用于比较 DAS-CMOP1-9 和 DAS-CMaOP1-9 与各种难度三元组的性能。实验结果表明，MOEA/D-CDP 机制可能更有效地解决收敛困难的 DAS-CMOPs，而 NSGA-II-CDP 机制可能更有效地解决同时具有多样性-、可行性-、和收敛性。C-NSGA-III 中的机制可能更有效地解决可行性困难的 CMaOPs，而 C-MOEA/DD 的机制可能更有效地解决具有收敛性的 CMaOPs。此外，