Difficulty Adjustable and Scalable Constrained Multi-objective Test Problem Toolkit

Multi-objective evolutionary algorithms (MOEAs) have achieved great progress in recent decades, but most of them are designed to solve unconstrained multi-objective optimization problems. In fact, many real-world multi-objective problems usually contain a number of constraints. To promote the research of constrained multi-objective optimization, we first propose three primary types of difficulty, which reflect the challenges in the real-world optimization problems, to characterize the constraint functions in CMOPs, including feasibility-hardness, convergence-hardness and diversity-hardness. We then develop a general toolkit to construct difficulty adjustable and scalable constrained multi-objective optimization problems (CMOPs) with three types of parameterized constraint functions according to the proposed three primary types of difficulty. In fact, combination of the three primary constraint functions with different parameters can lead to construct a large variety of CMOPs, whose difficulty can be uniquely defined by a triplet with each of its parameter specifying the level of each primary difficulty type respectively. Furthermore, the number of objectives in this toolkit are able to scale to more than two. Based on this toolkit, we suggest nine difficulty adjustable and scalable CMOPs named DAS-CMOP1-9. To evaluate the proposed test problems, two popular CMOEAs - MOEA/D-CDP and NSGA-II-CDP are adopted to test their performances on DAS-CMOP1-9 with different difficulty triplets. The experiment results demonstrate that none of them can solve these problems efficiently, which stimulate us to develop new constrained MOEAs to solve the suggested DAS-CMOPs.