Difficulty Adjustable and Scalable Constrained Multi-objective Test Problem Toolkit

In order to better understand the advantages and disadvantages of a constrained multi-objective evolutionary algorithm (CMOEA), it is important to understand the nature of difficulty of a constrained multi-objective optimization problem (CMOP) that the CMOEA is going to deal with. A CMOP includes objectives and constraints, and a number of features, such as the multi-modality and the degeneracy of a Pareto front, have already been extracted to define the difficulty of objectives. However, an explicit and precise definition of difficulty of constraints in CMOPs is still lacking, which makes it difficult to evaluate the performance of CMOEAs tested on them. In this paper, we first propose three primary types of difficulty to characterize the constraints in CMOPs, including feasibility-hardness, convergence-hardness and diversity-hardness. We then develop a general toolkit to construct difficulty adjustable 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 and CMaOPs, whose difficulty can be uniquely defined by a triplet with each of its parameter specifying the level of each primary difficulty type respectively. Based on this toolkit, we suggest fifteen difficulty adjustable CMOPs named DAC-MOP1-15 with different types and levels of difficulty. To study the effectiveness of DAC-MOP115, two popular CMOEAs MOEA/D-CDP and NSGA-II-CDP are adopted to test their performances on them. Furthermore, this toolkit also has the ability to scale the number of objectives. Nine difficulty adjustable constrained many-objective optimization problems (DAC-MaOPs) named DAC-MaOP1-9 with the scalability to the number of objectives are also proposed using this toolkit. Two constrained many-objective evolutionary algorithms (CMaOEAs) CNSGA-III and CMOEA/DD are applied to test their performances on three, five, seven and ten-objective DAC-MaOP1-9 with different difficulty levels and types.