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Multi-objective evolutionary algorithms (MOEAs) have progressed significantlyin recent decades, but most of them are designed to solve unconstrainedmulti-objective optimization problems. In fact, many real-world multi-objectiveproblems contain a number of constraints. To promote research on constrainedmulti-objective optimization, we first propose a problem classification schemewith three primary types of difficulty, which reflect various types ofchallenges presented by real-world optimization problems, in order tocharacterize the constraint functions in constrained multi-objectiveoptimization problems (CMOPs). These are feasibility-hardness,convergence-hardness and diversity-hardness. We then develop a general toolkitto construct difficulty-adjustable and scalable CMOPs (DAS-CMOPs, or DAS-CMaOPswhen the number of objectives is greater than three) with three types ofparameterized constraint functions developed to capture the three proposedtypes of difficulty. Based on this toolkit, we suggest ninedifficulty-adjustable and scalable CMOPs and nine CMaOPs. The experimentalresults reveal that mechanisms in MOEA/D-CDP may be more effective in solvingconvergence-hard DAS-CMOPs, while mechanisms of NSGA-II-CDP may be moreeffective in solving DAS-CMOPs with simultaneous diversity-, feasibility- andconvergence-hardness. Mechanisms in C-NSGA-III may be more effective in solvingfeasibility-hard CMaOPs, while mechanisms of C-MOEA/DD may be more effective insolving CMaOPs with convergence-hardness. In addition, none of them can solvethese problems efficiently, which stimulates us to continue to develop newCMOEAs and CMaOEAs to solve the suggested DAS-CMOPs and DAS-CMaOPs.

Difficulty Adjustable and Scalable Constrained Multi-objective Test Problem Toolkit