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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. [ABSTRACT FROM AUTHOR]

This paper proposes an improved epsilon constraint-handling mechanism and combines it with a decomposition-based multi-objective evolutionary algorithm (MOEA/D) to solve constrained multi-objective optimization problems (CMOPs). The proposed constrained multi-objective evolutionary algorithm (CMOEA) is named MOEA/D-IEpsilon. It adjusts the epsilon level dynamically according to the ratio of feasible to total solutions in the current population. In order to evaluate the performance of MOEA/D-IEpsilon, a new set of CMOPs with two and three objectives is designed, having large infeasible regions (relative to the feasible regions), and they are called LIR-CMOPs. Then, the 14 benchmarks, including LIR-CMOP1-14, are used to test MOEA/D-IEpsilon and four other decomposition-based CMOEAs, including MOEA/D-Epsilon, MOEA/D-SR, MOEA/D-CDP and CMOEA/D. The experimental results indicate that MOEA/D-IEpsilon is significantly better than the other four CMOEAs on all of the test instances, which shows that MOEA/D-IEpsilon is more suitable for solving CMOPs with large infeasible regions. Furthermore, a real-world problem, namely the robot gripper optimization problem, is used to test the five CMOEAs. The experimental results demonstrate that MOEA/D-IEpsilon also outperforms the other four CMOEAs on this problem. [ABSTRACT FROM AUTHOR]

In dealing with constrained multi-objective optimization problems (CMOPs), a key issue of multi-objective evolutionary algorithms (MOEAs) is to balance the convergence and diversity of working populations. However, most state-of-the-art MOEAs show poor performance in balancing them, and can cause the working populations to concentrate on part of the Pareto fronts, leading to serious imbalanced searching between preserving diversity and achieving convergence. This paper proposes a method which combines a multi-objective to multi-objective (M2M) decomposition approach with the push and pull search (PPS) framework, namely PPS-M2M. To be more specific, the proposed algorithm decomposes a CMOP into a set of simple CMOPs. Each simple CMOP corresponds to a sub-population and is solved in a collaborative manner. When dealing with constraints, each sub-population follows a procedure of "ignore the constraints in the push stage and consider the constraints in the pull stage", which helps each working sub-population get across infeasible regions. In order to evaluate the performance of the proposed PPS-M2M, it is compared with the other nine algorithms, including CM2M, MOEA/D-Epsilon, MOEA/D-SR, MOEA/D-CDP, C-MOEA/D, NSGA-II-CDP, MODE-ECHM, CM2M2 and MODE-SaE on a set of benchmark CMOPs. The experimental results show that the proposed PPS-M2M is significantly better than the other nine algorithms. In addition, a set of constrained and imbalanced multi-objective optimization problems (CIMOPs) are suggested to compare PPS-M2M and PPS-MOEA/D. The experimental results show that the proposed PPS-M2M outperforms PPS-MOEA/D on CIMOPs. [ABSTRACT FROM AUTHOR]

Designing and setting manipulator trajectories in a programming system can be a tedious and time-consuming task for manufacturers. In this paper, one kind of six degree-of-freedom (DOF) teaching manipulator is designed and developed for conveniently setting and recording trajectories for industrial robots. A constrained multi-objective optimization problem is formulated to optimize the design of the teaching manipulator. Two performance indexes, i.e. the magnitude of the peak operating force and difference between the maximum and minimum magnitude of operating forces are adopted as the objectives. Two PPS-based (push and pull search) algorithms, including PPS-MOEA/D and PPS-M2M, are suggested to solve the formulated CMOP. Several state-of-the-art CMOEAs, including MOEA/D-ACDP, MOEA/D-CDP, NSGA-II-CDP and CM2M, are also tested. The experimental results indicate that PPS-MOEA/D has the best performance among the six compared algorithms, and the PPS-based methods as a group outperform their counterparts without adopting the PPS framework, which demonstrates the superiority of the PPS framework for solving real-world optimization problems. [ABSTRACT FROM AUTHOR]