This paper proposes a push and pull search method in the framework of differential evolution (PPS-DE) to solve constrained singleobjective optimization problems (CSOPs). More specifically, two sub-populations, including the top and bottom sub-populations, are collaborated with each other to search global optimal solutions efficiently. The top sub-population adopts the pull and pull search (PPS) mechanism to deal with constraints, while the bottom sub-population use the superiority of feasible solutions (SF) technique to deal with constraints. In the top sub-population, the search process is divided into two different stages -push and pull stages. In the push stage, a CSOP is optimized without considering any constraints, which can help to get across to infeasible regions. In the pull stage, the CSOP is optimized with an improved epsilon constraint-handling method, which can help the population to search for feasible solutions. An adaptive DE variant with three trial vector generation strategies -DE /rand/1, DE/current-to-rand/1, and DE/current-to-pbest/1 is employed in the proposed PPS-DE. In the top sub-population, all the three trial vector generation strategies are used to generate offsprings, just like in CoDE. In the bottom sub-population, a strategy adaptation, in which the trial vector generation strategies are periodically self-adapted by learning from their experiences in generating promising solutions in the top sub-population, is used to choose a suitable trial vector generation strategy to generate one offspring. Furthermore, a parameter adaptation strategy from LSHADE44 is employed in both sup-populations to generate scale factor F and crossover rate CR for each trial vector generation strategy. Twenty-eight CSOPs with 10-, 30-, and 50-dimensional decision variables provided in the CEC2018 competition on real parameter single objective optimization are optimized by the proposed PPS-DE. The experimental results demonstrate that the proposed PPS-DE has the best performance compared with the other seven state-of-the-art algorithms, including AGA-PPS, LSHADE44, LSHADE44+IDE, UDE, IUDE, ǫMAg-ES and C 2 oDE.

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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 (RFS) 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 fourteen 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 C-MOEA/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.

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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.