A Survey of Multiobjective Evolutionary Algorithms Based on Decomposition: Variants, Challenges and Future Directions <s> 1) PENALTY FUNCTIONS <s> This paper modifies the replacement and update scheme in MOEA/D-DE developed in [1] for dealing with constraints in multiobjective optimization problems. The modified scheme introduces a penalty function to penalize infeasible solutions. The penalty function uses a threshold to control the amount of penalty to infeasible solutions. Experimental results have shown that this penalty method is very promising. <s> BIB001 </s> A Survey of Multiobjective Evolutionary Algorithms Based on Decomposition: Variants, Challenges and Future Directions <s> 1) PENALTY FUNCTIONS <s> This paper proposes a push and pull search (PPS) framework for solving constrained multi-objective optimization problems (CMOPs). To be more specific, the proposed PPS divides the search process into two different stages, including the push and pull search stages. In the push stage, a multi-objective evolutionary algorithm (MOEA) is adopted to explore the search space without considering any constraints, which can help to get across infeasible regions very fast and approach the unconstrained Pareto front. Furthermore, the landscape of CMOPs with constraints can be probed and estimated in the push stage, which can be utilized to conduct the parameters setting for constraint-handling approaches applied in the pull stage. Then, a constrained multi-objective evolutionary algorithm (CMOEA) equipped with an improved epsilon constraint-handling is applied to pull the infeasible individuals achieved in the push stage to the feasible and non-dominated regions. Compared with other CMOEAs, the proposed PPS method can more efficiently get across infeasible regions and converge to the feasible and non-dominated regions by applying push and pull search strategies at different stages. To evaluate the performance regarding convergence and diversity, a set of benchmark CMOPs is used to test the proposed PPS and compare with other five CMOEAs, including MOEA/D-CDP, MOEA/D-SR, C-MOEA/D, MOEA/D-Epsilon and MOEA/D-IEpsilon. The comprehensive experimental results demonstrate that the proposed PPS achieves significantly better or competitive performance than the other five CMOEAs on most of the benchmark set. <s> BIB002 </s> A Survey of Multiobjective Evolutionary Algorithms Based on Decomposition: Variants, Challenges and Future Directions <s> 1) PENALTY FUNCTIONS <s> This paper proposes a learning guided parameter setting method for constrained multi-objective optimization. To be more specific, the proposed method can generate penalty factors adaptively, which is inspired by the learning rate setting from deep learning. The suggested penalty function employs an exponential decay model by integrating constraint violation values, objectives values, the current generation counter and the maximum number of generations. Furthermore, the proposed self-adaptive penalty method is embedded in the push and pull search framework (PPS-SA) to deal with constrained multi-objective optimization problems (CMOPs). In PPS-SA, the search process is divided into two different stages — push and pull search stages. In the push stage, a CMOP is optimized without considering any constraints. In the pull stage, the CMOP is optimized with a self-adaptive penalty constraint-handling method. To evaluate the performance regarding convergence and diversity, two commonly used metrics, including IGD and HV, are used to test the proposed PPS-SA and other four state-of-the-art CMOEAs, including PPS-MOEA/D, MOEA/D-IEpsilon, MOEA/D-Epsilon and MOEA/D-CDP. The experimental results indicate that the proposed PPS-SA outperforms the other four algorithms in most of the test cases, which demonstrates the superiority of the proposed PPS-SA. <s> BIB003
The method based on the penalty function adds one penalty term to the objective functions, transforming the constrained optimization problem into an unconstrained one. In evolutionary computation, researchers choose the external penalty function method to solve the problem, mainly because it does not need to initially provide a feasible solution. Jan and Zhang BIB001 presented a penalty function to punish infeasible solutions and introduced it into the update scheme of MOEA/D-DE. The proposed algorithm calculates the degree of constraint violation V (x) for the solution x, and if V (x) = 0, x is feasible, otherwise it is infeasible. The proposed penalty function uses the threshold related to the maximum and minimum values of V (x) to dynamically control the penalty amount which increases sharply when V (x) exceeds the threshold, encouraging the algorithm to search in both the feasible region and the infeasible region near the feasible one. The implementation process of the resulting algorithm is similar to that of MOEA/D, while has the following main variations: a) the calculation of V (x) for solution x; b) using a new aggregation function (penalty function) to compare the new solution with the old one; and c) update of the V (x) value. The experiments showed that the algorithm performs well in six out of ten constrained multiobjective optimization test instances. To keep a balance between objective minimization and constraint satisfaction, Fan et al. BIB002 presented a push and pull search (PPS) framework, and combined it with MOEA/D. As its name suggests, the proposed PPS divides the search process into two different stages of push and pull search. In the first stage, a CMOP is solved by considering only objectives without any constraints, which can help to get across to infeasible regions. In the pull stage, an adaptive penalty-based function is applied to pull the infeasible individuals achieved in the push stage to the feasible and nondominated regions. To improve the PPS framework BIB002 further, Fan et al. BIB003 proposed a self-adaptive penalty-based constraint handling method and embedded it in PPS. More specifically, inspired by the learning rate in deep leaning, the penalty factor is dynamically adjusted according to the ratio of feasible solutions to a population, the constraint violation value, the target value, etc. In the original PPS, the Epsilon constraint handling method first considers constraint violation and then considers objectives. However, the proposed algorithm deals with objectives and constraints simultaneously, thus it can maintain better population diversity.