A Survey of Multiobjective Evolutionary Algorithms Based on Decomposition: Variants, Challenges and Future Directions <s> 2) SEPARATION OF OBJECTIVES AND CONSTRAINTS <s> This paper proposes an improved epsilon constraint handling method embedded in the multi-objective evolutionary algorithm based on decomposition (MOEA/D) to solve constrained multi-objective optimization problems (CMOPs). More specifically, it dynamically adjusts the epsilon level, which is a critical parameter in the epsilon constraint method, according to the feasible ratio of solutions in the current population. In order to verify the effect of the improved epsilon constraint handling method, three algorithms - MOEA/D-CDP, MOEA/D-Epsilon, and MOEA/D-IEpsilon (MOEA/D with the improved epsilon constraint handling mechanism) are tested on nine CMOPs (CMOP1-CMOP9). The comprehensive experimental results indicate that the proposed epsilon constraint handling method is very effective on the performance of both convergence and diversity. <s> BIB001 </s> A Survey of Multiobjective Evolutionary Algorithms Based on Decomposition: Variants, Challenges and Future Directions <s> 2) SEPARATION OF OBJECTIVES AND CONSTRAINTS <s> Solving constrained multi-objective optimization problems is a difficult task, it needs to simultaneously optimize multiple conflicting objectives and a number of constraints. This paper first reviews a number of popular constrained multi-objective evolutionary algorithms (CMOEAs) and twenty-three widely used constrained multi-objective optimization problems (CMOPs) (including CF1-10, CTP1-8, BNH, CONSTR, OSY, SRN and TNK problems). Then eight popular CMOEAs with simulated binary crossover (SBX) and differential evolution (DE) operators are selected to test their performance on the twenty-three CMOPs. The eight CMOEAs can be classified into domination-based CMOEAs (including ATM, IDEA, NSGA-II-CDP and SP) and decomposition-based CMOEAs (including CMOEA/D, MOEA/D-CDP, MOEA/D-SR and MOEA/D-IEpsilon). The comprehensive experimental results indicate that IDEA has the best performance in the domination-based CMOEAs and MOEA/D-IEpsilon has the best performance in the decomposition-based CMOEAs. Among the eight CMOEAs, MOEA/D-IEpsilon with both SBX and DE operators has the best performance on the twenty-three test problems. <s> BIB002 </s> A Survey of Multiobjective Evolutionary Algorithms Based on Decomposition: Variants, Challenges and Future Directions <s> 2) SEPARATION OF OBJECTIVES AND CONSTRAINTS <s> Abstract This paper proposes a novel constraint-handling mechanism, namely the angle-based constrained dominance principle (ACDP), to solve constrained multi-objective optimization problems (CMOPs). In this work, the mechanism of ACDP is embedded in a decomposition-based multi-objective evolutionary algorithm (MOEA/D). ACDP uses the angle information among solutions of a population and the proportion of feasible solutions to adjust the dominance relationship, so that it can maintain good convergence, diversity and feasibility of a population, simultaneously. To evaluate the performance of the proposed MOEA/D-ACDP, fourteen benchmark instances and an engineering optimization problem are studied. Six state-of-the-art CMOEAs, including C-MOEA/D, MOEA/D-CDP, MOEA/D-Epsilon, MOEA/D-SR, NSGA-II-CDP and SP, are compared. The experimental results illustrate that MOEA/D-ACDP is significantly better than the other six CMOEAs on these benchmark problems and the real-world case, which demonstrates the effectiveness of ACDP. <s> BIB003
For the penalty function method, it is not easy to select the appropriate penalty factor for a specific problem, because the factor is too small to be an effective punishment and/or too large to enable an algorithm to search near the feasible region. The separation mechanism of objectives and constraints, as its name suggests, treats objectives and constraints separately, which does not require any additional parameters while they are needed for the penalty function. Fan et al. BIB001 , BIB003 improved two typical methods of this mechanism, epsilon constraint handling method and constrained dominance principle (CDP), respectively. More specifically, Ref. BIB001 dynamically adjusted the epsilon level, which is a parameter to control the relaxation of constraints in the epsilon constraint handling method. Then the improved approach was embedded in the MOEA/D framework, leading to a new algorithm called MOEA/D-IEpsilon. This algorithm compares the epsilon level of the newly generated solution with that of its neighborhood solutions and updates the population by replacing the inferior solutions. Then the epsilon level is updated according to the proposed equations. The comprehensive experimental results verified that the MOEA/D-IEpsilon outperforms the MOEA/D-CDP and the MOEA/D-Epsilon in the performance of both convergence and diversity. Although the CDP is a popular constraint handling technique, it may neglect many infeasible solutions that have the potential to improve population diversity. Fan et al. BIB003 proposed an angle-based constrained dominance principle (ACDP), which is specifically designed to solve CMOPs with a low ratio of feasible solutions to all ones. The ACDP adds the angle information of objective functions to the original CDP to enhance population diversity in the infeasible region. For the comparison between the feasible solution and the infeasible one, if the angle of the peer solutions remains less than a user-defined threshold, the feasible solution is better, otherwise they are nondominated. This differs from the original CDP, which considers that any feasible solution is better than any infeasible solution. Experimental results indicated that the proposed ACDP method is very effective in the MOEA/D framework by comparing it with four popular constrained MOEA/Ds, including CMOEA/D, MOEA/D-CDP, MOEA/D-Epsilon and MOEA/D-SR (stochastic ranking). Fan et al. BIB002 reviewed a number of popular constrained MOEAs based on decomposition, including MOEA/D-IEpsilon, MOEA/D-CDP, CMOEA/D that embeds the epsilon constraint handling approach in MOEA/D, and MOEA/D-SR that applies SR to handle constraints in MOEA/D. The comprehensive experimental results indicated that the MOEA/D-IEpsilon has the best performance in the decomposition-based CMOEAs.