A Survey of Multiobjective Evolutionary Algorithms Based on Decomposition: Variants, Challenges and Future Directions <s> E. CONVERGENCE AND DIVERSITY <s> Decomposition-Based Multi-Objective Evolutionary Algorithms (DBMOEA), such as Multiple Single Objective Pareto Sampling (MSOPS) and Multiobjective Evolutionary Algorithm based on Decomposition (MOEA/D), have been successfully applied in finding Pareto-optimal fronts in Multiobjective Optimization Problems (MOPs), two or three-objective in general. DBMOEA decomposes one MOP into multiple Single-objective Optimization Problems (SOPs) where the convergence of approximated front is facilitated by finding the optimal solution of each SOP and its diversity is preserved by a group of well distributed SOPs. However, when solving problems with many objectives, one single solution can be the optimal solution of multiple SOPs which inadvertently leads to a severe loss of population diversity. In this paper, we propose a new diversity improvement method incorporated into a modified DBMOEA to directly handle this challenge. The design includes two steps. First, a few number of weight vectors guide the whole population towards a small number of solutions nearby the true Pareto front. Afterwards, initialize a subpopulation around each solution and diversify them toward well distribution. As a case study, a new algorithm based on this design is compared with three state-of-the-art DBMOEAs, MOEA/D, MSOPS, and MO-NSGA-II. Experimental results show that the proposed methods exhibit better performance in both convergence and diversity than the chosen competitors for solving many-objective optimization problems. <s> BIB001 </s> A Survey of Multiobjective Evolutionary Algorithms Based on Decomposition: Variants, Challenges and Future Directions <s> E. CONVERGENCE AND DIVERSITY <s> This paper studies how to use two reproduction operators with different characteristics for balancing the convergence and the diversity in MOEA/D. We consider two operators. One is a differential evolution and polynomial mutation, and the other is a neighbor learning and inversion mutation. We show that these two operators have different search abilities. Then we propose a scheme to use these two operators in our recently proposed MOEA/D-GR framework. We test the proposed algorithm on some benchmark problems to demonstrate its effectiveness. <s> BIB002 </s> A Survey of Multiobjective Evolutionary Algorithms Based on Decomposition: Variants, Challenges and Future Directions <s> E. CONVERGENCE AND DIVERSITY <s> A decomposition approach decomposes a multiobjective optimization problem into a number of scalar objective optimization subproblems. It plays a key role in decomposition-based multiobjective evolutionary algorithms. However, many widely used decomposition approaches, originally proposed for mathematical programming algorithms, may not be very suitable for evolutionary algorithms. To help decomposition-based multiobjective evolutionary algorithms balance the population diversity and convergence in an appropriate manner, this letter proposes to impose some constraints on the subproblems. Experiments have been conducted to demonstrate that our proposed constrained decomposition approach works well on most test instances. We further propose a strategy for adaptively adjusting constraints by using information collected from the search. Experimental results show that it can significantly improve the algorithm performance. <s> BIB003 </s> A Survey of Multiobjective Evolutionary Algorithms Based on Decomposition: Variants, Challenges and Future Directions <s> E. CONVERGENCE AND DIVERSITY <s> In evolutionary multi-objective optimization, each solution in the population generally has two roles. The first one is to approximate a part of the Pareto front, and the second one is to be a variable information resource to generate offspring. In many-objective optimization involving four or more conflicting objectives, solutions in the population have to be sparsely distributed in the objective space and the variable space to approximate a high-dimensional Pareto front, and each solution faces the difficulty to play the second role since variables are drastically individualized in the population. To overcome this problem, we focus on MOEA/D algorithm framework and propose a method to introduce supplemental weight vectors and solutions which maintain variable information resource to enhance the solution search for each part of the Pareto front. Experimental results using many-objective knapsack problems show that the supplemental weight vectors and solutions improves the search performance of MOEA/D by improving the diversity of the obtained solutions. <s> BIB004
A lot of studies have been carried out on algorithm convergence and population diversity. Different constraint values are needed for different subproblems, even for a specific problem at different stages BIB003 . Two evolutionary operators with different search characteristics of the solution uniformity and quality are used to generate promising solutions BIB002 . A population is initialized around each solution located on the boundary of the PS BIB001 . Supplemental weight vectors and the corresponding supplemental solutions are generated to enhance the search in a wider region BIB004 .