Two-Layered Weight Vector Specification in Decomposition-Based Multi-Objective Algorithms for Many-Objective Optimization Problems

Recently high performance of decomposition-based algorithms such as MOEA/D, NSGA-III and MOEA/DD for many-objective optimization has been repeatedly reported. When they are applied to many-objective problems, weight vectors are usually generated by a two-layered approach with boundary and inside layers. However, the specification of the two layers has not been discussed in detail in the literature. They are usually intuitively specified: More weight vectors are included in the boundary layer than the inside layer, and the size of the simplex in the inside layer is a half of that in the boundary layer. In this paper, we discuss the following two issues about the specification of the two layers: (i) how to specify the number of weight vectors in each layer, and (ii) how to specify the size of the simplex in the inside layer. We address these two issues through computational experiments on many-objective problems with six types of Pareto fronts: linear triangular, convex triangular, concave triangular, linear inverted triangular, convex inverted triangular, and concave inverted triangular. Our experimental results clearly demonstrate that the appropriate specification of the two layers strongly depends on the problem (i.e., the shape of the Pareto front) and the performance indicator. We also address the two issues from a viewpoint of the relation between the weight vector distribution for each shape of the Pareto front and the optimal distribution of solutions for each performance indicator.