Use of inverted triangular weight vectors in decomposition-based multiobjective algorithms

Recently a number of evolutionary multiobjective optimization algorithms have been proposed in the framework of MOEA/D (Multi-Objective Evolutionary Algorithm based on Decomposition). A multiobjective problem is decomposed into multiple single-objective problems using a set of weight vectors in MOEA/D. The number of single-objective problems is the same as the number of weight vectors, which is also the same as the population size. It is well known that the performance of MOEA/D depends on the shape of the Pareto front. Weight vectors in MOEA/D are specified by using a triangular simplex lattice structure. Thus MOEA/D works well on multiobjective problems with triangular Pareto fronts, and has difficulties in handling inverted triangular Pareto fronts. One may wonder what happens if an inverted triangular simplex lattice structure is used for generating weight vectors in MOEA/D. This is our research question. In this paper, first we explain how to use an inverted triangular simplex lattice structure for generating weight vectors. Next we analytically explain inherent difficulties in the use of inverted triangular weight vectors in MOEA/D. Then, through computational experiments, we examine the performance of two types of MOEA/D: One is with triangular weight vectors, and the other is with inverted triangular weight vectors. Our discussions show that the use of inverted triangular weight vectors is not a good choice except for some special cases.