Use of Piecewise Linear and Nonlinear Scalarizing Functions in MOEA/D

A number of weight vector-based algorithms have been proposed for many-objective optimization using the framework of MOEA/D (multi-objective evolutionary algorithm based on decomposition). Those algorithms are characterized by the use of uniformly distributed normalized weight vectors, which are also referred to as reference vectors, reference lines and search directions. Their common idea is to minimize the distance to the ideal point (i.e., convergence) and the distance to the reference line (i.e., uniformity). Each algorithm has its own mechanism for striking a convergence-uniformity balance. In the original MOEA/D with the PBI (penalty-based boundary intersection) function, this balance is handled by a penalty parameter. In this paper, we first discuss why an appropriate specification of the penalty parameter is difficult. Next we suggest a desired shape of contour lines of a scalarizing function in MOEA/D. Then we propose two ideas for modifying the PBI function. The proposed ideas generate piecewise linear and nonlinear contour lines. Finally we examine the effectiveness of the proposed ideas on the performance of MOEA/D for many-objective test problems.