Mathematical foundation, discussion and suggestion on penalty parameter setting of penalty-based boundary intersection method for many-objective optimization problems

Decomposition based method is a promising algorithmic framework for solving multi-objective optimization problems (MOPs). The target MOP is decomposed into a set of single-objective problems by using a scalarizing function with evenly specified weight vectors. Among the available scalarizing functions, penalty-based boundary intersection (PBI) with a proper penalty parameter is known to perform well. However, how to specify a penalty parameter θ that applies to an arbitrary Pareto front (PF) shape is not clear, let alone the theoretical basis behind the parameter setting. To the best of our knowledge, the theoretical basis has never been researched. The θ specification is basically a contour-setting problem. For a MOP with an unknown PF shape, one reliable practice is setting the contour on the border of the control area. To this end, a computational formulation of θ, which is determined by the angle between a given weight vector and the contour starting from this vector is deduced firstly. It provides a baseline (lower bound) of θ value and is a unified contour setting method applicable to various problems. Then, some fine-tuning methods are introduced for MOPs with a PF area that is hard to approach. Finally, a simple and classic minimal-PBI-function-first selection based genetic algorithm is used to illustrate the effectiveness of the proposed θ calculation. In the same algorithm frame, the performance of several representative θ setting methods are checked too. Their performances are compared on several test problems with different representative difficulties. The results show that the proposed θ setting can simultaneously ensure diversity and convergence.