An adaptive penalty-based boundary intersection method for many-objective optimization problem

Compared with domination-based methods, the multi-objective evolutionary algorithm based on decomposition (MOEA/D) is less prone to the difficulty caused by an increase in the number of objectives. It is a promising algorithmic framework for solving many-objective optimization problems (MaOPs). In MOEA/D, the target MaOP 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 an appropriate penalty parameter is known to perform well. However, its performance is heavily influenced by the setting of the penalty factor (0), which can take a value from zero to +infinity. A limited amount of work has thus far considered the choice of an appropriate value of theta. This paper presents a comprehensive experimental study on WFG and WFG-extend problems featuring two to 15 objectives. A range of values of theta is investigated to understand its influence on the performance of the PBI-based MOEA/D (MOEA/D-PBI). Based on the observations, the range of values of theta are divided into three sub-regions, and a two-stage adaptive penalty scheme is proposed to adaptively choose an appropriate value from 0.001 to 8000 during an optimization run. The results of experiments show that, the robustness of MOEA/D-PBI can be significantly enhanced using the proposed scheme. (C) 2019 Elsevier Inc. All rights reserved.