Pareto adaptive penalty-based boundary intersection method for multi-objective optimization

Penalty-based boundary intersection (PBI) method is a frequently used scalarizing method in decomposition based multi-objective evolutionary algorithms (MOEAs). It works well when a proper penalty value is provided, however, the determination of a suitable penalty value depends on the problem itself, more precisely, the Pareto optimal front (PF) shape. As the penalty value increases, the PBI method becomes less effective in terms of convergence, but is more capable of handling various PF shapes. In this study, a simple yet effective method called Pareto adaptive PBI (PaP) is proposed by which a suitable penalty value can be adaptively identified, which therefore can maintain fast convergence speed, meanwhile, leading to a good approximation of the PF. The PaP strategy integrated into the state-of-the-art decomposition algorithm, MOEA/D, denoted as MOEA/D-PaP, is examined on a set of multi-objective benchmarks with different PF shapes. Experimental results show that the PaP strategy is more effective than the weighted sum, the weighted Tchebycheff and the PBI method with (representative) fixed penalty values in general. In addition, the MOEA/D-PaP is examined on a real-world problem - multi-objective optimization of a hybrid renewable energy system whose PF is unknown. The outcome of the experiment further confirms its feasibility and superiority.