An improved multiobjective optimization evolutionary algorithm based on decomposition with hybrid penalty scheme

The multiobjective evolutionary algorithm based on decomposition (MOEA/D) decomposes a multiobjective optimization problem(MOP) into a number of single-objective subproblems. Penalty boundary intersection (PBI) in MOEA/D is one of the most popular decomposition approaches and has attracted significant attention. In this paper, we investigate two recent improvements on PBI, i.e. adaptive penalty scheme (APS) and subproblem-based penalty scheme (SPS), and demonstrate their strengths and weaknesses. Based on the observations, we further propose a hybrid penalty scheme (HPS), which adjusts the PBI penalty factor for each subproblem in two phases, to ensure the diversity of boundary solutions and good distribution of intermediate solutions. HPS specifies a distinct penalty value for each subproblem according to its weight vector. All the penalty values of subproblems increase with the same gradient during the first phase, and they are kept unchanged during the second phase