An Improved Penalty-Based Boundary Intersection Approach for Irregular Problems

The multi-objective evolutionary algorithm based on decomposition (MOEA/D) has achieved remarkable success in regular optimizing multi-objective problems. In practice, there are lots of irregular problems. The projection of their Pareto fronts (PF) may not cover the whole triangle on the first octant. There is an empty PF (EPF) or complex PF area. These problems significantly reduce the performance of MOEA/D. In this paper, an improved algorithm is proposed to deal with such complex problems. The basic idea is: (1) the PBI method is used to ensure the convergence and simultaneously keep diversity over the triangle; (2) a non-dominated and maximum distance based selection is used to replace the dominated objective individuals selected by the PBI method. It is referred to selection based on the PBI and Non-dominated Maximum distance (MOEAD-PNM). First, the mathematical foundation between the penalty factor and contour line is deduced. Based on the foundation, the penalty factor setting method is proposed and the PBI-based method is used to select an individual for each reference vector. The reference vectors directing to the EPF induce “bad” solutions. Then, these solutions are gradually replaced by better individuals selected according to the non-dominated and maximum distance-based selection strategy. The proposed two-step selection method can guarantee the wideness and uniformity of the “good” solution set. Finally, the performance of the MOEAD-PNM algorithm and five classic algorithms on more than ten test problems are compared. The experimental results show the competitiveness and effectiveness of the proposed algorithm on these challenging problems.