PBI function based evolutionary algorithm with precise penalty parameter for unconstrained many-objective optimization

Fixed or experiential penalty parameter of the penalty-based boundary intersection (PBI) function method cannot simultaneously ensure the convergence and diversity for all shape of Pareto front (PF). Too large penalty parameter may lead to bad convergence while too small parameter can not ensure the diversity. Specially, if the penalty parameter is too small, some reference weight vectors may have no solution on it. This error is hard to be rectified. In this paper, we prove that the lower bound of the penalty parameter is determined by three factors. The first one is the shape of the PF. The second one is the cosine distance between two adjacent reference vectors. The third one is the number of objectives. We deduce the lower bound of the penalty parameter. Once the penalty parameter was calculated, an individual with minimal PBI function is attached to the corresponding reference vector. The minimal-PBI-function-first principle is used in the environmental selection to guarantee the wideness and uniformity of the solution set. The time complexity is low. The proposed method is compared with other three state-of-the-art many-objective evolutionary algorithms on the unconstrained test problems MaOP, DTLZ and WFG with up to fifteen objectives. The experimental results show the competitiveness and effectiveness of the proposed algorithm in both time efficiency and accuracy.