A two-stage adaptive penalty method based on co-evolution for constrained evolutionary optimization

Penalty function method is popular for constrained evolutionary optimization. However, it is non-trivial to set a proper penalty factor for a constrained optimization problem. This paper takes advantage of co-evolution to adjust the penalty factor and proposes a two-stage adaptive penalty method. In the co-evolution stage, the population is divided into multiple subpopulations, each of which is associated with a penalty factor. Through the co-evolution of these subpopulations, the performance of penalty factors can be evaluated. Since different penalty factors are used, the subpopulations will evolve along different directions. Thus, exploration can be enhanced. In the shuffle stage, all subpopulations are merged into a population and the best penalty factor from the co-evolution stage is used to guide the evolution. In this manner, the information interaction among subpopulations can be facilitated; thus, exploitation can be promoted. By executing these two stages iteratively, the feasible optimum could be obtained finally. In the two-stage evolutionary process, the search algorithm is designed based on two trial vector generation strategies of differential evolution. Additionally, a restart mechanism is designed to help the population avoid stagnating in the infeasible region. Extensive experiments demonstrate the effectiveness of the proposed method.