Two-Stage Probabilistic Penalty for Expensive Constrained Multiobjective Optimization Problems

Handling constraints is the key of solving expensive constrained multiobjective optimization problems (ECMOPs) by evolutionary algorithms. However, existing constraint-handling techniques (CHTs) struggle to effectively explore multiple feasible parts in the early stage and exploit them in the later stage. Therefore, this paper proposes a new CHT for ECMOPs, namedtwo-stage probabilisticpenalty (TSP). TSP constructs a penalty objective for each original objective to compare two solutions, defined as the weighted sum of two probabilities (named PoB and PoF) calculated based on Kriging models. For each solution, PoB and PoF measure its optimality on the original objective and its feasibility, respectively. In the first stage, the weights are defined based on the objective-space distance between the two compared solutions. The bigger the distance, the higher the weight of PoB, thereby maintaining some valuable infeasible solutions with good diversity to explore unknown feasible parts. In the second stage, PoB and PoF are assigned equal weights, but their values are modified based on the feasibility of solutions, in order to give feasible solutions higher priority. When enough truly evaluated feasible solutions have been obtained, their decision-space distribution information is integrated into the penalty objectives to further emphasize feasibility. As a result, the penalty objectives of the second stage can effectively guide the population to converge in the already located feasible parts. Moreover, two conditions are designed for the stage transition. Further, we incorporate TSP into a Kriging-assisted evolutionary framework to solve ECMOPs, obtaining TSP+. Extensive experiments confirm the effectiveness of TSP and TSP+.