On the Stopping Criterion of Adaptive Weighted Aggregation for Multiobjective Continuous Optimization

This paper proposes a reasonable stopping criterion for Adaptive Weighted Aggregation (AWA), which is a scalarization-based multi-start framework developed in our previous study on continuous multiobjective optimization. Our previous study shows that AWA yields good solutions covering the entire Pareto set and front within a small consumption of running time and function evaluation on 2- to 6-objective benchmark problems. The experimental results also indicate, however, that the number of solutions generated by AWA is multiplied every iteration. The rapid increase of solutions requires a careful choice of the stopping criterion: even one iteration of shortage may deteriorate the coverage of solutions into an unsatisfactory level and one of excess gives rise to a significant waste of computational resources. We therefore discuss the minimum iteration that AWA yields an enough solution set to cover the Pareto set and front in the sense that the set contains at least one interior point of each of their non-empty "faces", that is, boundary submanifolds induced from the Pareto sets of subproblems with the same inclusion relation as faces of the simplex. Then, such an iteration, named the representing iteration, is proposed as a stopping criterion for AWA, and the number of solutions found by the representing iteration, named the representing number, is derived to analyze the space complexity of AWA. We also discuss the time complexity of AWA based on numerical experiments. The distribution of obtained solutions and its coverage measure show the usefulness of the proposed stopping criterion.