Topology of Pareto Sets of Strongly Convex Problems
Topology of Pareto Sets of Strongly Convex Problems
复制标题
DOI：
10.1137/19m1271439
复制
发表时间：
2019-04
期刊：
影响因子：
--
通讯作者：
Naoki Hamada;Kenta Hayano;S. Ichiki;Y. Kabata;H. Teramoto
中科院分区：
文献类型：
--
作者：
Naoki Hamada;Kenta Hayano;S. Ichiki;Y. Kabata;H. Teramoto
如果Pareto集合和前与单纯形同胚，并且在同胚下，单纯形的每个面对应于子问题的Pareto集合和前，则多目标优化问题是简单的。在本文中，我们证明了强凸问题在对目标映射的微分秩的温和假设下是简单的。我们进一步证明，只要源的维数比目标的维数足够大，就可以用一般的线性扰动使任何强凸问题满足这个假设。通过适当的变换，我们证明了位置问题、生物模型和脊回归可以简化为多目标强凸问题，并保持了Pareto排序和拓扑结构。
A multiobjective optimization problem is simplicial if the Pareto set and front are homeomorphic to a simplex and, under the homeomorphisms, each face of the simplex corresponds to the Pareto set and front of a subproblem. In this paper, we show that strongly convex problems are simplicial under a mild assumption on the ranks of the differentials of the objective mappings. We further prove that one can make any strongly convex problem satisfy the assumption by a generic linear perturbation, provided that the dimension of the source is sufficiently larger than that of the target. We demonstrate that the location problems, a biological modeling, and the ridge regression can be reduced to multiobjective strongly convex problems via appropriate transformations preserving the Pareto ordering and the topology.