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Topology of Pareto Sets of Strongly Convex Problems
SIAM Journal on Optimization ( IF 2.6 ) Pub Date : 2020-09-28 , DOI: 10.1137/19m1271439
Naoki Hamada , Kenta Hayano , Shunsuke Ichiki , Yutaro Kabata , Hiroshi Teramoto

SIAM Journal on Optimization, Volume 30, Issue 3, Page 2659-2686, January 2020.
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 that treats a subset of objective functions. 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.


中文翻译:

强凸问题的帕累托集的拓扑

SIAM优化杂志,第30卷,第3期,第2659-2686页,2020年1月。
如果Pareto集和front对单形同胚,并且在同胚性下,单形的每个面都对应于Pareto集和一个处理目标函数子集的子问题的front,那么多目标优化问题就很简单。在本文中,我们表明,在对目标映射的微分等级进行温和假设的情况下,强凸问题是简单的。我们进一步证明,只要源的尺寸比目标的尺寸足够大,就可以通过一般的线性摄动使任何强凸问题都满足假设。我们证明,可以通过保留帕累托有序和拓扑的适当转换,将位置问题,生物学模型和岭回归简化为多目标强凸问题。
更新日期:2020-11-13
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