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：

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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。