Simpliciality of strongly convex problems

Abstract

A multiobjective optimization problem is $C^r$ simplicial if the Pareto set and the Pareto front are $C^r$ diffeomorphic to a simplex and, under the $C^r$ diffeomorphisms, each face of the simplex corresponds to the Pareto set and the Pareto front of a subproblem, where $0\le r\le \infty$. In the paper titled "Topology of Pareto sets of strongly convex problems," it has been shown that a strongly convex $C^r$ problem is $C^{r-1}$ simplicial under a mild assumption on the ranks of the differentials of the mapping for $2\le r\le \infty$. On the other hand, in this paper, we show that a strongly convex $C^1$ problem is $C^0$ simplicial under the same assumption. Moreover, we establish a specialized transversality theorem on generic linear perturbations of a strongly convex $C^r$ mapping $(r\ge 2)$. By the transversality theorem, we also give an application of singularity theory to a strongly convex $C^r$ problem for $2\le r\le \infty$.

17 pages, to appear in Journal of the Mathematical Society of Japan