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 $0 \leq r \leq \infty$. 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 $2 \leq r \leq \infty$. 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 $(r \geq 2)$. By the transversality theorem, we also give an application of singularity theory to a strongly convex $C^r$ problem for $2 \leq r \leq \infty$.
Funding Statement
The second authour was supported by JSPS KAKENHI Grant Numbers JP19J00650 and JP17H06128.
Citation
Naoki HAMADA. Shunsuke ICHIKI. "Simpliciality of strongly convex problems." J. Math. Soc. Japan 73 (3) 965 - 982, July, 2021. https://doi.org/10.2969/jmsj/83918391
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