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July, 2021 Simpliciality of strongly convex problems
Naoki HAMADA, Shunsuke ICHIKI
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J. Math. Soc. Japan 73(3): 965-982 (July, 2021). DOI: 10.2969/jmsj/83918391


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.


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Naoki HAMADA. Shunsuke ICHIKI. "Simpliciality of strongly convex problems." J. Math. Soc. Japan 73 (3) 965 - 982, July, 2021.


Received: 17 December 2019; Revised: 18 March 2020; Published: July, 2021
First available in Project Euclid: 12 November 2020

Digital Object Identifier: 10.2969/jmsj/83918391

Primary: 90C25
Secondary: 57R45

Rights: Copyright ©2021 Mathematical Society of Japan


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Vol.73 • No. 3 • July, 2021
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