2021 年 73 巻 3 号 p. 965-982
A multiobjective optimization problem is 𝐶𝑟 simplicial if the Pareto set and the Pareto front are 𝐶𝑟 diffeomorphic to a simplex and, under the 𝐶𝑟 diffeomorphisms, each face of the simplex corresponds to the Pareto set and the Pareto front of a subproblem, where 0 ≤ 𝑟 ≤ ∞. In the paper titled “Topology of Pareto sets of strongly convex problems”, it has been shown that a strongly convex 𝐶𝑟 problem is 𝐶𝑟 −1 simplicial under a mild assumption on the ranks of the differentials of the mapping for 2 ≤ 𝑟 ≤ ∞. On the other hand, in this paper, we show that a strongly convex 𝐶1 problem is 𝐶0 simplicial under the same assumption. Moreover, we establish a specialized transversality theorem on generic linear perturbations of a strongly convex 𝐶𝑟 mapping (𝑟 ≥ 2). By the transversality theorem, we also give an application of singularity theory to a strongly convex 𝐶𝑟 problem for 2 ≤ 𝑟 ≤ ∞.
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