TY - JOUR
T1 - Topology of pareto sets of strongly convex problems
AU - Hamada, Naoki
AU - Hayano, Kenta
AU - Ichiki, Shunsuke
AU - Kabata, Yutaro
AU - Teramoto, Hiroshi
N1 - Funding Information:
\ast Received by the editors July 1, 2019; accepted for publication (in revised form) June 28, 2020; published electronically September 28, 2020. https://doi.org/10.1137/19M1271439 Funding: The second author was supported by JSPS KAKENHI grant JP17K14194. The third author was supported by JSPS KAKENHI grants JP16J06911, JP19J00650, and JP17H06128. The fourth author was supported by JSPS KAKENHI grant JP16J02200. The fifth author was supported by JSPS KAKENHI grant JP19K03484 and by JST PRESTO grant JPMJPR16E8. The authors were supported by the RIKEN Center for Advanced Intelligence Project. This work was also supported by the Research Institute for Mathematical Sciences, a Joint Usage/Research Center located in Kyoto University.
Publisher Copyright:
© 2020 Society for Industrial and Applied Mathematics
PY - 2020
Y1 - 2020
N2 - 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.
AB - 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.
KW - Multiobjective optimization
KW - Simplicial problem
KW - Strongly convex mapping
KW - Topology of differentiable mapping
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U2 - 10.1137/19M1271439
DO - 10.1137/19M1271439
M3 - Article
AN - SCOPUS:85093118162
SN - 1052-6234
VL - 30
SP - 2659
EP - 2686
JO - SIAM Journal on Optimization
JF - SIAM Journal on Optimization
IS - 3
ER -