Study of topological structure of solution sets of multi-objective optimization problems by using mapping on the solution sets

Reference No.	2025a028
Type/Category	Grant for Project Research- Short-term Visiting Researcher
Title of Research Project	Study of topological structure of solution sets of multi-objective optimization problems by using mapping on the solution sets
Principal Investigator	Takahiro Yamamoto（Department of Mathematics, Tokyo Gakugei University・Professor）
Research Period	September 8,2025. - September 12,2025. February 16,2026. - February 21,2025.
Keyword(s) of Research Fields	simplicial map, fiber singularity, singular fiber
Abstract for Research Report	To understand the topological structure of modelings of the solution set of a multi-objective optimization problem, we study the singularity of the mapping on the modeling. However, the modelings are not necessarily a smooth manifold; furthermore, the mappings on the modelings are not expected to be differentiable. So, we can not apply the singularity theory of smooth maps on smooth manifolds to study the topological structure of the modeling. On the other hand, the applicant recently defined the notion of fiber singularity for a continuous map of a topological space by focusing on fibers which are map germs along inverse images introduced by Osamu Saeki in 2006. In this project, we study the singularity of continuous maps by using fiber singularities. Furthermore, we establish a mathematical base that characterizes the topological structure of modelings of the solution set of multi-objective optimization problems by fiber singularities of continuous maps on the modeling. We expect to define a measure of the complexity of the topological structure of the modeling by the continuous maps on the modeling. For example, when the modeling is fixed, consider a continuous map from the modeling to a k-dimensional Euclidean space. One example is to define a set of such continuous maps whose behavior is not changed by small perturbations (or a set of continuous maps that only allow good fiber singularities) and the minimum number of certain fiber singularity types that a continuous map belonging to the set has.
Organizing Committee Members (Workshop) Participants (Short-term Joint Usage)	Takahiro Yammaoto（Tokyo Gakugei University・Professor）