Applied Mathematics Seminar

DATE2024-02-20 15:00-16:00

PLACE數學系館 1F3174教室

SPEAKERProfessor Tamaki Tanaka (田中環)(Niigata University 新瀉大學

TITLEGeneralized Cone-continuity of Set-Valued Maps with Scalarization

ABSTRACT A composite function is a function which is the nesting of two or more functions to form a single new function. Such operation frequently preserves several mathematical properties of each nested function. For instance, a composition of continuous maps is continuous on topological spaces. From the view point of vector optimization and set optimization, this kind of inheritance by composite operations is important and useful to prove extended results and to get characterizations of optimal solutions through scalarization. This is a typical approach by which optimization problems with vector-valued or set-valued maps can be easily handled by converting vectors or sets into real numbers. Recently, Ike, Liu, Ogata and Tanaka show certain results on the inheritance property of some kinds of continuity of set-valued maps via scalarization functions for sets: if a set-valued map has a kind of continuity (lower continuity or upper continuity), then the composition of its set-valued map and a certain scalarization function assures a similar semicontinuity to its scalarization function defined on the family of nonempty subsets of a real topological vector space. Their results are generalizations of results in earlier study by Kuwano, Tanaka and Yamada. However, the statements of inheritance are confined to four types out of the six set-relations proposed by Kuroiwa, Tanaka and Ha. On the other hand, Sonda, Kuwano, and Tanaka introduce two kinds of continuity with respect to cone, called ``cone continuity,'' for set-valued maps by analogy with semicontinuity for real-valued functions, and they investigate the inheritance properties on cone continuity of parent set-valued maps via scalarization. Therefore, it is interesting to investigate the inheritance of cone continuity for set-valued maps via general scalarization functions for sets in the same manner.
The aim of this talk is to introduce the mechanism by which composite functions of a set-valued map and a scalarization function transmit semicontinuity of parent set-valued maps through several scalarization for sets.