RCTS Optimization Seminar



SPEAKERProf. Tamaki Tanaka°]Niigata University°^

TITLEGeneralized Alternative Theorems for Set-Valued Maps

ABSTRACT Alternative theorems such as Farkas°¶ lemma and Gordan°¶s theorem usually play important roles in considering optimality conditions and many kinds of valuable extensions intended to find other similar conditions have been established. Jeyakumar [1] produced a generalized Gordan°¶s theorem for a vector-valued function in 1986. In 1999, Li [3] extended it to the case of set-valued maps, and also Yang, Yang, and Chen [5] generalized in 2000. In this talk, we would like to introduce alternative theorems from a set-valued analytic point of view, using the set-relations proposed by Kuroiwa, Tanaka, and Ha [2] in 1997. They can be considered in a topological vector space with the set-relations induced by a convex ordering cone. A similar approach had been done by Nishizawa, Onodsuka, and Tanaka [4] in 2005 by using scalarizing functions for vectors. We show 12 types of alternative theorems given by scalarizing functions for sets. Comparing with several types of former alternative theorems, these ones achieve the subdivision of the cases and the simplification of the forms at the same time. By reducting some conditions, we can recognize some of 12 types of theorems imply several former ones including Gordan°¶s theorem.