Joint Seminar on Numerical Analysis at Niigata University
Alternative theorems via set-valued analysis
*Yuto Ogata (Graduate School of Science and technology)
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  produced a generalized Gordan’s theorem for a vector-valued function in 1986. In 1999, Li  extended it to the case of set-valued maps.
In this talk, I would like to introduce alternative theorems from a set-valued analytic point of view, using the set-relations proposed by Kuroiwa, Tanaka, and Ha  in 1997. They can be considered in a topological vector space with the set-relations induced by a convex ordering cone. A similar approach with scalarizing functions for vectors had been done by Nishizawa, Onodsuka, and Tanaka  in 2005. Accordingly, I show 12 types of alternative theorems given by scalarizing functions for sets in order to extend them to more general forms. Comparing with a lot of former alternative theorems, these ones achieve the subdivision of the case and the simplification of the forms of them at the same time.
Reducing some conditions, we can recognize some of 12 types of theorems imply several former ones including Gordan’s theorem.
 V. Jeyakumar, A generalization of a minimax theorem of Fan via a theorem of the alternative, J. Optim. Theory Appl. 48 (1986), 525-533.
 Z. Li, A theorem of the alternative and its application to the optimization of set-valued maps, J. Optim. Theory Appl. 100 (1999), 365-375.
 D. Kuroiwa, T. Tanaka, and T.X.D. Ha, On convexity of set-valued maps, Nonlinear Anal. 30 (1997), 1487-1496.
 S. Nishizawa, M. Onodsuka, and T. Tanaka, Alternative theorems for set-valued maps based on a nonlinear scalarization, Pacific J. of Optim. 1 (2005), 147-159.