STUDY ON A RELAXATION FOR THEOREMS OF THE ALTERNATIVE FOR SETS (Study on Nonlinear Analysis and Convex Analysis)

Abstract

In the literature, characterization for set relations are of various application to optimality conditions of set optimization problems, variational principles for set-valued maps, theorems of the alternative, certain robustness of vector optimization problems, and so on. In this paper, the author presents properties of scalarization functions as dual expression of set relations. Comparing to existing results, one can confirm their uniqueness in their relaxed conditions using convex cone-compactness and closedness. Also, we show the results implies generalized Gordan's theorems of the alternative at the last part of the thesis.