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

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

Sublinear scalarizations for proper and approximate proper efficient points in nonconvex vector optimization

We show that under a separation property, a Q-minimal point in a normed space is the minimum of a given sublinear function. This fact provides sufficient conditions, via scalarization, for nine types of proper efficient points; establishing a characterization in the particular case of Benson proper efficient points. We also obtain necessary and sufficient conditions in terms of scalarization for approximate Benson and Henig proper efficient points. The separation property we handle is a variation of another known property and our scalarization results do not require convexity or boundedness assumptions.Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. Fernando García-Castaño and Miguel Ángel Melguizo-Padial acknowledge the financial support from the Spanish Ministry of Science, Innovation and Universities (MCIN/AEI) under grant PID2021-122126NB-C32, co-funded by the European Regional Development Fund (ERDF) under the slogan “A way of making Europe”