Abstract
The variational inequality problem with set-valued mappings is very useful in economics and nonsmooth optimization. In this paper, we study the existence of solutions and the formulation of solution methods for vector variational inequalities (VVI) with set-valued mappings. We introduce gap functions and establish necessary and sufficient conditions for the existence of a solution of the VVI. It is shown that the optimization problem formulated by using gap functions can be transformed into a semi-infinite programming problem. We investigate also the existence of a solution for the generalized VVI with a set-valued mapping by virtue of the existence of a solution of the VVI with a single-valued function and a continuous selection theorem.
Original language | English |
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Pages (from-to) | 407-417 |
Number of pages | 11 |
Journal | Journal of Optimization Theory and Applications |
Volume | 115 |
Issue number | 2 |
DOIs | |
Publication status | Published - 1 Nov 2002 |
Keywords
- existence of a solution
- gap functions
- semi-infinite programming
- set-valued mappings
- Vector variational inequalities
ASJC Scopus subject areas
- Control and Optimization
- Management Science and Operations Research
- Applied Mathematics