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**Gap functions and existence of solutions to set-valued vector variational inequalities.**
*(English)*
Zbl 1027.49003

Summary: 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.

### Keywords:

vector variational inequalities; set-valued mappings; gap functions; existence of a solution; semi-infinite programming; continuous selection
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\textit{X. Q. Yang} and \textit{J. C. Yao}, J. Optim. Theory Appl. 115, No. 2, 407--417 (2002; Zbl 1027.49003)

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### References:

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