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Gap Functions and Existence of Solutions to Set-Valued Vector Variational Inequalities

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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.

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Authors and Affiliations

  1. Department of Applied Mathematics, Hong Kong Polytechnic University, Kowloon, Hong Kong

    X.Q. Yang (Associate Professor)

  2. Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung, Taiwan

    J.C. Yao (Professor)

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  1. X.Q. Yang
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  2. J.C. Yao
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Yang, X., Yao, J. Gap Functions and Existence of Solutions to Set-Valued Vector Variational Inequalities. Journal of Optimization Theory and Applications 115, 407–417 (2002). https://doi.org/10.1023/A:1020844423345

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