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Generalization of an Existence Theorem for Variational Inequalities

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  • Z.H. Huang

    (Institute of Applied Mathematics)

Abstract

By using the concept of exceptional family of elements, Zhao proposed a new existence theorem for variational inequalities over a general nonempty closed convex set (Ref. 1, Theorem 2.3), which is a generalization of the well-known Moré's existence theorem for nonlinear complementarity problems. The proof of Theorem 2.3 in Ref. 1 depends strongly on the condition 0∈K. Since this condition is rather strict for a general variational inequality, Zhao proposed an open question at the end of Ref. 1: Can the condition 0∈K in Theorem 2.3 be removed? In this paper, we answer this open question. Furthermore, we present the new notion of exceptional family of elements and establish a theorem of the alternative, by which we develop two new existence theorems for variational inequalities. Our results generalize the Zhao existence result.

Suggested Citation

  • Z.H. Huang, 2003. "Generalization of an Existence Theorem for Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 118(3), pages 567-585, September.
    • Handle: RePEc:spr:joptap:v:118:y:2003:i:3:d:10.1023_b:jota.0000004871.55273.15
    DOI: 10.1023/B:JOTA.0000004871.55273.15
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    References listed on IDEAS

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    1. G. Isac, 2000. "Exceptional Families of Elements, Feasibility and Complementarity," Journal of Optimization Theory and Applications, Springer, vol. 104(3), pages 577-588, March.
    2. Y. B. Zhao & G. Isac, 2000. "Quasi-P*-Maps, P(τ, α, β)-Maps, Exceptional Family of Elements, and Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 105(1), pages 213-231, April.
    3. G. Isac & W. T. Obuchowska, 1998. "Functions Without Exceptional Family of Elements and Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 99(1), pages 147-163, October.
    4. Y. B. Zhao & J. Y. Han & H. D. Qi, 1999. "Exceptional Families and Existence Theorems for Variational Inequality Problems," Journal of Optimization Theory and Applications, Springer, vol. 101(2), pages 475-495, May.
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    Cited by:

    1. J. Han & Z. H. Huang & S. C. Fang, 2004. "Solvability of Variational Inequality Problems," Journal of Optimization Theory and Applications, Springer, vol. 122(3), pages 501-520, September.
    2. Tran Nghi & Nguyen Nang Tam, 2020. "A Frank–Wolfe-Type Theorem for Cubic Programs and Solvability for Quadratic Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 187(2), pages 448-468, November.
    3. Zheng-Hai Huang & Liqun Qi, 2019. "Tensor Complementarity Problems—Part I: Basic Theory," Journal of Optimization Theory and Applications, Springer, vol. 183(1), pages 1-23, October.
    4. Yong Wang & Zheng-Hai Huang & Liqun Qi, 2018. "Global Uniqueness and Solvability of Tensor Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 177(1), pages 137-152, April.

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