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

Author

Listed:
  • X.Q. Yang

    (Hong Kong Polytechnic University)

  • J.C. Yao

    (National Sun Yat-Sen University)

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.

Suggested Citation

  • X.Q. Yang & J.C. Yao, 2002. "Gap Functions and Existence of Solutions to Set-Valued Vector Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 115(2), pages 407-417, November.
    • Handle: RePEc:spr:joptap:v:115:y:2002:i:2:d:10.1023_a:1020844423345
    DOI: 10.1023/A:1020844423345
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    References listed on IDEAS

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    1. Goh, C. J. & Yang, X. Q., 1999. "Vector equilibrium problem and vector optimization," European Journal of Operational Research, Elsevier, vol. 116(3), pages 615-628, August.
    2. Q. H. Ansari & J> C> Yao, 2000. "On Nondifferentiable and Nonconvex Vector Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 106(3), pages 475-488, September.
    3. K. L. Lin & D. P. Yang & J. C. Yao, 1997. "Generalized Vector Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 92(1), pages 117-125, January.
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    Cited by:

    1. Xing Wang & Nan-jing Huang, 2014. "A Class of Differential Vector Variational Inequalities in Finite Dimensional Spaces," Journal of Optimization Theory and Applications, Springer, vol. 162(2), pages 633-648, August.
    2. J. Li & G. Mastroeni, 2010. "Vector Variational Inequalities Involving Set-valued Mappings via Scalarization with Applications to Error Bounds for Gap Functions," Journal of Optimization Theory and Applications, Springer, vol. 145(2), pages 355-372, May.
    3. N. J. Huang & J. Li & J. C. Yao, 2007. "Gap Functions and Existence of Solutions for a System of Vector Equilibrium Problems," Journal of Optimization Theory and Applications, Springer, vol. 133(2), pages 201-212, May.
    4. Ren-you Zhong & Nan-jing Huang, 2011. "Lower Semicontinuity for Parametric Weak Vector Variational Inequalities in Reflexive Banach Spaces," Journal of Optimization Theory and Applications, Springer, vol. 150(2), pages 317-326, August.
    5. Xiao-bo Li & Li-wen Zhou & Nan-jing Huang, 2016. "Gap Functions and Global Error Bounds for Generalized Mixed Variational Inequalities on Hadamard Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 168(3), pages 830-849, March.
    6. Anurag Jayswal & Shipra Singh, 2018. "Characterization of weakly sharp solutions of a variational-type inequality with convex functional," Annals of Operations Research, Springer, vol. 269(1), pages 297-315, October.
    7. Y. C. Lin, 2009. "On F-Implicit Generalized Vector Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 142(3), pages 557-568, September.
    8. Li, J. & Huang, N.J. & Yang, X.Q., 2010. "Weak sharp minima for set-valued vector variational inequalities with an application," European Journal of Operational Research, Elsevier, vol. 205(2), pages 262-272, September.
    9. Y. Chiang & J. C. Yao, 2004. "Vector Variational Inequalities and the (S)+ Condition," Journal of Optimization Theory and Applications, Springer, vol. 123(2), pages 271-290, November.
    10. G. Mastroeni, 2012. "On the image space analysis for vector quasi-equilibrium problems with a variable ordering relation," Journal of Global Optimization, Springer, vol. 53(2), pages 203-214, June.
    11. T. Jabarootian & J. Zafarani, 2008. "Generalized Vector Variational-Like Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 136(1), pages 15-30, January.
    12. S. K. Mishra & S. Y. Wang & K. K. Lai, 2008. "Gap Function for Set-Valued Vector Variational-Like Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 138(1), pages 77-84, July.
    13. S. Al-Homidan & Q. H. Ansari & S. Schaible, 2007. "Existence of Solutions of Systems of Generalized Implicit Vector Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 134(3), pages 515-531, September.
    14. Cunlin Li & Mihai Postolache & Zhifu Jia, 2019. "Weighted Method for Uncertain Nonlinear Variational Inequality Problems," Mathematics, MDPI, vol. 7(10), pages 1-22, October.
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    16. Yong Zhao & Jin Zhang & Xinmin Yang & Gui-Hua Lin, 2017. "Expected Residual Minimization Formulation for a Class of Stochastic Vector Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 175(2), pages 545-566, November.

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