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Nonemptiness and boundedness of solution sets for vector variational inequalities via topological method

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  • Jiang-hua Fan
  • Yan Jing
  • Ren-you Zhong

Abstract

In this paper, some characterizations of nonemptiness and boundedness of solution sets for vector variational inequalities are studied in finite and infinite dimensional spaces, respectively. By using a new proof method which is different from the one used in Huang et al. (J Optim Theory Appl 162:548–558 2014 ), a sufficient and necessary condition for the nonemptiness and boundedness of solution sets is established. Basing on this result, some new characterizations of nonemptiness and boundedness of solution sets for vector variational inequalities are proved. Compared with the known results in Huang et al. ( 2014 ), the key assumption that $$K_\infty \cap (F(K))^{w\circ }_C=\{0\}$$ K ∞ ∩ ( F ( K ) ) C w ∘ = { 0 } is not required in finite dimensional spaces. Furthermore, the corresponding result of Huang et al. ( 2014 ) is extended to the case of infinite dimensional spaces. Some examples are also given to illustrated the main results. Copyright Springer Science+Business Media New York 2015

Suggested Citation

  • Jiang-hua Fan & Yan Jing & Ren-you Zhong, 2015. "Nonemptiness and boundedness of solution sets for vector variational inequalities via topological method," Journal of Global Optimization, Springer, vol. 63(1), pages 181-193, September.
    • Handle: RePEc:spr:jglopt:v:63:y:2015:i:1:p:181-193
    DOI: 10.1007/s10898-015-0279-2
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    References listed on IDEAS

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    1. X. X. Huang & X. Q. Yang & K. L. Teo, 2004. "Characterizing Nonemptiness and Compactness of the Solution Set of a Convex Vector Optimization Problem with Cone Constraints and Applications," Journal of Optimization Theory and Applications, Springer, vol. 123(2), pages 391-407, November.
    2. X. X. Huang & Y. P. Fang & X. Q. Yang, 2014. "Characterizing the Nonemptiness and Compactness of the Solution Set of a Vector Variational Inequality by Scalarization," Journal of Optimization Theory and Applications, Springer, vol. 162(2), pages 548-558, August.
    3. F. Flores-Bazán & C. Vera, 2006. "Characterization of the Nonemptiness and Compactness of Solution Sets in Convex and Nonconvex Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 130(2), pages 185-207, August.
    4. S. Deng, 2009. "Characterizations of the Nonemptiness and Boundedness of Weakly Efficient Solution Sets of Convex Vector Optimization Problems in Real Reflexive Banach Spaces," Journal of Optimization Theory and Applications, Springer, vol. 140(1), pages 1-7, January.
    5. Massimo Marinacci & Luigi Montrucchio, 2011. "Finitely Well-Positioned Sets," Working Papers 386, IGIER (Innocenzo Gasparini Institute for Economic Research), Bocconi University.
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