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Exceptional Families of Elements for Optimization Problems in Reflexive Banach Spaces with Applications

Author

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  • Ren-you Zhong

    (Guangxi Normal University)

  • Huan-xia Lian

    (Guangxi Normal University)

  • Jiang-hua Fan

    (Guangxi Normal University)

Abstract

In this paper, we propose a new notion of ‘exceptional family of elements’ for convex optimization problems. By employing the notion of ‘exceptional family of elements’, we establish some existence results for convex optimization problem in reflexive Banach spaces. We show that the nonexistence of an exceptional family of elements is a sufficient and necessary condition for the solvability of the optimization problem. Furthermore, we establish several equivalent conditions for the solvability of convex optimization problems. As applications, the notion of ‘exceptional family of elements’ for convex optimization problems is applied to the constrained optimization problem and convex quadratic programming problem and some existence results for solutions of these problems are obtained.

Suggested Citation

  • Ren-you Zhong & Huan-xia Lian & Jiang-hua Fan, 2013. "Exceptional Families of Elements for Optimization Problems in Reflexive Banach Spaces with Applications," Journal of Optimization Theory and Applications, Springer, vol. 159(2), pages 341-359, November.
    • Handle: RePEc:spr:joptap:v:159:y:2013:i:2:d:10.1007_s10957-013-0342-4
    DOI: 10.1007/s10957-013-0342-4
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    References listed on IDEAS

    as
    1. J. H. Fan & X. G. Wang, 2009. "Solvability of Generalized Variational Inequality Problems for Unbounded Sets in Reflexive Banach Spaces," Journal of Optimization Theory and Applications, Springer, vol. 143(1), pages 59-74, October.
    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. 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.
    5. M. Bianchi & N. Hadjisavvas & S. Schaible, 2006. "Exceptional Families of Elements for Variational Inequalities in Banach Spaces," Journal of Optimization Theory and Applications, Springer, vol. 129(1), pages 23-31, April.
    6. 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.
    7. G. Isac & V. V. Kalashnikov, 2001. "Exceptional Family of Elements, Leray–Schauder Alternative, Pseudomonotone Operators and Complementarity," Journal of Optimization Theory and Applications, Springer, vol. 109(1), pages 69-83, April.
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