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Functions Without Exceptional Family of Elements and Complementarity Problems

Author

Listed:
  • G. Isac

    (Royal Military College of Canada)

  • W. T. Obuchowska

    (University of Nebraska at Omaha)

Abstract

In Ref. 1, Isac, Bulavski, and Kalashnikov introduced the concept of exceptional family of elements for a continuous function f: R n→R n. It is known that, if there does not exist an exceptional family of elements for f, then the corresponding complementarity problem has a solution. In this paper, we show that several classes of nonlinear functions, known in complementarity theory or other domains, are functions without exceptional family of elements and consequently the corresponding complementarity problem is solvable. It is evident that the notion of exceptional family of elements provides an alternative way of determining whether or not the complementarity problem has a solution.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:99:y:1998:i:1:d:10.1023_a:1021704311867
    DOI: 10.1023/A:1021704311867
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    References listed on IDEAS

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    1. M. Seetharama Gowda & Jong-Shi Pang, 1992. "On Solution Stability of the Linear Complementarity Problem," Mathematics of Operations Research, INFORMS, vol. 17(1), pages 77-83, February.
    2. Jorge J. Moré, 1996. "Global Methods for Nonlinear Complementarity Problems," Mathematics of Operations Research, INFORMS, vol. 21(3), pages 589-614, August.
    3. J. M. Borwein & M. A. H. Dempster, 1989. "The Linear Order Complementarity Problem," Mathematics of Operations Research, INFORMS, vol. 14(3), pages 534-558, August.
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    Citations

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    Cited by:

    1. Y. B. Zhao & D. Li, 2000. "Strict Feasibility Conditions in Nonlinear Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 107(3), pages 641-664, December.
    2. 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.
    3. 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.
    4. Y. Chiang, 2010. "Vectorial exceptional families of elements," Journal of Global Optimization, Springer, vol. 47(1), pages 53-62, May.
    5. 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.
    6. M. Bianchi & N. Hadjisavvas & S. Schaible, 2004. "Minimal Coercivity Conditions and Exceptional Families of Elements in Quasimonotone Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 122(1), pages 1-17, July.
    7. G. Isac, 2000. "Exceptional Families of Elements, Feasibility and Complementarity," Journal of Optimization Theory and Applications, Springer, vol. 104(3), pages 577-588, March.
    8. 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.

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