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Quasi-P*-Maps, P(τ, α, β)-Maps, Exceptional Family of Elements, and Complementarity Problems

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  • Y. B. Zhao

    (Chinese Academy of Sciences)

  • G. Isac

    (Royal Military College of Canada)

Abstract

Quasi-P*-maps and P(τ, α, β)-maps defined in this paper are two large classes of nonlinear mappings which are broad enough to include P*-maps as special cases. It is of interest that the class of quasi-P*-maps also encompasses quasimonotone maps (in particular, pseudomonotone maps) as special cases. Under a strict feasibility condition, it is shown that the nonlinear complementarity problem has a solution if the function is a nonlinear quasi-P*-map or P(τ, α, β)-map. This result generalizes a classical Karamardian existence theorem and a recent result concerning quasimonotone maps established by Hadjisawas and Schaible, but restricted to complementarity problems. A new existence result under an exceptional regularity condition is also established. Our method is based on the concept of exceptional family of elements for a continuous function, which is a powerful tool for investigating the solvability of complementarity problems.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:105:y:2000:i:1:d:10.1023_a:1004674330768
    DOI: 10.1023/A:1004674330768
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    References listed on IDEAS

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    1. F. A. Potra & R. Sheng, 1998. "Superlinearly Convergent Infeasible-Interior-Point Algorithm for Degenerate LCP," Journal of Optimization Theory and Applications, Springer, vol. 97(2), pages 249-269, May.
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    3. 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.
    4. G. Isac, 2000. "Exceptional Families of Elements, Feasibility and Complementarity," Journal of Optimization Theory and Applications, Springer, vol. 104(3), pages 577-588, March.
    5. Jen-Chih Yao, 1994. "Variational Inequalities with Generalized Monotone Operators," Mathematics of Operations Research, INFORMS, vol. 19(3), pages 691-705, August.
    6. 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.
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    Cited by:

    1. G. Isac & S. Z. Németh, 2006. "Duality of Implicit Complementarity Problems by Using Inversions and Scalar Derivatives," Journal of Optimization Theory and Applications, Springer, vol. 128(3), pages 621-633, March.
    2. 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.
    3. 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.
    4. 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.
    5. 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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