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Equivalence of Equilibrium Problems and Least Element Problems

Author

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  • Y.-P. Fang

    (Sichuan University, Chengdu)

  • N.-J. Huang

    (Sichuan University, Chengdu)

Abstract

In this paper, we introduce the concept of feasible set for an equilibrium problem with a convex cone and generalize the notion of a Z-function for bifunctions. Under suitable assumptions, we derive some equivalence results of equilibrium problems, least element problems, and nonlinear programming problems. The results presented extend some results of [Riddell, R.C.: Equivalence of nonlinear complementarity problems and least element problems in Banach lattices. Math. Oper. Res. 6, 462–474 (1981)] to equilibrium problems.

Suggested Citation

  • Y.-P. Fang & N.-J. Huang, 2007. "Equivalence of Equilibrium Problems and Least Element Problems," Journal of Optimization Theory and Applications, Springer, vol. 132(3), pages 411-422, March.
    • Handle: RePEc:spr:joptap:v:132:y:2007:i:3:d:10.1007_s10957-007-9186-0
    DOI: 10.1007/s10957-007-9186-0
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    References listed on IDEAS

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    1. O. L. Mangasarian, 1979. "Simplified Characterizations of Linear Complementarity Problems Solvable as Linear Programs," Mathematics of Operations Research, INFORMS, vol. 4(3), pages 268-273, August.
    2. 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.
    3. R. C. Riddell, 1981. "Equivalence of Nonlinear Complementarity Problems and Least Element Problems in Banach Lattices," Mathematics of Operations Research, INFORMS, vol. 6(3), pages 462-474, August.
    4. N. Hadjisavvas & S. Schaible, 1998. "From Scalar to Vector Equilibrium Problems in the Quasimonotone Case," Journal of Optimization Theory and Applications, Springer, vol. 96(2), pages 297-309, February.
    5. Q. H. Ansari & T. C. Lai & J. C. Yao, 1999. "On the Equivalence of Extended Generalized Complementarity and Generalized Least-Element Problems," Journal of Optimization Theory and Applications, Springer, vol. 102(2), pages 277-288, August.
    6. M. Bianchi & N. Hadjisavvas & S. Schaible, 1997. "Vector Equilibrium Problems with Generalized Monotone Bifunctions," Journal of Optimization Theory and Applications, Springer, vol. 92(3), pages 527-542, March.
    7. Richard W. Cottle & Jong-Shi Pang, 1978. "A Least-Element Theory of Solving Linear Complementarity Problems as Linear Programs," Mathematics of Operations Research, INFORMS, vol. 3(2), pages 155-170, May.
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    Cited by:

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    2. R. Hu & Y. P. Fang, 2009. "Feasibility-Solvability Theorem for a Generalized System," Journal of Optimization Theory and Applications, Springer, vol. 142(3), pages 493-499, September.

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