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New Relaxation Method for Mathematical Programs with Complementarity Constraints

Author

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  • G.H. Lin

    (Kyoto University
    Dalian University of Technology)

  • M. Fukushima

    (Kyoto University)

Abstract

In this paper, we present a new relaxation method for mathematical programs with complementarity constraints. Based on the fact that a variational inequality problem defined on a simplex can be represented by a finite number of inequalities, we use an expansive simplex instead of the nonnegative orthant involved in the complementarity constraints. We then remove some inequalities and obtain a standard nonlinear program. We show that the linear independence constraint qualification or the Mangasarian–Fromovitz constraint qualification holds for the relaxed problem under some mild conditions. We consider also a limiting behavior of the relaxed problem. We prove that any accumulation point of stationary points of the relaxed problems is a weakly stationary point of the original problem and that, if the function involved in the complementarity constraints does not vanish at this point, it is C-stationary. We obtain also some sufficient conditions of B-stationarity for a feasible point of the original problem. In particular, some conditions described by the eigenvalues of the Hessian matrices of the Lagrangian functions of the relaxed problems are new and can be verified easily. Our limited numerical experience indicates that the proposed approach is promising.

Suggested Citation

  • G.H. Lin & M. Fukushima, 2003. "New Relaxation Method for Mathematical Programs with Complementarity Constraints," Journal of Optimization Theory and Applications, Springer, vol. 118(1), pages 81-116, July.
    • Handle: RePEc:spr:joptap:v:118:y:2003:i:1:d:10.1023_a:1024739508603
    DOI: 10.1023/A:1024739508603
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    References listed on IDEAS

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    1. Holger Scheel & Stefan Scholtes, 2000. "Mathematical Programs with Complementarity Constraints: Stationarity, Optimality, and Sensitivity," Mathematics of Operations Research, INFORMS, vol. 25(1), pages 1-22, February.
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    Cited by:

    1. Nie, Pu-yan & Chen, Li-hua & Fukushima, Masao, 2006. "Dynamic programming approach to discrete time dynamic feedback Stackelberg games with independent and dependent followers," European Journal of Operational Research, Elsevier, vol. 169(1), pages 310-328, February.
    2. Gui-Hua Lin & Masao Fukushima, 2005. "A Modified Relaxation Scheme for Mathematical Programs with Complementarity Constraints," Annals of Operations Research, Springer, vol. 133(1), pages 63-84, January.
    3. Gemayqzel Bouza & Georg Still, 2007. "Mathematical Programs with Complementarity Constraints: Convergence Properties of a Smoothing Method," Mathematics of Operations Research, INFORMS, vol. 32(2), pages 467-483, May.

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