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Stationarity Conditions and Their Reformulations for Mathematical Programs with Vertical Complementarity Constraints

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  • Yan-Chao Liang

    (Dalian University of Technology)

  • Gui-Hua Lin

    (Dalian University of Technology)

Abstract

We consider the mathematical program with vertical complementarity constraints. We show that the min-max-min problems and the problems with max-min constraints can be reformulated as the above problem. As a complement of the work of Scheel and Scholtes in 2000, we derive the Mordukhovich-type stationarity conditions for the considered problem. We further reformulate various popular stationarity systems as nonlinear equations with simple constraints. A modified Levenberg–Marquardt method is employed to solve these constrained equations.

Suggested Citation

  • Yan-Chao Liang & Gui-Hua Lin, 2012. "Stationarity Conditions and Their Reformulations for Mathematical Programs with Vertical Complementarity Constraints," Journal of Optimization Theory and Applications, Springer, vol. 154(1), pages 54-70, July.
    • Handle: RePEc:spr:joptap:v:154:y:2012:i:1:d:10.1007_s10957-012-9992-x
    DOI: 10.1007/s10957-012-9992-x
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    References listed on IDEAS

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    1. E. Polak & J. O. Royset, 2003. "Algorithms for Finite and Semi-Infinite Min–Max–Min Problems Using Adaptive Smoothing Techniques," Journal of Optimization Theory and Applications, Springer, vol. 119(3), pages 421-457, December.
    2. 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.
    3. Hui-juan Xiong & Bo Yu, 2010. "An aggregate deformation homotopy method for min-max-min problems with max-min constraints," Computational Optimization and Applications, Springer, vol. 47(3), pages 501-527, November.
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