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Duality for Nonsmooth Optimization Problems with Equilibrium Constraints, Using Convexificators

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  • Yogendra Pandey

    (Banaras Hindu University)

  • Shashi Kant Mishra

    (Banaras Hindu University)

Abstract

In this paper, we consider optimization problems with equilibrium constraints. We study the Wolfe-type dual problem for the optimization problems with equilibrium constraints under the convexity assumptions using convexificators. A Mond–Weir-type dual problem is also formulated and studied for the optimization problems with equilibrium constraints under convexity and generalized convexity assumptions using convexificators. Weak duality theorems are established to relate the optimization problems with equilibrium constraints and two dual programs in the framework of convexificators. Further, strong duality theorems are obtained under generalized standard Abadie constraint qualification.

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  • Yogendra Pandey & Shashi Kant Mishra, 2016. "Duality for Nonsmooth Optimization Problems with Equilibrium Constraints, Using Convexificators," Journal of Optimization Theory and Applications, Springer, vol. 171(2), pages 694-707, November.
    • Handle: RePEc:spr:joptap:v:171:y:2016:i:2:d:10.1007_s10957-016-0885-2
    DOI: 10.1007/s10957-016-0885-2
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    References listed on IDEAS

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    1. Lei Guo & Gui-Hua Lin, 2013. "Notes on Some Constraint Qualifications for Mathematical Programs with Equilibrium Constraints," Journal of Optimization Theory and Applications, Springer, vol. 156(3), pages 600-616, March.
    2. Lei Guo & Gui-Hua Lin & Jane J. Ye, 2015. "Solving Mathematical Programs with Equilibrium Constraints," Journal of Optimization Theory and Applications, Springer, vol. 166(1), pages 234-256, July.
    3. V. Jeyakumar & D. T. Luc, 1999. "Nonsmooth Calculus, Minimality, and Monotonicity of Convexificators," Journal of Optimization Theory and Applications, Springer, vol. 101(3), pages 599-621, June.
    4. Nguyen Huy Chieu & Gue Myung Lee, 2014. "Constraint Qualifications for Mathematical Programs with Equilibrium Constraints and their Local Preservation Property," Journal of Optimization Theory and Applications, Springer, vol. 163(3), pages 755-776, December.
    5. Jane J. Ye & Jin Zhang, 2014. "Enhanced Karush–Kuhn–Tucker Conditions for Mathematical Programs with Equilibrium Constraints," Journal of Optimization Theory and Applications, Springer, vol. 163(3), pages 777-794, December.
    6. S. K. Suneja & B. Kohli, 2011. "Optimality and Duality Results for Bilevel Programming Problem Using Convexifactors," Journal of Optimization Theory and Applications, Springer, vol. 150(1), pages 1-19, July.
    7. M.L. Flegel & C. Kanzow, 2005. "Abadie-Type Constraint Qualification for Mathematical Programs with Equilibrium Constraints," Journal of Optimization Theory and Applications, Springer, vol. 124(3), pages 595-614, March.
    8. X. F. Li & J. Z. Zhang, 2006. "Necessary Optimality Conditions in Terms of Convexificators in Lipschitz Optimization," Journal of Optimization Theory and Applications, Springer, vol. 131(3), pages 429-452, December.
    9. Nguyen Huy Chieu & Gue Myung Lee, 2013. "A Relaxed Constant Positive Linear Dependence Constraint Qualification for Mathematical Programs with Equilibrium Constraints," Journal of Optimization Theory and Applications, Springer, vol. 158(1), pages 11-32, July.
    10. Lei Guo & Gui-Hua Lin & Jane J. Ye, 2013. "Second-Order Optimality Conditions for Mathematical Programs with Equilibrium Constraints," Journal of Optimization Theory and Applications, Springer, vol. 158(1), pages 33-64, July.
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    Cited by:

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    3. Yogendra Pandey & S. K. Mishra, 2018. "Optimality conditions and duality for semi-infinite mathematical programming problems with equilibrium constraints, using convexificators," Annals of Operations Research, Springer, vol. 269(1), pages 549-564, October.
    4. Qingjie Hu & Jiguang Wang & Yu Chen, 2020. "New dualities for mathematical programs with vanishing constraints," Annals of Operations Research, Springer, vol. 287(1), pages 233-255, April.
    5. Bhuwan Chandra Joshi, 2021. "Some Results on Mathematical Programs with Equilibrium Constraints," SN Operations Research Forum, Springer, vol. 2(4), pages 1-18, December.
    6. Kin Keung Lai & Shashi Kant Mishra & Sanjeev Kumar Singh & Mohd Hassan, 2022. "Stationary Conditions and Characterizations of Solution Sets for Interval-Valued Tightened Nonlinear Problems," Mathematics, MDPI, vol. 10(15), pages 1-16, August.
    7. Tran Van Su, 2023. "Optimality and duality for nonsmooth mathematical programming problems with equilibrium constraints," Journal of Global Optimization, Springer, vol. 85(3), pages 663-685, March.

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