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Optimality conditions and duality for semi-infinite mathematical programming problems with equilibrium constraints, using convexificators

Author

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  • Yogendra Pandey

    (Indian Institute of Technology)

  • S. K. Mishra

    (Banaras Hindu University)

Abstract

In this paper, we consider semi-infinite mathematical programming problems with equilibrium constraints (SIMPEC). We establish necessary and sufficient optimality conditions for the SIMPEC, using convexificators. We study the Wolfe type dual problem for the SIMPEC under the $$\partial ^{*}$$ ∂ ∗ -convexity assumption. A Mond–Weir type dual problem is also formulated and studied for the SIMPEC under the $$\partial ^{*}$$ ∂ ∗ -convexity, $$\partial ^{*}$$ ∂ ∗ -pseudoconvexity and $$\partial ^{*}$$ ∂ ∗ -quasiconvexity assumptions. Weak duality theorems are established to relate the SIMPEC and two dual programs in the framework of convexificators. Further, strong duality theorems are obtained under generalized standard Abadie constraint qualification.

Suggested Citation

  • 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.
  • Handle: RePEc:spr:annopr:v:269:y:2018:i:1:d:10.1007_s10479-017-2422-6
    DOI: 10.1007/s10479-017-2422-6
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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. Yohan Shim & Marte Fodstad & Steven Gabriel & Asgeir Tomasgard, 2013. "A branch-and-bound method for discretely-constrained mathematical programs with equilibrium constraints," Annals of Operations Research, Springer, vol. 210(1), pages 5-31, November.
    5. 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.
    6. Benoît Colson & Patrice Marcotte & Gilles Savard, 2007. "An overview of bilevel optimization," Annals of Operations Research, Springer, vol. 153(1), pages 235-256, September.
    7. Adil Bagirov & Napsu Karmitsa & Marko M. Mäkelä, 2014. "Introduction to Nonsmooth Optimization," Springer Books, Springer, edition 127, number 978-3-319-08114-4, October.
    8. S. Dempe & A. Zemkoho, 2012. "Bilevel road pricing: theoretical analysis and optimality conditions," Annals of Operations Research, Springer, vol. 196(1), pages 223-240, July.
    9. 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.
    10. 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.
    11. 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.
    12. 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.
    13. Lopez, Marco & Still, Georg, 2007. "Semi-infinite programming," European Journal of Operational Research, Elsevier, vol. 180(2), pages 491-518, July.
    14. S. K. Mishra & Vinay Singh & Vivek Laha, 2016. "On duality for mathematical programs with vanishing constraints," Annals of Operations Research, Springer, vol. 243(1), pages 249-272, August.
    15. 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.
    16. 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.
    17. 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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