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Variational Analysis of Marginal Functions with Applications to Bilevel Programming

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  • Boris S. Mordukhovich

    (Wayne State University)

  • Nguyen Mau Nam

    (University of Texas-Pan American)

  • Hung M. Phan

    (Wayne State University)

Abstract

This paper pursues a twofold goal. First goal is to derive new results on generalized differentiation in variational analysis focusing mainly on a broad class of intrinsically nondifferentiable marginal/value functions. Then the results established in this direction are applied to deriving necessary optimality conditions for the optimistic version of bilevel programs, which occupy a remarkable place in optimization theory and its various applications. We obtain new sets of optimality conditions in both smooth and nonsmooth settings of finite-dimensional and infinite-dimensional spaces.

Suggested Citation

  • Boris S. Mordukhovich & Nguyen Mau Nam & Hung M. Phan, 2012. "Variational Analysis of Marginal Functions with Applications to Bilevel Programming," Journal of Optimization Theory and Applications, Springer, vol. 152(3), pages 557-586, March.
    • Handle: RePEc:spr:joptap:v:152:y:2012:i:3:d:10.1007_s10957-011-9940-1
    DOI: 10.1007/s10957-011-9940-1
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    References listed on IDEAS

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    1. Boris S. Mordukhovich & Nguyen Mau Nam, 2005. "Variational Stability and Marginal Functions via Generalized Differentiation," Mathematics of Operations Research, INFORMS, vol. 30(4), pages 800-816, November.
    2. Jane J. Ye, 2011. "Necessary Optimality Conditions for Multiobjective Bilevel Programs," Mathematics of Operations Research, INFORMS, vol. 36(1), pages 165-184, February.
    3. Joydeep Dutta & Stephan Dempe, 2006. "Bilevel programming with convex lower level problems," Springer Optimization and Its Applications, in: Stephan Dempe & Vyacheslav Kalashnikov (ed.), Optimization with Multivalued Mappings, pages 51-71, Springer.
    4. T. Q. Bao & P. Gupta & B. S. Mordukhovich, 2007. "Necessary Conditions in Multiobjective Optimization with Equilibrium Constraints," Journal of Optimization Theory and Applications, Springer, vol. 135(2), pages 179-203, November.
    5. Jane J. Ye, 2006. "Constraint Qualifications and KKT Conditions for Bilevel Programming Problems," Mathematics of Operations Research, INFORMS, vol. 31(4), pages 811-824, November.
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    Cited by:

    1. Antonino Maugeri & Laura Scrimali, 2016. "A New Approach to Solve Convex Infinite-Dimensional Bilevel Problems: Application to the Pollution Emission Price Problem," Journal of Optimization Theory and Applications, Springer, vol. 169(2), pages 370-387, May.
    2. S. Dempe & N. Gadhi & A. B. Zemkoho, 2013. "New Optimality Conditions for the Semivectorial Bilevel Optimization Problem," Journal of Optimization Theory and Applications, Springer, vol. 157(1), pages 54-74, April.
    3. Mengwei Xu & Jane J. Ye, 2020. "Relaxed constant positive linear dependence constraint qualification and its application to bilevel programs," Journal of Global Optimization, Springer, vol. 78(1), pages 181-205, September.

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