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Necessary Conditions in Multiobjective Optimization with Equilibrium Constraints

Author

Listed:
  • T. Q. Bao

    (Wayne State University)

  • P. Gupta

    (University of Delhi)

  • B. S. Mordukhovich

    (Wayne State University)

Abstract

We study multiobjective optimization problems with equilibrium constraints (MOPECs) described by parametric generalized equations in the form $$0\in G(x,y)+Q(x,y),$$ where both mappings G and Q are set-valued. Such models arise particularly from certain optimization-related problems governed by variational inequalities and first-order optimality conditions in nondifferentiable programming. We establish verifiable necessary conditions for the general problems under consideration and for their important specifications by using modern tools of variational analysis and generalized differentiation. The application of the obtained necessary optimality conditions is illustrated by a numerical example from bilevel programming with convex while nondifferentiable data.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:135:y:2007:i:2:d:10.1007_s10957-007-9209-x
    DOI: 10.1007/s10957-007-9209-x
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    References listed on IDEAS

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    1. J. V. Outrata, 1999. "Optimality Conditions for a Class of Mathematical Programs with Equilibrium Constraints," Mathematics of Operations Research, INFORMS, vol. 24(3), pages 627-644, August.
    2. D. Aussel & N. Hadjisavvas, 2004. "On Quasimonotone Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 121(2), pages 445-450, May.
    3. J. J. Ye & X. Y. Ye, 1997. "Necessary Optimality Conditions for Optimization Problems with Variational Inequality Constraints," Mathematics of Operations Research, INFORMS, vol. 22(4), pages 977-997, November.
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    Cited by:

    1. Aram V. Arutyunov & Boris S. Mordukhovich & Sergey E. Zhukovskiy, 2023. "Coincidence Points of Parameterized Generalized Equations with Applications to Optimal Value Functions," Journal of Optimization Theory and Applications, Springer, vol. 196(1), pages 177-198, January.
    2. M. Durea & R. Strugariu, 2011. "On parametric vector optimization via metric regularity of constraint systems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 74(3), pages 409-425, December.
    3. L. Q. Anh & P. Q. Khanh & D. T. M. Van, 2012. "Well-Posedness Under Relaxed Semicontinuity for Bilevel Equilibrium and Optimization Problems with Equilibrium Constraints," Journal of Optimization Theory and Applications, Springer, vol. 153(1), pages 42-59, April.
    4. Nguyen Quang Huy & Do Sang Kim & Nguyen Van Tuyen, 2017. "Existence Theorems in Vector Optimization with Generalized Order," Journal of Optimization Theory and Applications, Springer, vol. 174(3), pages 728-745, September.
    5. Thai Doan Chuong, 2020. "Optimality conditions for nonsmooth multiobjective bilevel optimization problems," Annals of Operations Research, Springer, vol. 287(2), pages 617-642, April.
    6. Vo Duc Thinh & Thai Doan Chuong, 2018. "Directionally generalized differentiation for multifunctions and applications to set-valued programming problems," Annals of Operations Research, Springer, vol. 269(1), pages 727-751, October.
    7. N. Gadhi & S. Dempe, 2012. "Necessary Optimality Conditions and a New Approach to Multiobjective Bilevel Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 155(1), pages 100-114, October.
    8. Thai Doan Chuong & Do Sang Kim, 2016. "A class of nonsmooth fractional multiobjective optimization problems," Annals of Operations Research, Springer, vol. 244(2), pages 367-383, September.
    9. Jane J. Ye, 2011. "Necessary Optimality Conditions for Multiobjective Bilevel Programs," Mathematics of Operations Research, INFORMS, vol. 36(1), pages 165-184, February.
    10. 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.
    11. P. Khanh & L. Tung, 2015. "First- and second-order optimality conditions for multiobjective fractional programming," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 23(2), pages 419-440, July.
    12. A. J. Zaslavski, 2009. "Existence of Solutions of a Vector Optimization Problem with a Generic Lower Semicontinuous Objective Function," Journal of Optimization Theory and Applications, Springer, vol. 141(1), pages 217-230, April.

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