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Necessary Optimality Conditions in Terms of Convexificators in Lipschitz Optimization

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  • X. F. Li

    (Jilin University)

  • J. Z. Zhang

    (City University of Hong Kong)

Abstract

This study is devoted to constraint qualifications and Kuhn-Tucker type necessary optimality conditions for nonsmooth optimization problems involving locally Lipschitz functions. The main tool of the study is the concept of convexificators. First, the case of a minimization problem in the presence of an arbitrary set constraint is considered by using the contingent cone and the adjacent cone to the constraint set. Then, in the case of a minimization problem with inequality constraints, Abadie type constraint qualifications and several other qualifications are proposed; Kuhn-Tucker type necessary optimality conditions are derived under the qualifications.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:131:y:2006:i:3:d:10.1007_s10957-006-9155-z
    DOI: 10.1007/s10957-006-9155-z
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    References listed on IDEAS

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    1. ,, 2001. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 17(6), pages 1157-1160, December.
    2. D. H. Martin & G. G. Watkins, 1985. "Cores of Tangent Cones and Clarke's Tangent Cone," Mathematics of Operations Research, INFORMS, vol. 10(4), pages 565-575, November.
    3. V. Preda & I. Chiţescu, 1999. "On Constraint Qualification in Multiobjective Optimization Problems: Semidifferentiable Case," Journal of Optimization Theory and Applications, Springer, vol. 100(2), pages 417-433, February.
    4. 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.
    5. ,, 2001. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 17(5), pages 1025-1031, October.
    6. J. Dutta & S. Chandra, 2002. "Convexifactors, Generalized Convexity, and Optimality Conditions," Journal of Optimization Theory and Applications, Springer, vol. 113(1), pages 41-64, April.
    7. D. E. Ward & G. M. Lee, 2001. "Generalized properly efficient solutions of vector optimization problems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 53(2), pages 215-232, June.
    8. X. F. Li, 2000. "Constraint Qualifications in Nonsmooth Multiobjective Optimization," Journal of Optimization Theory and Applications, Springer, vol. 106(2), pages 373-398, August.
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    Cited by:

    1. 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.
    2. Bhawna Kohli, 2012. "Optimality Conditions for Optimistic Bilevel Programming Problem Using Convexifactors," Journal of Optimization Theory and Applications, Springer, vol. 152(3), pages 632-651, March.
    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. 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.

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