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Constraint Qualifications in Nonsmooth Multiobjective Optimization

Author

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  • X. F. Li

    (Jilin University of Technology
    City University of Hong Kong)

Abstract

For an inequality constrained nonsmooth multiobjective optimization problem involving locally Lipschitz functions, stronger KT-type necessary conditions and KT necessary conditions (which in the continuously differentiable case reduce respectively to the stronger KT conditions studied recently by Maeda and the usual KT conditions) are derived for efficiency and weak efficiency under several constraint qualifications. Stimulated by the stronger KT-type conditions, the notion of core of the convex hull of the union of finitely many convex sets is introduced. As main tool in the derivation of the necessary conditions, a theorem of the alternatives and a core separation theorem are also developed which are respectively extensions of the Motzkin transposition theorem and the Tucker theorem.

Suggested Citation

  • X. F. Li, 2000. "Constraint Qualifications in Nonsmooth Multiobjective Optimization," Journal of Optimization Theory and Applications, Springer, vol. 106(2), pages 373-398, August.
    • Handle: RePEc:spr:joptap:v:106:y:2000:i:2:d:10.1023_a:1004607615343
    DOI: 10.1023/A:1004607615343
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    Citations

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    Cited by:

    1. Do Van Luu & Manh-Hung Nguyen, 2006. "On alternative theorems and necessary conditions for efficiency," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00112454, HAL.
    2. Min Feng & Shengjie Li & Jie Wang, 2022. "On Tucker-Type Alternative Theorems and Necessary Optimality Conditions for Nonsmooth Multiobjective Optimization," Journal of Optimization Theory and Applications, Springer, vol. 195(2), pages 480-503, November.
    3. C. Cromvik & M. Patriksson, 2010. "On the Robustness of Global Optima and Stationary Solutions to Stochastic Mathematical Programs with Equilibrium Constraints, Part 1: Theory," Journal of Optimization Theory and Applications, Springer, vol. 144(3), pages 461-478, March.
    4. Balendu Bhooshan Upadhyay & Shubham Kumar Singh & Ioan Stancu-Minasian, 2024. "Constraint Qualifications and Optimality Conditions for Nonsmooth Semidefinite Multiobjective Programming Problems with Mixed Constraints Using Convexificators," Mathematics, MDPI, vol. 12(20), pages 1-21, October.
    5. Manh-Hung Nguyen & Do Van Luu, 2006. "On necessary conditions for efficiency in directionally differentiable optimization problems," Post-Print halshs-00118977, HAL.
    6. S. Nobakhtian, 2008. "Generalized (F,ρ)-Convexity and Duality in Nonsmooth Problems of Multiobjective Optimization," Journal of Optimization Theory and Applications, Springer, vol. 136(1), pages 61-68, January.
    7. 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.
    8. Ali Sadeghieh & Nader Kanzi & Giuseppe Caristi & David Barilla, 2022. "On stationarity for nonsmooth multiobjective problems with vanishing constraints," Journal of Global Optimization, Springer, vol. 82(4), pages 929-949, April.
    9. Hui Huang & Haole Zhu, 2022. "Stationary Condition for Borwein Proper Efficient Solutions of Nonsmooth Multiobjective Problems with Vanishing Constraints," Mathematics, MDPI, vol. 10(23), pages 1-18, December.

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