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On Constraint Qualifications in Nonsmooth Optimization

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  • O. Stein

    (RWTH Aachen University)

Abstract

We introduce extensions of the Mangasarian-Fromovitz and Abadie constraint qualifications to nonsmooth optimization problems with feasibility given by means of lower-level sets. We do not assume directional differentiability, but only upper semicontinuity of the defining functions. By deriving and reviewing primal first-order optimality conditions for nonsmooth problems, we motivate the formulations of the constraint qualifications. Then, we study their interrelation, and we show how they are related to the Slater condition for nonsmooth convex problems, to nonsmooth reverse-convex problems, to the stability of parametric feasible set mappings, and to alternative theorems for the derivation of dual first-order optimality conditions. In the literature on general semi-infinite programming problems, a number of formally different extensions of the Mangasarian-Fromovitz constraint qualification have been introduced recently under different structural assumptions. We show that all these extensions are unified by the constraint qualification presented here.

Suggested Citation

  • O. Stein, 2004. "On Constraint Qualifications in Nonsmooth Optimization," Journal of Optimization Theory and Applications, Springer, vol. 121(3), pages 647-671, June.
    • Handle: RePEc:spr:joptap:v:121:y:2004:i:3:d:10.1023_b:jota.0000037607.48762.45
    DOI: 10.1023/B:JOTA.0000037607.48762.45
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    References listed on IDEAS

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    1. Oliver Stein, 2001. "First-Order Optimality Conditions for Degenerate Index Sets in Generalized Semi-Infinite Optimization," Mathematics of Operations Research, INFORMS, vol. 26(3), pages 565-582, August.
    2. G. Stein & G. Still, 2000. "On Optimality Conditions for Generalized Semi-Infinite Programming Problems," Journal of Optimization Theory and Applications, Springer, vol. 104(2), pages 443-458, February.
    3. J. J. Rückmann & A. Shapiro, 1999. "First-Order Optimality Conditions in Generalized Semi-Infinite Programming," Journal of Optimization Theory and Applications, Springer, vol. 101(3), pages 677-691, June.
    4. William Hogan, 1973. "Directional Derivatives for Extremal-Value Functions with Applications to the Completely Convex Case," Operations Research, INFORMS, vol. 21(1), pages 188-209, February.
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    Cited by:

    1. Xiaoqi Yang & Zhangyou Chen & Jinchuan Zhou, 2016. "Optimality Conditions for Semi-Infinite and Generalized Semi-Infinite Programs Via Lower Order Exact Penalty Functions," Journal of Optimization Theory and Applications, Springer, vol. 169(3), pages 984-1012, June.
    2. Oliver Stein & Nathan Sudermann-Merx, 2014. "On smoothness properties of optimal value functions at the boundary of their domain under complete convexity," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 79(3), pages 327-352, June.
    3. Nader Kanzi, 2011. "Necessary optimality conditions for nonsmooth semi-infinite programming problems," Journal of Global Optimization, Springer, vol. 49(4), pages 713-725, April.
    4. Li-Ping Pang & Jian Lv & Jin-He Wang, 2016. "Constrained incremental bundle method with partial inexact oracle for nonsmooth convex semi-infinite programming problems," Computational Optimization and Applications, Springer, vol. 64(2), pages 433-465, June.
    5. 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.
    6. Jian Lv & Li-Ping Pang & Fan-Yun Meng, 2018. "A proximal bundle method for constrained nonsmooth nonconvex optimization with inexact information," Journal of Global Optimization, Springer, vol. 70(3), pages 517-549, March.
    7. J. J. Ye & S. Y. Wu, 2008. "First Order Optimality Conditions for Generalized Semi-Infinite Programming Problems," Journal of Optimization Theory and Applications, Springer, vol. 137(2), pages 419-434, May.

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