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On Constraint Qualifications for Mathematical Programming Problems with Vanishing Constraints on Hadamard Manifolds

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  • Balendu Bhooshan Upadhyay

    (Indian Institute of Technology Patna)

  • Arnav Ghosh

    (Indian Institute of Technology Patna)

Abstract

This article is devoted to the study of mathematical programming problems with vanishing constraints on Hadamard manifolds (in short, MPVC-HM). We present the Abadie constraint qualification (in short, ACQ) and (MPVC-HM)-tailored ACQ for MPVC-HM and provide some necessary conditions for the satisfaction of ACQ for MPVC-HM. Moreover, we demonstrate that the Guignard constraint qualification (in short, GCQ) is satisfied for MPVC-HM under certain mild restrictions. We introduce several (MPVC-HM)-tailored constraint qualifications in the framework of Hadamard manifolds that ensure satisfaction of GCQ. Moreover, we refine our analysis and present some modified sufficient conditions which guarantee that GCQ is satisfied. Several non-trivial examples are incorporated to illustrate the significance of the derived results. To the best of our knowledge, constraint qualifications for mathematical programming problems with vanishing constraints in manifold setting have not been explored before.

Suggested Citation

  • Balendu Bhooshan Upadhyay & Arnav Ghosh, 2023. "On Constraint Qualifications for Mathematical Programming Problems with Vanishing Constraints on Hadamard Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 199(1), pages 1-35, October.
    • Handle: RePEc:spr:joptap:v:199:y:2023:i:1:d:10.1007_s10957-023-02207-2
    DOI: 10.1007/s10957-023-02207-2
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    References listed on IDEAS

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    1. M.L. Flegel & C. Kanzow, 2005. "Abadie-Type Constraint Qualification for Mathematical Programs with Equilibrium Constraints," Journal of Optimization Theory and Applications, Springer, vol. 124(3), pages 595-614, March.
    2. 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.
    3. Mohammad Mahdi Karkhaneei & Nezam Mahdavi-Amiri, 2019. "Nonconvex Weak Sharp Minima on Riemannian Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 183(1), pages 85-104, October.
    4. Lei Guo & Gui-Hua Lin, 2013. "Notes on Some Constraint Qualifications for Mathematical Programs with Equilibrium Constraints," Journal of Optimization Theory and Applications, Springer, vol. 156(3), pages 600-616, March.
    5. Qingjie Hu & Jiguang Wang & Yu Chen, 2020. "New dualities for mathematical programs with vanishing constraints," Annals of Operations Research, Springer, vol. 287(1), pages 233-255, April.
    6. Glaydston C. Bento & Jefferson G. Melo, 2012. "Subgradient Method for Convex Feasibility on Riemannian Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 152(3), pages 773-785, March.
    7. S. K. Mishra & Vinay Singh & Vivek Laha, 2016. "On duality for mathematical programs with vanishing constraints," Annals of Operations Research, Springer, vol. 243(1), pages 249-272, August.
    8. Savin Treanţă & Balendu Bhooshan Upadhyay & Arnav Ghosh & Kamsing Nonlaopon, 2022. "Optimality Conditions for Multiobjective Mathematical Programming Problems with Equilibrium Constraints on Hadamard Manifolds," Mathematics, MDPI, vol. 10(19), pages 1-20, September.
    9. Dominik Dorsch & Vladimir Shikhman & Oliver Stein, 2012. "Mathematical programs with vanishing constraints: critical point theory," Journal of Global Optimization, Springer, vol. 52(3), pages 591-605, March.
    10. Sajjad Kazemi & Nader Kanzi, 2018. "Constraint Qualifications and Stationary Conditions for Mathematical Programming with Non-differentiable Vanishing Constraints," Journal of Optimization Theory and Applications, Springer, vol. 179(3), pages 800-819, December.
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    Cited by:

    1. 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.
    2. Harry Oviedo, 2023. "Proximal Point Algorithm with Euclidean Distance on the Stiefel Manifold," Mathematics, MDPI, vol. 11(11), pages 1-17, May.
    3. Arnav Ghosh & Balendu Bhooshan Upadhyay & I. M. Stancu-Minasian, 2023. "Pareto Efficiency Criteria and Duality for Multiobjective Fractional Programming Problems with Equilibrium Constraints on Hadamard Manifolds," Mathematics, MDPI, vol. 11(17), pages 1-28, August.
    4. Balendu Bhooshan Upadhyay & Arnav Ghosh & Savin Treanţă, 2024. "Efficiency conditions and duality for multiobjective semi-infinite programming problems on Hadamard manifolds," Journal of Global Optimization, Springer, vol. 89(3), pages 723-744, July.
    5. Balendu Bhooshan Upadhyay & Arnav Ghosh & Savin Treanţă & Jen-Chih Yao, 2024. "Constraint Qualifications and Optimality Conditions for Multiobjective Mathematical Programming Problems with Vanishing Constraints on Hadamard Manifolds," Mathematics, MDPI, vol. 12(19), pages 1-24, September.

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