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

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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.

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Acknowledgements

The authors would like to thank the anonymous referees for their careful reading of the paper and constructive suggestions, which have substantially improved the paper in its present form. The first author is supported by Science and Engineering Research Board (SERB), Government of India, under Mathematical Research Impact Centric Support (MATRICS) scheme, through Grant Number MTR/2022/000925. The second author is supported by the Council of Scientific and Industrial Research (CSIR), New Delhi, India, through Grant Number 09/1023(0044)/2021-EMR-I.

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  1. Department of Mathematics, Indian Institute of Technology Patna, Patna, India

    Balendu Bhooshan Upadhyay & Arnav Ghosh

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  1. Balendu Bhooshan Upadhyay
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Correspondence to Balendu Bhooshan Upadhyay.

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Upadhyay, B.B., Ghosh, A. On Constraint Qualifications for Mathematical Programming Problems with Vanishing Constraints on Hadamard Manifolds. J Optim Theory Appl 199, 1–35 (2023). https://doi.org/10.1007/s10957-023-02207-2

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