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Stability of Implicit Multifunctions via Point-Based Criteria and Applications

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  • Thai Doan Chuong

    (Ton Duc Thang University
    Ton Duc Thang University)

Abstract

We first establish new point-based sufficient conditions for an implicit multifunction to achieve the metric subregularity. These conditions are expressed in terms of limit set critical for metric subregularity of the corresponding parametric multifunction formulated the implicit multifunction. We then show that the sufficient conditions obtained turn out to be also necessary for the metric subregularity of the implicit multifunction in the case, where the corresponding parametric multifunction is (locally) convex and closed. In this way, we give criteria ensuring the calmness for the implicit multifunction. As applications, we derive point-based sufficient and necessary conditions for a multifunction (resp., its inverse multifunction) to have the metric subregularity (resp., the calmness) and for the efficient solution map of a parametric vector optimization problem to admit the metric subregularity as well as the calmness.

Suggested Citation

  • Thai Doan Chuong, 2019. "Stability of Implicit Multifunctions via Point-Based Criteria and Applications," Journal of Optimization Theory and Applications, Springer, vol. 183(3), pages 920-943, December.
    • Handle: RePEc:spr:joptap:v:183:y:2019:i:3:d:10.1007_s10957-019-01562-3
    DOI: 10.1007/s10957-019-01562-3
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    References listed on IDEAS

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    1. Thai Doan Chuong & Do Sang Kim, 2016. "Hölder-Like Property and Metric Regularity of a Positive-Order for Implicit Multifunctions," Mathematics of Operations Research, INFORMS, vol. 41(2), pages 596-611, May.
    2. N. Q. Huy & D. S. Kim & K. V. Ninh, 2012. "Stability of Implicit Multifunctions in Banach Spaces," Journal of Optimization Theory and Applications, Springer, vol. 155(2), pages 558-571, November.
    3. T. Chuong & A. Kruger & J.-C. Yao, 2011. "Calmness of efficient solution maps in parametric vector optimization," Journal of Global Optimization, Springer, vol. 51(4), pages 677-688, December.
    4. Huynh Van Ngai & Phan Nhat Tinh, 2015. "Metric Subregularity of Multifunctions: First and Second Order Infinitesimal Characterizations," Mathematics of Operations Research, INFORMS, vol. 40(3), pages 703-724, March.
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