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Stability of Implicit Multifunctions via Point-Based Criteria and Applications

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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.

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Acknowledgements

The author would like to thank the referees for the valuable comments and suggestions which have improved the final preparation of the paper.

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Authors and Affiliations

  1. Optimization and Applications Research Group, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Thai Doan Chuong

  2. Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Thai Doan Chuong

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  1. Thai Doan Chuong
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Correspondence to Thai Doan Chuong.

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Communicated by Regina S. Burachik.

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Dedicated to Professor Do Sang Kim on the occasion of his 65th birthday.

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Chuong, T.D. Stability of Implicit Multifunctions via Point-Based Criteria and Applications. J Optim Theory Appl 183, 920–943 (2019). https://doi.org/10.1007/s10957-019-01562-3

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