Extending a Continuous Convex Function Over a Closed Subset

• Published: 02 June 2020

Lipschitz Continuity of Convex Functions

Applied Mathematics & Optimization volume 84,pages 1623–1640 (2021)Cite this article

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Abstract

We provide some necessary and sufficient conditions for a proper lower semicontinuous convex function, defined on a real Banach space, to be locally or globally Lipschitz continuous. Our criteria rely on the existence of a bounded selection of the subdifferential mapping and the intersections of the subdifferential mapping and the normal cone operator to the domain of the given function. Moreover, we also point out that the Lipschitz continuity of the given function on an open and bounded (not necessarily convex) set can be characterized via the existence of a bounded selection of the subdifferential mapping on the boundary of the given set and as a consequence it is equivalent to the local Lipschitz continuity at every point on the boundary of that set. Our results are applied to extend a Lipschitz and convex function to the whole space and to study the Lipschitz continuity of its Moreau envelope functions.

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Acknowledgements

The authors are grateful to the editors and two anonymous referees for constructive comments and suggestions, which greatly improved the paper. Pham Duy Khanh was supported, in part, by the Fondecyt Postdoc Project 3180080, the Basal Program CMM-AFB 170001 from CONICYT-Chile, and the National Foundation for Science and Technology Development (NAFOSTED) under Grant Number 101.01-2017.325.

Author information

Authors and Affiliations

1. Universidad de O'Higgins, Rancagua, Chile

   Bao Tran Nguyen

2. Quy Nhon University, Quy Nhon, Vietnam

   Bao Tran Nguyen

3. Department of Mathematics, HCMC University of Education, Ho Chi Minh, Vietnam

   Pham Duy Khanh

4. Center for Mathematical Modeling, Universidad de Chile, Santiago, Chile

   Pham Duy Khanh

Corresponding author

Correspondence to Pham Duy Khanh.

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Nguyen, B.T., Khanh, P.D. Lipschitz Continuity of Convex Functions. Appl Math Optim 84, 1623–1640 (2021). https://doi.org/10.1007/s00245-020-09689-w

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• Published: 02 June 2020

• Issue Date: October 2021

• DOI : https://doi.org/10.1007/s00245-020-09689-w

Keywords

• Convex function
• Lipschitz continuity
• Calmness
• Subdifferential
• Normal cone
• Moreau envelope function

Mathematics Subject Classification

• 26A16
• 46N10
• 52A41

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