Lipschitz-like property relative to a set and the generalized Mordukhovich criterion

In this paper we will establish some necessary condition and sufficient condition respectively for a set-valued mapping to have the Lipschitz-like property relative to a closed set by employing regular normal cone and limiting normal cone of a restricted graph of the set-valued mapping. We will obta...

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Bibliographic Details
Published in:Mathematical programming Vol. 189; no. 1-2; pp. 455 - 489
Main Authors: Meng, K. W., Li, M. H., Yao, W. F., Yang, X. Q.
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.09.2021
Springer Nature B.V
Subjects:
Calculus of Variations and Optimal Control; Optimization
Combinatorics
Criteria
Full Length Paper
Graphical representations
Mapping
Mathematical and Computational Physics
Mathematical functions
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Mathematics of Computing
Numerical Analysis
Theoretical
90C31
90C30
49J53
Mordukhovich criterion
Lipschitz-like property relative to a set
Graphical modulus
Projectional coderivative
Level-set mapping
ISSN:0025-5610, 1436-4646
Online Access:Get full text
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Summary:In this paper we will establish some necessary condition and sufficient condition respectively for a set-valued mapping to have the Lipschitz-like property relative to a closed set by employing regular normal cone and limiting normal cone of a restricted graph of the set-valued mapping. We will obtain a complete characterization for a set-valued mapping to have the Lipschitz-property relative to a closed and convex set by virtue of the projection of the coderivative onto a tangent cone. Furthermore, by introducing a projectional coderivative of set-valued mappings, we establish a verifiable generalized Mordukhovich criterion for the Lipschitz-like property relative to a closed and convex set. We will study the representation of the graphical modulus of a set-valued mapping relative to a closed and convex set by using the outer norm of the corresponding projectional coderivative value. For an extended real-valued function, we will apply the obtained results to investigate its Lipschitz continuity relative to a closed and convex set and the Lipschitz-like property of a level-set mapping relative to a half line.
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ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-020-01568-0