Subgradient of distance functions with applications to Lipschitzian stability

The paper is devoted to studying generalized differential properties of distance functions that play a remarkable role in variational analysis, optimization, and their applications. The main object under consideration is the distance function of two variables in Banach spaces that signifies the dist...

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Bibliographic Details
Published in:Mathematical programming Vol. 104; no. 2-3; pp. 635 - 668
Main Authors: Mordukhovich, Boris S., Nam, Nguyen Mau
Format: Journal Article
Language:English
Published: Heidelberg Springer 01.11.2005
Springer Nature B.V
Subjects:
Applied sciences
Approximation
Banach spaces
Digital Object Identifier
Exact sciences and technology
Mapping
Mathematical programming
Operational research and scientific management
Operational research. Management science
Optimization
Studies
Variational analysis and optimization
Subgradient optimization
Fix point
Subdifferential
Parametric programming
Lipschitzian stability
Banach space
Generalized differentiation
Constrained optimization
Mathematical programming
Distance functions
ISSN:0025-5610, 1436-4646
Online Access:Get full text
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Summary:The paper is devoted to studying generalized differential properties of distance functions that play a remarkable role in variational analysis, optimization, and their applications. The main object under consideration is the distance function of two variables in Banach spaces that signifies the distance from a point to a moving set. We derive various relationships between Frechet-type subgradients and limiting (basic and singular) subgradients of this distance function and corresponding generalized normals to sets and coderivatives of set-valued mappings. These relationships are essentially different depending on whether or not the reference point belongs to the graph of the involved set-valued mapping. Our major results are new even for subdifferentiation of the standard distance function signifying the distance between a point and a fixed set in finite-dimensional spaces. The subdifferential results obtained are applied to deriving efficient dual-space conditions for the local Lipschitz continuity of distance functions generated by set-valued mappings, in particular, by those arising in parametric constrained optimization. [PUBLICATION ABSTRACT]
Bibliography:SourceType-Scholarly Journals-1
ObjectType-Feature-1
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ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-005-0632-1