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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Veröffentlicht in:	Mathematical programming Jg. 104; H. 2-3; S. 635 - 668
Hauptverfasser:	Mordukhovich, Boris S., Nam, Nguyen Mau
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	Heidelberg Springer 01.11.2005 Springer Nature B.V
Schlagworte:	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
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Abstract	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]
AbstractList	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]
Author	Mordukhovich, Boris S. Nam, Nguyen Mau
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Cites_doi	10.1007/b97543 10.1023/A:1011291608129 10.1090/S0002-9947-1987-0887515-5 10.1090/S0002-9947-96-01543-7 10.1016/0362-546X(85)90024-0 10.1137/S1052623402414701 10.1007/BF01033643 10.1016/0022-247X(92)90336-C 10.1007/978-3-642-02431-3 10.1090/S0002-9947-1993-1156300-4 10.1112/jlms/s2-41.1.175 10.1137/0329056 10.1023/B:JOGO.0000026454.56343.b9 10.1287/moor.9.1.87 10.1016/0021-8928(76)90136-2 10.1016/S0022-0396(03)00129-3 10.1016/0022-247X(74)90025-0 10.1016/S0362-546X(00)00183-8
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Keywords	Variational analysis and optimization Subgradient optimization Fix point Subdifferential Parametric programming Lipschitzian stability Banach space Generalized differentiation Constrained optimization Mathematical programming Distance functions
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SubjectTerms	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
Title	Subgradient of distance functions with applications to Lipschitzian stability
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