Coderivative calculus and metric regularity for constraint and variational systems
This paper provides new developments in generalized differentiation theory of variational analysis with their applications to metric regularity of parameterized constraint and variational systems in finite-dimensional and infinite-dimensional spaces. Our approach to the study of metric regularity fo...
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| Published in: | Nonlinear analysis Vol. 70; no. 1; pp. 529 - 552 |
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| Main Authors: | , , |
| Format: | Journal Article |
| Language: | English |
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2009
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| ISSN: | 0362-546X, 1873-5215 |
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| Abstract | This paper provides new developments in
generalized differentiation theory of variational analysis with their applications to
metric regularity of parameterized
constraint and
variational systems in finite-dimensional and infinite-dimensional spaces. Our approach to the study of metric regularity for these two major classes of parametric systems is based on appropriate
coderivative constructions for set-valued mappings and on extended calculus rules supporting their computation and estimation. The main attention is paid in this paper to the so-called
reversed mixed coderivative, which is of crucial importance for efficient
pointwise characterizations of metric regularity in the general framework of set-valued mappings between
infinite-dimensional spaces. We develop new
calculus results for the latter coderivative that allow us to compute it for large classes of parametric constraint and variational systems. On this basis we derive verifiable
sufficient conditions,
necessary conditions as well as
complete characterizations for metric regularity of such systems with computing the corresponding
exact bounds of metric regularity constants/moduli. This approach allows us to reveal general settings in which
metric regularity fails for major classes of parametric variational systems. Furthermore, the developed coderivative calculus leads us also to establishing new formulas for computing the
radius of metric regularity for constraint and variational systems, which characterize the maximal region of preserving metric regularity under linear (and other types of)
perturbations and are closely related to
conditioning aspects of optimization. |
|---|---|
| AbstractList | This paper provides new developments in
generalized differentiation theory of variational analysis with their applications to
metric regularity of parameterized
constraint and
variational systems in finite-dimensional and infinite-dimensional spaces. Our approach to the study of metric regularity for these two major classes of parametric systems is based on appropriate
coderivative constructions for set-valued mappings and on extended calculus rules supporting their computation and estimation. The main attention is paid in this paper to the so-called
reversed mixed coderivative, which is of crucial importance for efficient
pointwise characterizations of metric regularity in the general framework of set-valued mappings between
infinite-dimensional spaces. We develop new
calculus results for the latter coderivative that allow us to compute it for large classes of parametric constraint and variational systems. On this basis we derive verifiable
sufficient conditions,
necessary conditions as well as
complete characterizations for metric regularity of such systems with computing the corresponding
exact bounds of metric regularity constants/moduli. This approach allows us to reveal general settings in which
metric regularity fails for major classes of parametric variational systems. Furthermore, the developed coderivative calculus leads us also to establishing new formulas for computing the
radius of metric regularity for constraint and variational systems, which characterize the maximal region of preserving metric regularity under linear (and other types of)
perturbations and are closely related to
conditioning aspects of optimization. This paper provides new developments in generalized differentiation theory of variational analysis with their applications to metric regularity of parameterized constraint and variational systems in finite-dimensional and infinite-dimensional spaces. Our approach to the study of metric regularity for these two major classes of parametric systems is based on appropriate coderivative constructions for set-valued mappings and on extended calculus rules supporting their computation and estimation. The main attention is paid in this paper to the so-called reversed mixed coderivative, which is of crucial importance for efficient pointwise characterizations of metric regularity in the general framework of set-valued mappings between infinite-dimensional spaces. We develop new calculus results for the latter coderivative that allow us to compute it for large classes of parametric constraint and variational systems. On this basis we derive verifiable sufficient conditions, necessary conditions as well as complete characterizations for metric regularity of such systems with computing the corresponding exact bounds of metric regularity constants/moduli. This approach allows us to reveal general settings in which metric regularity fails for major classes of parametric variational systems. Furthermore, the developed coderivative calculus leads us also to establishing new formulas for computing the radius of metric regularity for constraint and variational systems, which characterize the maximal region of preserving metric regularity under linear (and other types of) perturbations and are closely related to conditioning aspects of optimization. |
| Author | Geremew, W. Nam, N.M. Mordukhovich, B.S. |
| Author_xml | – sequence: 1 givenname: W. surname: Geremew fullname: Geremew, W. email: wondi.geremew@stockton.edu organization: Department of Mathematics, The Richard Stockton College of New Jersey, PO Box 195, Pomona, NJ 08240, USA – sequence: 2 givenname: B.S. surname: Mordukhovich fullname: Mordukhovich, B.S. email: boris@math.wayne.edu organization: Department of Mathematics, Wayne State University, Detroit, MI 48202, USA – sequence: 3 givenname: N.M. surname: Nam fullname: Nam, N.M. email: nguyenmn@utpa.edu organization: Department of Mathematics, The University of Texas-Pan American, 1201 W. University Drive, Edinburg, TX 78539-2999, USA |
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| Cites_doi | 10.2307/2154544 10.1023/B:JOGO.0000026454.56343.b9 10.1137/S1052623400377153 10.1007/BFb0120850 10.1007/s10107-004-0519-6 10.1137/0805026 10.1090/S0002-9947-02-03088-X 10.1016/S0024-3795(03)00392-6 10.1070/RM2000v055n03ABEH000292 |
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| Keywords | Variational analysis and optimization Calculus rules Metric regularity Exact bounds Conditioning Generalized differentiation Coderivatives Nonlinear analysis Optimization method Mathematical model |
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| References | Mordukhovich (b8) 1993; 340 Canovas, Lopez, Parra, Toledo (b2) 2005; 103 Phelps (b15) 1993 Ioffe (b4) 2000; 55 Mordukhovich (b11) 2006 Mordukhovich, Outrata (b12) 2001; 12 Dontchev, Lewis, Rockafellar (b3) 2003; 355 Peña (b14) 2003; 370 Rockafellar, Wets (b18) 1998 Facchinei, Pang (b6) 2003 Borwein, Zhu (b1) 2005 Mordukhovich (b7) 1980; 22 Mordukhovich (b10) 2006 Robinson (b17) 1979; 10 Mordukhovich (b9) 2004; 28 Renegar (b16) 1995; 5 Ioffe (b5) 2003; 32 Outrata (b13) 2006; vol. 485 Peña (10.1016/j.na.2007.12.025_b14) 2003; 370 Mordukhovich (10.1016/j.na.2007.12.025_b11) 2006 Facchinei (10.1016/j.na.2007.12.025_b6) 2003 Ioffe (10.1016/j.na.2007.12.025_b4) 2000; 55 Mordukhovich (10.1016/j.na.2007.12.025_b8) 1993; 340 Mordukhovich (10.1016/j.na.2007.12.025_b9) 2004; 28 Robinson (10.1016/j.na.2007.12.025_b17) 1979; 10 Phelps (10.1016/j.na.2007.12.025_b15) 1993 Mordukhovich (10.1016/j.na.2007.12.025_b7) 1980; 22 Outrata (10.1016/j.na.2007.12.025_b13) 2006; vol. 485 Renegar (10.1016/j.na.2007.12.025_b16) 1995; 5 Canovas (10.1016/j.na.2007.12.025_b2) 2005; 103 Dontchev (10.1016/j.na.2007.12.025_b3) 2003; 355 Ioffe (10.1016/j.na.2007.12.025_b5) 2003; 32 Rockafellar (10.1016/j.na.2007.12.025_b18) 1998 Mordukhovich (10.1016/j.na.2007.12.025_b12) 2001; 12 Borwein (10.1016/j.na.2007.12.025_b1) 2005 Mordukhovich (10.1016/j.na.2007.12.025_b10) 2006 |
| References_xml | – year: 1998 ident: b18 article-title: Variational Analysis – year: 2003 ident: b6 article-title: Finite-Dimensional Variational Inequalities and Complementarity Problems – volume: 355 start-page: 493 year: 2003 end-page: 517 ident: b3 article-title: The radius of metric regularity publication-title: Trans. Amer. Math. Soc. – year: 1993 ident: b15 article-title: Convex Functions, Monotone Operators and Differentiability – volume: vol. 485 start-page: 221 year: 2006 end-page: 274 ident: b13 article-title: Mathematical programs with equilibrium constraints: Theory and numerical methods publication-title: Nonsmooth Mechanics of Solids – year: 2006 ident: b11 article-title: Variational Analysis and Generalized Differentiation, II: Applications – volume: 10 start-page: 128 year: 1979 end-page: 141 ident: b17 article-title: Generalized equations and their solutions, I: Basic theory publication-title: Math. Prog. 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generalized differentiation theory of variational analysis with their applications to
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| SubjectTerms | Algebra Calculus of variations and optimal control Calculus rules Coderivatives Conditioning Exact bounds Exact sciences and technology Generalized differentiation Global analysis, analysis on manifolds Linear and multilinear algebra, matrix theory Mathematical analysis Mathematics Metric regularity Numerical analysis Numerical analysis. Scientific computation Numerical methods in mathematical programming, optimization and calculus of variations Sciences and techniques of general use Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds Variational analysis and optimization |
| Title | Coderivative calculus and metric regularity for constraint and variational systems |
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