Second-Order Analysis of Polyhedral Systems in Finite and Infinite Dimensions with Applications to Robust Stability of Variational Inequalities
This paper concerns second-order analysis for a remarkable class of variational systems in finite-dimensional and infinite-dimensional spaces, which is particularly important for the study of optimization and equilibrium problems with equilibrium constraints. Systems of this type are described via v...
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| Vydáno v: | SIAM journal on optimization Ročník 20; číslo 5; s. 2199 - 2227 |
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| Médium: | Journal Article |
| Jazyk: | angličtina |
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Society for Industrial and Applied Mathematics
01.01.2010
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| ISSN: | 1052-6234, 1095-7189 |
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| Abstract | This paper concerns second-order analysis for a remarkable class of variational systems in finite-dimensional and infinite-dimensional spaces, which is particularly important for the study of optimization and equilibrium problems with equilibrium constraints. Systems of this type are described via variational inequalities over polyhedral convex sets and allow us to provide a comprehensive local analysis by using appropriate generalized differentiation of the normal cone mappings for such sets. In this paper we efficiently compute the required coderivatives of the normal cone mappings exclusively via the initial data of polyhedral sets in reflexive Banach spaces. This provides the main tools of second-order variational analysis allowing us, in particular, to derive necessary and sufficient conditions for robust Lipschitzian stability of solution maps to parameterized variational inequalities with evaluating the exact bound of the corresponding Lipschitzian moduli. The efficient coderivative calculations and characterizations of robust stability obtained in this paper are the first results in the literature for the problems under consideration in infinite-dimensional spaces. Most of them are also new in finite dimension [PUBLICATION ABSTRACT] |
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| AbstractList | This paper concerns second-order analysis for a remarkable class of variational systems in finite-dimensional and infinite-dimensional spaces, which is particularly important for the study of optimization and equilibrium problems with equilibrium constraints. Systems of this type are described via variational inequalities over polyhedral convex sets and allow us to provide a comprehensive local analysis by using appropriate generalized differentiation of the normal cone mappings for such sets. In this paper we efficiently compute the required coderivatives of the normal cone mappings exclusively via the initial data of polyhedral sets in reflexive Banach spaces. This provides the main tools of second-order variational analysis allowing us, in particular, to derive necessary and sufficient conditions for robust Lipschitzian stability of solution maps to parameterized variational inequalities with evaluating the exact bound of the corresponding Lipschitzian moduli. The efficient coderivative calculations and characterizations of robust stability obtained in this paper are the first results in the literature for the problems under consideration in infinite-dimensional spaces. Most of them are also new in finite dimension [PUBLICATION ABSTRACT] This paper concerns second-order analysis for a remarkable class of variational systems in finite-dimensional and infinite-dimensional spaces, which is particularly important for the study of optimization and equilibrium problems with equilibrium constraints. Systems of this type are described via variational inequalities over polyhedral convex sets and allow us to provide a comprehensive local analysis by using appropriate generalized differentiation of the normal cone mappings for such sets. In this paper we efficiently compute the required coderivatives of the normal cone mappings exclusively via the initial data of polyhedral sets in reflexive Banach spaces. This provides the main tools of second-order variational analysis allowing us, in particular, to derive necessary and sufficient conditions for robust Lipschitzian stability of solution maps to parameterized variational inequalities with evaluating the exact bound of the corresponding Lipschitzian moduli. The efficient coderivative calculations and characterizations of robust stability obtained in this paper are the first results in the literature for the problems under consideration in infinite-dimensional spaces. Most of them are also new in finite dimension |
| Author | Henrion, René Mordukhovich, Boris S. Nam, Nguyen Mau |
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| Cites_doi | 10.1137/060665609 10.1016/0022-247X(78)90260-3 10.1137/S1052623495284029 10.1007/s10492-007-0028-z 10.1090/S0002-9947-1993-1156300-4 10.1070/RM2000v055n03ABEH000292 10.1137/S1052623496309296 10.1007/s10107-003-0452-0 10.1137/S1052623400377153 10.1137/06067438 10.1016/j.na.2007.12.025 10.1287/moor.5.1.43 10.1137/S105262349834847X 10.1016/0021-8928(76)90136-2 10.1007/s11228-008-0092-x 10.3792/pjaa.56.462 10.1016/j.na.2008.11.089 |
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| References | Mordukhovich B. S. (R12) 1980; 22 R3 R5 R6 R7 R8 Yao J.-C. (R28) 2009; 5 R30 R10 R21 R20 R11 R22 R14 R24 R18 R29 R17 Yao J.-C. (R27) 2009; 34 R1 |
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| SubjectTerms | Banach space Convex analysis Differentiation Inequalities Mapping Mathematical analysis Optimization Stability Studies |
| Title | Second-Order Analysis of Polyhedral Systems in Finite and Infinite Dimensions with Applications to Robust Stability of Variational Inequalities |
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