Several approaches for the derivation of stationarity conditions for elliptic MPECs with upper-level control constraints

The derivation of multiplier-based optimality conditions for elliptic mathematical programs with equilibrium constraints (MPEC) is essential for the characterization of solutions and development of numerical methods. Though much can be said for broad classes of elliptic MPECs in both polyhedric and...

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Veröffentlicht in:	Mathematical programming Jg. 146; H. 1-2; S. 555 - 582
Hauptverfasser:	Hintermüller, Michael, Mordukhovich, Boris S., Surowiec, Thomas M.
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	Berlin/Heidelberg Springer Berlin Heidelberg 01.08.2014 Springer Nature B.V
Schlagworte:	Analysis Calculus of Variations and Optimal Control; Optimization Combinatorics Constraints Control theory Convex analysis Derivation Equilibrium Full Length Paper Inequality Mathematical analysis Mathematical and Computational Physics Mathematical Methods in Physics Mathematical models Mathematical programming Mathematics Mathematics and Statistics Mathematics of Computing Numerical Analysis Optimization Partial differential equations Regularization Studies Theoretical 49K21 90C46 Stationarity conditions Elliptic mathematical programs with equilibrium constraints Control constraints 90C33 65K10 Optimal control of variational inequalities Coderivatives
ISSN:	0025-5610, 1436-4646
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Abstract	The derivation of multiplier-based optimality conditions for elliptic mathematical programs with equilibrium constraints (MPEC) is essential for the characterization of solutions and development of numerical methods. Though much can be said for broad classes of elliptic MPECs in both polyhedric and non-polyhedric settings, the calculation becomes significantly more complicated when additional constraints are imposed on the control. In this paper we develop three derivation methods for constrained MPEC problems: via concepts from variational analysis, via penalization of the control constraints, and via penalization of the lower-level problem with the subsequent regularization of the resulting nonsmoothness. The developed methods and obtained results are then compared and contrasted.
AbstractList	The derivation of multiplier-based optimality conditions for elliptic mathematical programs with equilibrium constraints (MPEC) is essential for the characterization of solutions and development of numerical methods. Though much can be said for broad classes of elliptic MPECs in both polyhedric and non-polyhedric settings, the calculation becomes significantly more complicated when additional constraints are imposed on the control. In this paper we develop three derivation methods for constrained MPEC problems: via concepts from variational analysis, via penalization of the control constraints, and via penalization of the lower-level problem with the subsequent regularization of the resulting nonsmoothness. The developed methods and obtained results are then compared and contrasted. The derivation of multiplier-based optimality conditions for elliptic mathematical programs with equilibrium constraints (MPEC) is essential for the characterization of solutions and development of numerical methods. Though much can be said for broad classes of elliptic MPECs in both polyhedric and non-polyhedric settings, the calculation becomes significantly more complicated when additional constraints are imposed on the control. In this paper we develop three derivation methods for constrained MPEC problems: via concepts from variational analysis, via penalization of the control constraints, and via penalization of the lower-level problem with the subsequent regularization of the resulting nonsmoothness. The developed methods and obtained results are then compared and contrasted.[PUBLICATION ABSTRACT]
Author	Hintermüller, Michael Surowiec, Thomas M. Mordukhovich, Boris S.
Author_xml	– sequence: 1 givenname: Michael surname: Hintermüller fullname: Hintermüller, Michael email: hint@math.hu-berlin.de organization: Institut für Mathematik, Humboldt-Universität zu Berlin – sequence: 2 givenname: Boris S. surname: Mordukhovich fullname: Mordukhovich, Boris S. organization: Department of Mathematics, Wayne State University – sequence: 3 givenname: Thomas M. surname: Surowiec fullname: Surowiec, Thomas M. organization: Institut für Mathematik, Humboldt-Universität zu Berlin
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Cites_doi	10.1287/moor.25.1.1.15213 10.1007/s10589-009-9307-9 10.1137/0322028 10.1016/0022-1236(76)90017-3 10.1016/j.na.2006.05.021 10.1137/100802396 10.1007/3-540-31246-3 10.1137/080720681 10.1007/s11228-010-0158-4 10.1007/978-1-4612-1394-9 10.1090/gsm/112
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Keywords	49K21 90C46 Stationarity conditions Elliptic mathematical programs with equilibrium constraints Control constraints 90C33 65K10 Optimal control of variational inequalities Coderivatives
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References	Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation. vol. 2: Applications. Springer, Berlin (2006) AdamsRAFournierJJFSobolev Spaces20082AmsterdamElsevier MignotFContrôle dans les inéquations variationelles elliptiquesJ. Funct. Anal.197622213018510.1016/0022-1236(76)90017-30364.49003423155 ScheelHScholtesSMathematical programs with complementarity constraints: stationarity, optimality, and sensitivityMath. Oper. Res.200025112210.1287/moor.25.1.1.152131073.905571854317 BonnansJFShapiroAPerturbation Analysis of Optimization Problems2000BerlinSpringer10.1007/978-1-4612-1394-90966.49001 DunfordNSchwartzJTLinear Operators. Part I. Wiley Classics Library1988New YorkWiley AubinJPFrankowskaHSet-Valued Analysis1990BostonBirkhäuser0713.49021 OutrataJVJařusekJStaráJOn optimality conditions in control of elliptic variational inequalitiesSet-Valued Var. Anal.201119234210.1007/s11228-010-0158-41211.490362770895 OutrataJVJařusekJOn sharp necessary optimality conditions in control of contact problems with stringsNonlinear Anal.20076741117112810.1016/j.na.2006.05.0211113.490242325366 MignotFPuelJPOptimal control in some variational inequalitiesSIAM J. Control Optim.198422346647610.1137/03220280561.49007739836 HintermüllerMKopackaIMathematical programs with complementarity constraints in function space: C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C$$\end{document}- and strong stationarity and a path-following algorithmSIAM J. Optim.200920286890210.1137/0807206811189.490322515801 AttouchHVariational Convergence for Functions and Operators1984BostonPitman Advanced Publishing Program0561.49012 RodriguesJFObstacle Problems in Mathematical Physics. No. 134 in North-Holland Mathematics Studies1984AmsterdamNorth-Holland Publishing Co. KinderlehrerDStampacchiaGAn Introduction to Variational Inequalities and Their Applications1980New YorkAcademic Press0457.35001 BarbuVOptimal Control of Variational Inequalities, Research Notes in Mathematics1984BostonPitman (Advanced Publishing Program) Hintermüller, M., Kopacka, I.: A smooth penalty approach and a nonlinear multigrid algorithm for elliptic MPECs. Comput. Optim. Appl. (2009). doi:10.1007/s10589-009-9307-9 Tröltzsch, F.: Optimal Control of Partial Differential Equations, Graduate Studies in Mathematics, vol. 112. American Mathematical Society, Providence (2010). Theory, methods and applications, Translated from the 2005 German original by Jürgen Sprekels Glowinski, R., Lions, J.L., Trémolières, R.: Analyse numérique des inéquations variationnelles. Tome 1. Dunod, Paris (1976). Théorie générale premiéres applications, Méthodes Mathématiques de l’Informatique, 5 HintermüllerMSurowiecTFirst-order optimality conditions for elliptic mathematical programs with equilibrium constraints via variational analysisSIAM J. Optim.20112141561159310.1137/1008023961248.490302869508 Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation. vol. 1: Basic Theory. Springer, Berlin (2006) V Barbu (704_CR4) 1984 M Hintermüller (704_CR10) 2011; 21 N Dunford (704_CR6) 1988 RA Adams (704_CR1) 2008 JF Bonnans (704_CR5) 2000 F Mignot (704_CR13) 1984; 22 JP Aubin (704_CR3) 1990 704_CR7 704_CR9 D Kinderlehrer (704_CR11) 1980 H Attouch (704_CR2) 1984 704_CR20 704_CR14 704_CR15 F Mignot (704_CR12) 1976; 22 JF Rodrigues (704_CR18) 1984 H Scheel (704_CR19) 2000; 25 JV Outrata (704_CR16) 2007; 67 M Hintermüller (704_CR8) 2009; 20 JV Outrata (704_CR17) 2011; 19
References_xml	– reference: HintermüllerMSurowiecTFirst-order optimality conditions for elliptic mathematical programs with equilibrium constraints via variational analysisSIAM J. Optim.20112141561159310.1137/1008023961248.490302869508 – reference: AttouchHVariational Convergence for Functions and Operators1984BostonPitman Advanced Publishing Program0561.49012 – reference: DunfordNSchwartzJTLinear Operators. Part I. Wiley Classics Library1988New YorkWiley – reference: OutrataJVJařusekJStaráJOn optimality conditions in control of elliptic variational inequalitiesSet-Valued Var. Anal.201119234210.1007/s11228-010-0158-41211.490362770895 – reference: OutrataJVJařusekJOn sharp necessary optimality conditions in control of contact problems with stringsNonlinear Anal.20076741117112810.1016/j.na.2006.05.0211113.490242325366 – reference: AdamsRAFournierJJFSobolev Spaces20082AmsterdamElsevier – reference: AubinJPFrankowskaHSet-Valued Analysis1990BostonBirkhäuser0713.49021 – reference: KinderlehrerDStampacchiaGAn Introduction to Variational Inequalities and Their Applications1980New YorkAcademic Press0457.35001 – reference: BarbuVOptimal Control of Variational Inequalities, Research Notes in Mathematics1984BostonPitman (Advanced Publishing Program) – reference: RodriguesJFObstacle Problems in Mathematical Physics. No. 134 in North-Holland Mathematics Studies1984AmsterdamNorth-Holland Publishing Co. – reference: BonnansJFShapiroAPerturbation Analysis of Optimization Problems2000BerlinSpringer10.1007/978-1-4612-1394-90966.49001 – reference: MignotFContrôle dans les inéquations variationelles elliptiquesJ. Funct. Anal.197622213018510.1016/0022-1236(76)90017-30364.49003423155 – reference: Hintermüller, M., Kopacka, I.: A smooth penalty approach and a nonlinear multigrid algorithm for elliptic MPECs. Comput. Optim. Appl. (2009). doi:10.1007/s10589-009-9307-9 – reference: Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation. vol. 2: Applications. Springer, Berlin (2006) – reference: Glowinski, R., Lions, J.L., Trémolières, R.: Analyse numérique des inéquations variationnelles. Tome 1. Dunod, Paris (1976). Théorie générale premiéres applications, Méthodes Mathématiques de l’Informatique, 5 – reference: Tröltzsch, F.: Optimal Control of Partial Differential Equations, Graduate Studies in Mathematics, vol. 112. American Mathematical Society, Providence (2010). Theory, methods and applications, Translated from the 2005 German original by Jürgen Sprekels – reference: HintermüllerMKopackaIMathematical programs with complementarity constraints in function space: C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C$$\end{document}- and strong stationarity and a path-following algorithmSIAM J. Optim.200920286890210.1137/0807206811189.490322515801 – reference: ScheelHScholtesSMathematical programs with complementarity constraints: stationarity, optimality, and sensitivityMath. Oper. Res.200025112210.1287/moor.25.1.1.152131073.905571854317 – reference: Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation. vol. 1: Basic Theory. Springer, Berlin (2006) – reference: MignotFPuelJPOptimal control in some variational inequalitiesSIAM J. Control Optim.198422346647610.1137/03220280561.49007739836 – volume-title: Linear Operators. Part I. Wiley Classics Library year: 1988 ident: 704_CR6 – volume-title: Variational Convergence for Functions and Operators year: 1984 ident: 704_CR2 – volume-title: Sobolev Spaces year: 2008 ident: 704_CR1 – volume-title: An Introduction to Variational Inequalities and Their Applications year: 1980 ident: 704_CR11 – volume-title: Set-Valued Analysis year: 1990 ident: 704_CR3 – volume: 25 start-page: 1 issue: 1 year: 2000 ident: 704_CR19 publication-title: Math. Oper. Res. doi: 10.1287/moor.25.1.1.15213 – ident: 704_CR9 doi: 10.1007/s10589-009-9307-9 – volume: 22 start-page: 466 issue: 3 year: 1984 ident: 704_CR13 publication-title: SIAM J. Control Optim. doi: 10.1137/0322028 – volume: 22 start-page: 130 issue: 2 year: 1976 ident: 704_CR12 publication-title: J. Funct. Anal. doi: 10.1016/0022-1236(76)90017-3 – volume: 67 start-page: 1117 issue: 4 year: 2007 ident: 704_CR16 publication-title: Nonlinear Anal. doi: 10.1016/j.na.2006.05.021 – ident: 704_CR7 – volume-title: Obstacle Problems in Mathematical Physics. No. 134 in North-Holland Mathematics Studies year: 1984 ident: 704_CR18 – volume: 21 start-page: 1561 issue: 4 year: 2011 ident: 704_CR10 publication-title: SIAM J. Optim. doi: 10.1137/100802396 – ident: 704_CR14 doi: 10.1007/3-540-31246-3 – ident: 704_CR15 doi: 10.1007/3-540-31246-3 – volume: 20 start-page: 868 issue: 2 year: 2009 ident: 704_CR8 publication-title: SIAM J. Optim. doi: 10.1137/080720681 – volume: 19 start-page: 23 year: 2011 ident: 704_CR17 publication-title: Set-Valued Var. Anal. doi: 10.1007/s11228-010-0158-4 – volume-title: Optimal Control of Variational Inequalities, Research Notes in Mathematics year: 1984 ident: 704_CR4 – volume-title: Perturbation Analysis of Optimization Problems year: 2000 ident: 704_CR5 doi: 10.1007/978-1-4612-1394-9 – ident: 704_CR20 doi: 10.1090/gsm/112
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SubjectTerms	Analysis Calculus of Variations and Optimal Control; Optimization Combinatorics Constraints Control theory Convex analysis Derivation Equilibrium Full Length Paper Inequality Mathematical analysis Mathematical and Computational Physics Mathematical Methods in Physics Mathematical models Mathematical programming Mathematics Mathematics and Statistics Mathematics of Computing Numerical Analysis Optimization Partial differential equations Regularization Studies Theoretical
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Title	Several approaches for the derivation of stationarity conditions for elliptic MPECs with upper-level control constraints
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