Primal–Dual Stability in Local Optimality
Much is known about when a locally optimal solution depends in a single-valued Lipschitz continuous way on the problem’s parameters, including tilt perturbations. Much less is known, however, about when that solution and a uniquely determined multiplier vector associated with it exhibit that depende...
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| Veröffentlicht in: | Journal of optimization theory and applications Jg. 203; H. 2; S. 1325 - 1354 |
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| Abstract | Much is known about when a locally optimal solution depends in a single-valued Lipschitz continuous way on the problem’s parameters, including tilt perturbations. Much less is known, however, about when that solution and a uniquely determined multiplier vector associated with it exhibit that dependence as a primal–dual pair. In classical nonlinear programming, such advantageous behavior is tied to the combination of the standard strong second-order sufficient condition (SSOC) for local optimality and the linear independent gradient condition (LIGC) on the active constraint gradients. But although second-order sufficient conditons have successfully been extended far beyond nonlinear programming, insights into what should replace constraint gradient independence as the extended dual counterpart have been lacking. The exact answer is provided here for a wide range of optimization problems in finite dimensions. Behind it are advances in how coderivatives and strict graphical derivatives can be deployed. New results about strong metric regularity in solving variational inequalities and generalized equations are obtained from that as well. |
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| AbstractList | Much is known about when a locally optimal solution depends in a single-valued Lipschitz continuous way on the problem’s parameters, including tilt perturbations. Much less is known, however, about when that solution and a uniquely determined multiplier vector associated with it exhibit that dependence as a primal–dual pair. In classical nonlinear programming, such advantageous behavior is tied to the combination of the standard strong second-order sufficient condition (SSOC) for local optimality and the linear independent gradient condition (LIGC) on the active constraint gradients. But although second-order sufficient conditons have successfully been extended far beyond nonlinear programming, insights into what should replace constraint gradient independence as the extended dual counterpart have been lacking. The exact answer is provided here for a wide range of optimization problems in finite dimensions. Behind it are advances in how coderivatives and strict graphical derivatives can be deployed. New results about strong metric regularity in solving variational inequalities and generalized equations are obtained from that as well. |
| Author | Benko, Matúš Rockafellar, R. Tyrrell |
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| Cites_doi | 10.1137/S1052623498348274 10.1137/22M1530161 10.1137/S1052623495284029 10.1007/978-1-4939-1037-3 10.1023/A:1008662404582 10.1007/s11228-015-0325-8 10.1137/S1052623496309296 10.1007/s10107-022-01768-w 10.1007/s11228-018-0496-1 10.1016/j.na.2014.10.013 10.1090/S0002-9947-96-01544-9 10.1137/120887722 10.1007/s10013-019-00339-5 10.1016/j.jmaa.2021.125895 10.1137/110852528 10.1287/moor.2014.0669 |
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| Keywords | 90C31 Second-order variational analysis Graphically Lipschitzian mappings Crypto-continuity Variational sufficiency Kummer’s inverse theorem Strict graphical derivatives Primal–dual stability Tilt stability, full stability, metric regularity Local optimality Implicit mapping theorems Coderivatives |
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| References | Levy, Poliquin, Rockafellar (CR8) 2000; 10 Mordukhovich, Nghia, Rockafellar (CR9) 2015; 40 Dontchev, Rockafellar (CR3) 2006; 5 CR6 Kummer (CR7) 1991; 158 Gfrerer, Outrata (CR5) 2022; 508 Dontchev, Rockafellar (CR4) 2014 Wang, Ding, Zhang, Zhao (CR21) 2023; 33 Rockafellar (CR16) 2019; 27 Mordukhovich, Rockafellar, Sarabi (CR12) 2013; 23 Poliquin, Rockafellar (CR13) 1998; 8 Rockafellar, Zagrodny (CR19) 1997; 5 Adam, Červinka, Pištěk (CR1) 2016; 24 Mordukhovich, Sarabi (CR10) 2015; 121 Mordukhovich, Rockafellar (CR11) 2012; 22 Rockafellar (CR15) 2019; 47 Poliquin, Rockafellar (CR14) 1996; 348 Rockafellar (CR17) 2023; 198 Rockafellar, Wets (CR18) 1997 Spanier (CR20) 1966 Dontchev, Rockafellar (CR2) 1996; 6 BS Mordukhovich (2467_CR9) 2015; 40 RT Rockafellar (2467_CR17) 2023; 198 RT Rockafellar (2467_CR15) 2019; 47 RA Poliquin (2467_CR13) 1998; 8 BS Mordukhovich (2467_CR11) 2012; 22 RA Poliquin (2467_CR14) 1996; 348 SW Wang (2467_CR21) 2023; 33 BS Mordukhovich (2467_CR12) 2013; 23 AD Dontchev (2467_CR3) 2006; 5 RT Rockafellar (2467_CR19) 1997; 5 AB Levy (2467_CR8) 2000; 10 BS Mordukhovich (2467_CR10) 2015; 121 EH Spanier (2467_CR20) 1966 AD Dontchev (2467_CR4) 2014 AD Dontchev (2467_CR2) 1996; 6 H Gfrerer (2467_CR5) 2022; 508 B Kummer (2467_CR7) 1991; 158 RT Rockafellar (2467_CR18) 1997 2467_CR6 L Adam (2467_CR1) 2016; 24 RT Rockafellar (2467_CR16) 2019; 27 |
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| SubjectTerms | Applications of Mathematics Calculus of Variations and Optimal Control; Optimization Engineering Mathematics Mathematics and Statistics Operations Research/Decision Theory Optimization Theory of Computation |
| Title | Primal–Dual Stability in Local Optimality |
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