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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Bibliographic Details
Published in:Journal of optimization theory and applications Vol. 203; no. 2; pp. 1325 - 1354
Main Authors: Benko, Matúš, Rockafellar, R. Tyrrell
Format: Journal Article
Language:English
Published: New York Springer US 01.11.2024
Subjects:
Applications of Mathematics
Calculus of Variations and Optimal Control; Optimization
Engineering
Mathematics
Mathematics and Statistics
Operations Research/Decision Theory
Optimization
Theory of Computation
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
ISSN:0022-3239, 1573-2878
Online Access:Get full text
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Summary: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.
ISSN:0022-3239
1573-2878
DOI:10.1007/s10957-024-02467-6