Primal–Dual Stability in Local Optimality

Matúš Benko () and R. Tyrrell Rockafellar ()
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Matúš Benko: Johann Radon Institute for Computational and Applied Mathematics
R. Tyrrell Rockafellar: University of Washington

Journal of Optimization Theory and Applications, 2024, vol. 203, issue 2, No 11, 1325-1354

Abstract: 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.

Keywords: Second-order variational analysis; Local optimality; Primal–dual stability; Tilt stability; full stability; metric regularity; Kummer’s inverse theorem; Implicit mapping theorems; Graphically Lipschitzian mappings; Crypto-continuity; Strict graphical derivatives; Coderivatives; Variational sufficiency; 90C31 (search for similar items in EconPapers)
Date: 2024
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10957-024-02467-6

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