On Quantitative Stability in Optimization and Optimal Control |
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Authors: | A. L. Dontchev W. W. Hager K. Malanowski V. M. Veliov |
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Affiliation: | (1) Mathematical Reviews, Ann Arbor, MI, 48107, U.S.A.;(2) Department of Mathematics, University of Florida, Gainesville, FL, 32611, U.S.A.;(3) Systems Research Institute, Polish Academy of Sciences, ul. Newelska 6, 01-447 Warsaw, Poland;(4) Institute of Mathematics and Informatics, Bulgarian Acad. of Sc., 1113 Sofia, Bulgaria;(5) Vienna University of Technology, Wiedner Hauptstr. 8-10/15, A-1040 Vienna, Austria |
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Abstract: | We study two continuity concepts for set-valued maps that play central roles in quantitative stability analysis of optimization problems: Aubin continuity and Lipschitzian localization. We show that various inverse function theorems involving these concepts can be deduced from a single general result on existence of solutions to an inclusion in metric spaces. As applications, we analyze the stability with respect to canonical perturbations of a mathematical program in a Hilbert space and an optimal control problem with inequality control constraints. For stationary points of these problems, Aubin continuity and Lipschitzian localization coincide; moreover, both properties are equivalent to surjectivity of the map of the gradients of the active constraints combined with a strong second-order sufficient optimality condition. |
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Keywords: | stability in optimization generalized equations Lipschitz continuity mathematical programming optimal control |
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