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Journal of Convex Analysis 19 (2012), No. 4, 1043--1072
Copyright Heldermann Verlag 2012

From Convergence Principles to Stability and Optimality Conditions

Diethard Klatte
Institut für Operations Research, Universität Zürich, Moussonstrasse 15, 8044 Zürich, Switzerland

Alexander Kruger
Centre for Informatics and Applied Optimization, University of Ballarat, POB 663, Ballarat, Vic. 3350, Australia

Bernd Kummer
Institut für Mathematik, Humboldt-Universität, Unter den Linden 6, 10099 Berlin, Germany

We show in a rather general setting that Hoelder and Lipschitz stability properties of solutions to variational problems can be characterized by convergence of more or less abstract iteration schemes. Depending on the principle of convergence, new and intrinsic stability conditions can be derived. Our most abstract models are (multi-) functions on complete metric spaces. The relevance of this approach is illustrated by deriving both classical and new results on existence and optimality conditions, stability of feasible and solution sets and convergence behavior of solution procedures.

Keywords: Generalized equations, Hoelder stability, iteration schemes, calmness, Aubin property, variational principles.

MSC: 49J53, 49K40, 90C31, 65J05

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