Abstract
Our paper deals with the interrelation of optimization methods and Lipschitz stability of multifunctions in arbitrary Banach spaces. Roughly speaking, we show that linear convergence of several first order methods and Lipschitz stability mean the same. Particularly, we characterize calmness and the Aubin property by uniformly (with respect to certain starting points) linear convergence of descent methods and approximate projection methods. So we obtain, e.g., solution methods (for solving equations or variational problems) which require calmness only. The relations of these methods to several known basic algorithms are discussed, and errors in the subroutines as well as deformations of the given mappings are permitted. We also recall how such deformations are related to standard algorithms like barrier, penalty or regularization methods in optimization.
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This paper is dedicated to Professor Stephen M. Robinson on the occasion of his 65th birthday.
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Klatte, D., Kummer, B. Optimization methods and stability of inclusions in Banach spaces. Math. Program. 117, 305–330 (2009). https://doi.org/10.1007/s10107-007-0174-9
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DOI: https://doi.org/10.1007/s10107-007-0174-9
Keywords
- Generalized equation
- Variational inequality
- Perturbation
- Regularization
- Stability criteria
- Metric regularity
- Calmness
- Approximate projections
- Penalization
- Successive approximation
- Newton’s method