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Abstract Newtonian Frameworks and Their Applications

A.F. Izmailov(izmaf***at***ccas.ru)
A.S. Kurennoy(alex-kurennoy***at***yandex.ru)

Abstract: We unify and extend some Newtonian iterative frameworks developed earlier in the literature, which results in a collection of convenient tools for local convergence analysis of various algorithms under various sets of assumptions including strong metric regularity, semistability, or upper-Lipschizt stability, the latter allowing for nonisolated solutions. These abstract schemes are further applied for deriving sharp local convergence results for some constrained optimization algorithms under the reduced smoothness hypotheses. Specifically, we consider applications to the augmented Lagrangian method and to the linearly constrained Lagrangian method for problems with Lipschitzian derivatives but possibly without second derivatives, and our local convergence analysis for these methods improves all the existing theories of this kind.

Keywords: generalized equation, abstract Newton scheme, augmented Lagrangian, multiplier method, local convergence, strong metric regularity, semistability, upper-Lipschitz stability

Category 1: Nonlinear Optimization (Constrained Nonlinear Optimization )

Category 2: Convex and Nonsmooth Optimization (Nonsmooth Optimization )

Category 3: Complementarity and Variational Inequalities

Citation: Moscow State University, Moscow, 11/2012

Download: [PDF]

Entry Submitted: 02/07/2013
Entry Accepted: 02/07/2013
Entry Last Modified: 02/07/2013

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