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Convergence and Stability of a Regularization Method for Maximal Monotone Inclusions and Its Applications to Convex Optimization

  • Ya. I. Alber
  • D. Butnariu
  • G. Kassay
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 79)

Abstract

In this paper we study the stability and convergence of a regularization method for solving inclusions fAx, where A is a maximal monotone point-to-set operator from a reflexive smooth Banach space X with the Kadec-Klee property to its dual. We assume that the data A and f involved in the inclusion are given by approximations A k and f k converging to A and f, respectively, in the sense of Mosco type topologies. We prove that the sequence x k = (A k + α k J μ)−1 f k which results from the regularization process converges weakly and, under some conditions, converges strongly to the minimum norm solution of the inclusion fAx, provided that the inclusion is consistent. These results lead to a regularization procedure for perturbed convex optimization problems whose objective functions and feasibility sets are given by approximations. In particular, we obtain a strongly convergent version of the generalized proximal point optimization algorithm which is applicable to problems whose feasibility sets are given by Mosco approximations

Key words

Maximal monotone inclusion Mosco convergence of sets regularization method convex optimization problem generalized proximal point method for optimization 

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Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • Ya. I. Alber
    • 1
  • D. Butnariu
    • 2
  • G. Kassay
    • 3
  1. 1.Faculty of MathematicsTechnion - Israel Institute of TechnologyHaifaIsrael
  2. 2.Dept. of MathematicsUniversity of HaifaHaifaIsrael
  3. 3.Faculty of MathematicsUniversity Babes-BolyaiCluj-NapocaRomania

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