Weak, strong and linear convergence of a doublelayer fixed point algorithm
Invited abstract in session MB54: Projection methods in optimization problems 2, stream Convex Optimization.
Area: Continuous Optimization
Room: Building PA, Room B
Authors (first author is the speaker)
1.  Rafal Zalas

Mathematics, Technion  Israel Institute of Technology 
Abstract
In this talk we consider consistent convex feasibility problems in a real Hilbert space defined by a finite family of sets Ci. In particular, we are interested in the case where Ci=Fix Ui={z: pi(z)=0}, Ui is a cutter and pi is a proximity function. Moreover, we make the following stateoftheart assumption: the computation of pi is at most as difficult as the evaluation of Ui and this is at most as difficult as projecting onto Ci.
The considered doublelayer fixed point algorithm, for every step k, applies two types of controls. The first one  the outer control  is assumed to be almost cyclic. The second one  the inner control  determines the most important sets from those offered by the first one. The selection is made in terms of proximity functions.
The convergence results presented in this talk depend on the conditions which first, bind together sets, operators and proximity functions and second, connect inner and outer controls. In particular, a demiclosedness principle, bounded regularity and bounded linear regularity imply weak, strong and linear convergence of our algorithm, respectively. The framework presented in this talk covers many known (subgradient) projection algorithms already existing in the literature; see, for example, those applied with (almost) cyclic, remotest set, maximum displacement, most violated constraint, parallel and block iterative controls.
Keywords
 Algorithms
 Complexity and Approximation
Status: accepted
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