Weak, strong and linear convergence of a double-layer fixed point algorithm

Invited abstract in session MB-54: Projection methods in optimization problems 2, stream Convex Optimization.

Area: Continuous Optimization

Monday, 10:30-12:00
Room: Building PA, Room B

Authors (first author is the speaker)

1. Rafal Zalas
Mathematics, Technion - Israel Institute of Technology


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 state-of-the-art 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 double-layer 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 demi-closedness 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.


Status: accepted

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