In this article we consider a consistent convex feasibility problem in a real Hilbert space defined by a finite family of sets @math . We are interested, in particular, in the case where for each @math , @math , @math is a cutter and @math is a proximity function. Moreover, we make the following assumption: the computation of @math is at most as difficult as the evaluation of @math and this is at most as difficult as projecting onto @math . We study a double-layer fixed point algorithm which applies two types of controls in every iteration step. 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 manuscript depend on the conditions which first, bind together the sets, the operators and the proximity functions and second, connect the inner and outer controls. In particular, weak regularity (demi-closedness principle), bounded regularity and bounded linear regularity imply weak, strong and linear convergence of our algorithm, respectively. The framework presented in this paper covers many known (subgradient) projection algorithms already existing in the literature; for example, those applied with (almost) cyclic, remotest-set, maximum displacement, most-violated constraint and simultaneous controls. In addition, we provide several new examples, where the double-layer approach indeed accelerates the convergence speed as we demonstrate numerically.
We would like to begin with two important general observations regarding the types of convergence one could expect. The first is that in the infinite dimensional setting, in view of Hundal's counterexample, it may happen that even for basic cyclic or parallel projection methods the convergence can only be in the weak topology; see @cite_29 and @cite_34 . Moreover, the result of [Theorem 1.4] BauschkeDeutschHundal2009 shows that norm convergence can be far away from a linear rate. Furthermore, it can be arbitrarily slow. See also @cite_2 in this connection. Thus both norm and linear convergence require some additional assumptions to which we refer in general as bounded regularity and bounded linear regularity.
@cite_29 @cite_34 @cite_2
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Recently, Hundal has constructed a hyperplane H , a cone K, and a starting point y0 in ‘2 such that the sequence of alternating projections ((PK PH ) n y0)n∈N converges weakly to some point in H ∩ K, but not in norm. We show how this construction results in a counterexample to norm convergence for iterates of averaged projections; hence, we give an a:rmative answer to a question raised by Reich two decades ago. Furthermore, new counterexamples to norm convergence for iterates of <rmly nonexpansive maps (=a la Genel and Lindenstrauss) and for the proximal point algorithm (= al a G@ uler) are provided. We also present a counterexample, along with some weak and norm convergence results, for the new framework of string-averaging projection methods introduced by Extensions to Banach spaces and the situation for the Hilbert ball are discussed as well. ? 2003 Elsevier Ltd. All rights reserved. MSC: 47H09; 47J25; 90C25 A generalization of the cosine of the Friedrichs angle between two subspaces to a parameter associated to several closed subspaces in a Hilbert space is given. This is used to analyze the rate of convergence in the von Neumann-Halperin method of cyclic alternating projections. General dichotomy theorems are proved, in the Hilbert or Banach space situation, providing conditions under which the alternative QUC ASC (quick uniform convergence versus arbitrarily slow convergence) holds. Several meanings for ASC are proposed.