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In this article we consider a consistent convex feasibility problem in a realHilbert space defined by a finite family of sets $C_i$. We are interested, inparticular, in the case where for each $i$, $C_i=Fix (U_i)=\{z\in \mathcalH\mid p_i(z)=0\}$, $U_i\colon\mathcal H\rightarrow \mathcal H$ is a cutter and$p_i\colon\mathcal H\rightarrow [0,\infty)$ is a proximity function. Moreover,we make the following assumption: the computation of $p_i$ is at most asdifficult as the evaluation of $U_i$ and this is at most as difficult asprojecting onto $C_i$. We study a double-layer fixed point algorithm whichapplies two types of controls in every iteration step. The first one -- theouter control -- is assumed to be almost cyclic. The second one -- the innercontrol -- determines the most important sets from those offered by the firstone. The selection is made in terms of proximity functions. The convergenceresults presented in this manuscript depend on the conditions which first, bindtogether 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 regularityimply weak, strong and linear convergence of our algorithm, respectively. Theframework presented in this paper covers many known (subgradient) projectionalgorithms already existing in the literature; for example, those applied with(almost) cyclic, remotest-set, maximum displacement, most-violated constraintand simultaneous controls. In addition, we provide several new examples, wherethe double-layer approach indeed accelerates the convergence speed as wedemonstrate numerically.

Weak, Strong and Linear Convergence of a Double-Layer Fixed Point Algorithm