  In this article we consider a consistent convex feasibility problem in a real
Hilbert space defined by a finite family of sets $C_i$. We are interested, in
particular, in the case where for each $i$, $C_i=Fix (U_i)=\{z\in \mathcal
H\mid p_i(z)=0\}$, $U_i\colon\mathcal H\rightarrow \mathcal H$ 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 as
difficult as the evaluation of $U_i$ and this is at most as difficult as
projecting onto $C_i$. 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.
