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A relaxed alternating CQ-algorithm for convex feasibility problems. (English) Zbl 1256.49044

Summary: Let ${H}_{1},{H}_{2},{H}_{3}$ be real Hilbert spaces, let $C\subset {H}_{1}$, $Q\subset {H}_{2}$ be two nonempty closed convex level sets, let $A:{H}_{1}\to {H}_{3}$, $B:{H}_{2}\to {H}_{3}$ be two bounded linear operators. Our interest is in solving the following new convex feasibility problem

$\text{Find}\phantom{\rule{4pt}{0ex}}x\in C,y\in Q\phantom{\rule{4pt}{0ex}}\text{such}\phantom{\rule{4.pt}{0ex}}\text{that}\phantom{\rule{4pt}{0ex}}Ax=By,$

which allows asymmetric and partial relations between the variables $x$ and $y$. In this paper, we present and study the convergence of a relaxed alternating CQ-algorithm (RACQA) and show that the sequences generated by such an algorithm weakly converge to a solution of the above problem. The interest of RACQA is that we just need projections onto half-spaces, thus making the relaxed CQ-algorithm implementable. Note that by taking $B=I$ we recover the split convex feasibility problem originally introduced by Y. Censor and J. Elfving [Numer. Algorithms 8, No. 2–4, 221–239 (1994; Zbl 0828.65065)] and used later in intensity-modulated radiation therapy [Y. Censor et al., “A unified approach for inversion problems in intensity-modulated radiation therapy”, Physics in Medicine and Biology 51, 2353–2365 (2006)]. We also recover the relaxed CQ-algorithm introduced by Q. Yang [Inverse Probl. 20, No. 4, 1261–1266 (2004; Zbl 1066.65047)] by particularizing both $B$ and a given parameter.

##### MSC:
 49M37 Methods of nonlinear programming type in calculus of variations 49J53 Set-valued and variational analysis 65K10 Optimization techniques (numerical methods) 90C25 Convex programming