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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 $$\text{Find}\ x\in C, y\in Q \ \text{such that} \ Ax=By,$$ 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 {\it Y. Censor} and {\it J. Elfving} [Numer. Algorithms 8, No. 2--4, 221--239 (1994; Zbl 0828.65065)] and used later in intensity-modulated radiation therapy [{\it 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 {\it Q. Yang} [Inverse Probl. 20, No. 4, 1261--1266 (2004; Zbl 1066.65047)] by particularizing both $B$ and a given parameter.

49M37Methods of nonlinear programming type in calculus of variations
49J53Set-valued and variational analysis
65K10Optimization techniques (numerical methods)
90C25Convex programming
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