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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 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 Numerical methods based on nonlinear programming
49J53 Set-valued and variational analysis
65K10 Numerical optimization and variational techniques
90C25 Convex programming
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