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# A relaxed alternating CQ-algorithm for convex feasibility

Abstract : Let H1,H2,H3 be real Hilbert spaces, let C⊂H1, Q⊂H2 be two nonempty closed convex level sets, let A:H1→H3, B:H2→H3 be two bounded linear operators. Our interest is in solving the following new convex feasibility problem Find x∈C,y∈Q 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 (1.1). 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, in (1.1), we recover the split convex feasibility problem originally introduced in Censor and Elfving (1994) [13] and used later in intensity-modulated radiation therapy (Censor et al. (2006) [11]). We also recover the relaxed CQ-algorithm introduced by Yang (2004) [8] by particularizing both B and a given parameter.
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https://hal.univ-antilles.fr/hal-00776638
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Submitted on : Tuesday, January 15, 2013 - 8:10:16 PM
Last modification on : Monday, May 23, 2022 - 2:06:49 PM

### Citation

Abdellatif Moudafi. A relaxed alternating CQ-algorithm for convex feasibility. Nonlinear Analysis: Theory, Methods and Applications, Elsevier, 2013, 79, pp.117-121. ⟨10.1016/j.na.2012.11.013⟩. ⟨hal-00776638⟩

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