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

Citations:

Zbl 0828.65065; Zbl 1066.65047
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