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A note on the $CQ$ algorithm for the split feasibility problem. (English) Zbl 1080.65033
The authors present modifications to the $CQ$ algorithm proposed by Ch. Byrne [Inverse Probl. 18, No. 2, 441–453 (2002; Zbl 0996.65048)] and to the relaxed $CQ$ algorithm proposed by Q. Z. Yang [Inverse Probl. 20, 1261–1266 (2004; Zbl 1066.65047)] to solve the split feasibility problem ${x}^{k+1}={P}_{C}\left({x}^{k}-y{A}^{T}\left({P}_{Q}-I\right)A{x}^{k}\right)$ by adopting Armijo-like searches. The modified algorithm need not compute matrix inverses and the largest eigenvalue of the matrix ${A}^{T}A$. It provides a sufficient decrease of the objective function at each iteration by a judicious choice of the stepsize and can identify the existence of solutions by the iterative sequence. The convergence of the modified algorithms is established under mild conditions.
##### MSC:
 65F30 Other matrix algorithms 65F10 Iterative methods for linear systems