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 \(x^{k+1}=P_C(x^k-yA^T(P_Q-I)Ax^k)\) by adopting Armijo-like searches. The modified algorithm need not compute matrix inverses and the largest eigenvalue of the matrix \(A^TA\). 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.


65F30 Other matrix algorithms (MSC2010)
65F10 Iterative numerical methods for linear systems
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