A note on the \(CQ\) algorithm for the split feasibility problem. (English) Zbl 1080.65033

Qu, Biao; Xiu, Naihua

Inverse Probl. 21, No. 5, 1655-1665 (2005).

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.

Reviewer: Rémi Vaillancourt (Ottawa)

MSC:

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

Keywords:

iterative oblique projection; split feasibility problem; \(CQ\) algorithm; convergence

Citations:

Zbl 0996.65048; Zbl 1066.65047