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The relaxed CQ algorithm solving the split feasibility problem. (English) Zbl 1066.65047
Let C and Q be nonempty closed convex sets in N and M , respectively, and A an M×N real matrix. The split feasibility problem (SFP) is to find xC with AxQ, if such x exits. The author presents the following relaxed CQ algorithm: x k+1 =P C k (x k +γA T (P Q k -I)Ax k ) for solving the SFP by replacing the sets C={x N |c(x)0} and Q={y M |q(y)0}, where c and q are convex functions, by the larger sets C k ={x N |c(x k )+ξ k ,x-x k 0} and Q k ={y M |q(Ax k )+η k ,y-Ax k 0} where the subgradients are ξ k c(x k ) and η k q(Ax k ). Here P C k and P Q k are projections onto C k and Q k , and γ(0,2/L) where L denotes the largest eigenvalue of A T A. The convergence of the algorithm is established and another algorithm for SFP is given.

MSC:
65F30Other matrix algorithms
65F10Iterative methods for linear systems