A note on the <i>CQ</i> algorithm for the split feasibility problem 论文

摘要

Let C and Q be nonempty closed convex sets in ℜN and ℜM, respectively, and A an M × N real matrix. The split feasibility problem (SFP) is to find x ∊ C with Ax ∊ Q, if such x exist. Byrne (2002 Inverse Problems 18 441–53) proposed a CQ algorithm with the following iterative scheme: where γ ∊ (0, 2/L), L denotes the largest eigenvalue of the matrix ATA, and PC and PQ denote the orthogonal projections onto C and Q, respectively. In his algorithm, Byrne assumed that the projections PC and PQ are easily calculated. However, in some cases it is impossible or needs too much work to exactly compute the orthogonal projection. Recently, Yang (2004 Inverse Problems 20 1261–6) presented a relaxed CQ algorithm, in which he replaced PC and PQ by and , that is, the orthogonal projections onto two halfspaces Ck and Qk, respectively. Clearly, the latter is easy to implement. One common advantage of the CQ algorithm and the relaxed CQ algorithm is that computation of the matrix inverses is not necessary. However, they use a fixed stepsize related to the largest eigenvalue of the matrix ATA, which sometimes affects convergence of the algorithms. In this paper, we present modifications of the CQ algorithm and the relaxed CQ algorithm by adopting Armijo-like searches. The modified algorithms need not compute the matrix inverses and the largest eigenvalue of the matrix ATA, and make a sufficient decrease of the objective function at each iteration. We also show convergence of the modified algorithms under mild conditions.