Several solution methods for the split feasibility problem

Let C and Q be nonempty, closed convex sets in R-n and R-m respectively, and A be an in x n real matrix. The split feasibility problem (SFP) is to find x is an element of C with Ax is an element of Q, if such points exist. Byrne proposed the following CQ algorithm to solve the SFP: x(k+1) = P-C(x(k) -gamma A(T)(I - P-Q) Ax(k)) where gamma is an element of (0, 2/rho (A(T) A)) with rho(A(T) A) the spectral radius of the matrix A(T) A and P-C and P-Q denote the orthogonal projections onto C and Q, respectively. However, in some cases, it is difficult or even impossible to compute PC and PQ exactly. In this paper, based on the CQ algorithm, we present several algorithms to solve the SFR Compared with the CQ algorithm, our algorithms are more practical and easier to implement. They can be regarded as improvements of the CQ algorithm.