Self-adaptive projection methods for the multiple-sets split feasibility problem

The multiple-sets split feasibility problem (MSFP) is to find a point closest to the intersection of a family of closed convex sets in one space, such that its image under a linear transformation will be closest to the intersection of another family of closed convex sets in the image space. This problem arises in many practical fields, and it can be a model for many inverse problems. Noting that some existing algorithms require estimating the Lipschitz constant or calculating the largest eigenvalue of the matrix, in this paper, we first introduce a self-adaptive projection method by adopting Armijo-like searches to solve the MSFP, then we focus on a special case of the MSFP and propose a relaxed self-adaptive method by using projections onto half-spaces instead of those onto the original convex sets, which is much more practical. Convergence results for both methods are analyzed. Preliminary numerical results show that our methods are practical and promising for solving larger scale MSFPs.