Self-Adaptive and Relaxed Self-Adaptive Projection Methods for Solving the Multiple-Set Split Feasibility Problem

الملخص EN

Given nonempty closed convex subsets Ci⊆Rm, i=1,2,…,t and nonempty closed convex subsets Qj⊆Rn, j=1,2,…,r, in the n- and m-dimensional Euclidean spaces, respectively.

The multiple-set split feasibility problem (MSSFP) proposed by Censor is to find a vector x∈⋂i=1tCi such that Ax∈⋂j=1rQj, where A is a given M×N real matrix.

It serves as a model for many inverse problems where constraints are imposed on the solutions in the domain of a linear operator as well as in the operator’s range.

MSSFP has a variety of specific applications in real world, such as medical care, image reconstruction, and signal processing.

In this paper, for the MSSFP, we first propose a new self-adaptive projection method by adopting Armijo-like searches, which dose not require estimating the Lipschitz constant and calculating the largest eigenvalue of the matrix ATA; besides, it makes a sufficient decrease of the objective function at each iteration.

Then we introduce a relaxed self-adaptive projection method by using projections onto half-spaces instead of those onto convex sets.

Obviously, the latter are easy to implement.

Global convergence for both methods is proved under a suitable condition.