Abstract
Given nonempty closed convex subsets , and nonempty closed convex subsets , , in the - and -dimensional Euclidean spaces, respectively. The multiple-set split feasibility problem (MSSFP) proposed by Censor is to find a vector such that , where is a given 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 ; 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.
Highlights
The multiple-sets split feasibility problem MSSFP requires to find a point closest to a family of closed convex sets in one space such that its image under a linear transformation will be closest to another family of closed convex sets in the image space
For a given nonempty closed convex set Ω in Rn, the orthogonal projection from Rn onto Ω is defined by PΩ x argmin x − y | y ∈ Ω, x ∈ Rn
Algorithm 3.1 need not estimate the largest eigenvalue of the matrix AT A, and the stepsize τk is chosen so that the objective function q x has a sufficient decrease. It is a special case of the standard gradient projection method with the Armijo-like search rule for solving the constrained optimization problem: min g x ; x ∈ Ω, 3.7 where Ω ⊆ Rn is a nonempty closed convex set, and the function g x is continuously differentiable on Ω, the following convergence result ensures the convergence of Algorithm 3.1
Summary
The multiple-sets split feasibility problem MSSFP requires to find a point closest to a family of closed convex sets in one space such that its image under a linear transformation will be closest to another family of closed convex sets in the image space. In 2005, Qu and Xiu 6 modified the CQ algorithm 7 and relaxed CQ algorithm 8 by adopting Armijo-like searches to solve the SFP, where the second algorithm used orthogonal projections onto half-spaces instead of projections onto the original convex sets, just as Yang’s relaxed CQ algorithm 8. This may reduce a lot of work for computing projections, since projections onto half-spaces can be directly calculated.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
