Abstract
In this paper, we propose a catalog of iterative methods for solving the Split Feasibility Problem in the non-convex setting. We study four different optimization formulations of the problem, where each model has advantages in different settings of the problem. For each model, we study relevant iterative algorithms, some of which are well-known in this area and some are new. All the studied methods, including the well-known CQ Algorithm, are proven to have global convergence guarantees in the non-convex setting under mild conditions on the problem’s data.
Highlights
In 1994 Censor and Elfving [17] introduced the Split Convex Feasibility (SCF) Problem, which is formulated as follows
Algorithms with a provable convergence theory, and since we focus on the non-convex setting, we mean by that algorithms with convergence guarantees to critical points of the used optimization model
Here we focus on the non-convex setting, we point interested readers to [7] and the recent book [6], where theoretical results on the Projected Gradient (PG) method can be found, including accelerations and various modifications
Summary
In 1994 Censor and Elfving [17] introduced the (two-sets) Split Convex Feasibility (SCF) Problem, which is formulated as follows. While the literature on iterative methods for solving the SCF Problem is focused on the optimization model (SF3) and to the best of our understanding most algorithms are modifications and/or generalizations of the CQ Algorithm for other settings of the problem or other related problems with similar affinities, there is one recent paper [18] (as far as we know) that study the CQ Algorithm in the non-convex setting. It should be noted that in this paper we focus on theoretical guarantees in the sense of global convergence of the whole generated sequence, in comparison to sub-sequences convergence, to critical points of the tackled model Another type of theoretical guarantees that can be achieved for such algorithms in the non-convex setting is of a local nature and consist of results of linear rate of convergence.
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