Abstract
In this paper, we propose an iterative algorithm with inertial extrapolation to approximate the solution of multiple-set split feasibility problem. Based on Lopez et al. (Inverse Probl. 28(8):085004, 2012), we have developed a self-adaptive technique to choose the stepsizes such that the implementation of our algorithm does not need any prior information about the operator norm. We then prove the strong convergence of a sequence generated by our algorithm. We also present numerical examples to illustrate that the acceleration of our algorithm is effective.
Highlights
Throughout the paper, unless otherwise stated, we assume H1 and H2 are two real Hilbert spaces, and A : H1 → H2 is a bounded linear operator.The split feasibility problem (SFP) is a problem of finding a point xwith a property x ∈ C such that Ax ∈ Q, (1)where C and Q are nonempty closed convex subsets of H1 and H2, respectively
This paper contributes a strongly convergent iterative algorithm for multipleset split feasibility problem (MSSFP) with inertial effect (extrapolated point xn + αn(xn – xn–1), rather than xn itself ) in the direction of half-space relaxation (assuming Ci and Qj are given as sublevel sets of convex functions (7)) where the projection onto half-spaces (8) and (9) is computed in parallel and a priori knowledge of the operator norm is not required
We introduce the extended form of the way of selecting stepsize used by Lopez et al [1], to work for MSSFP framework, and we analyze the strong convergence of our proposed algorithm
Summary
Throughout the paper, unless otherwise stated, we assume H1 and H2 are two real Hilbert spaces, and A : H1 → H2 is a bounded linear operator. This paper contributes a strongly convergent iterative algorithm for MSSFP with inertial effect (extrapolated point xn + αn(xn – xn–1), rather than xn itself ) in the direction of half-space relaxation (assuming Ci and Qj are given as sublevel sets of convex functions (7)) where the projection onto half-spaces (8) and (9) is computed in parallel and a priori knowledge of the operator norm is not required For this purpose, we introduce the extended form of the way of selecting stepsize used by Lopez et al [1], to work for MSSFP framework, and we analyze the strong convergence of our proposed algorithm.
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