Abstract
AbstractWe introduce an iterative algorithm for solving the multiple-sets split feasibility problem with splitting self-adaptive step size. This step size is calculated directly from the iteration process without need to know the spectral norm of linear operators. We also generalize the chosen step size to a relaxed iterative algorithm. Theoretical convergence is proved in an infinite dimensional Hilbert space. Some numerical experiments are presented to verify the effectiveness of our proposed methods.
Highlights
Linear inverse problems often arise in many real-world application problems such as signal and image processing, medical image reconstruction, etc
In Section, we propose an iterative algorithm with splitting self-adaptive step size and prove its convergence
We proposed a new iterative algorithm with splitting self-adaptive step size
Summary
Linear inverse problems often arise in many real-world application problems such as signal and image processing, medical image reconstruction, etc. In Section , we propose an iterative algorithm with splitting self-adaptive step size and prove its convergence. Letting t = r = , the MSSFP reduces to the split feasibility problem (SFP) as follows: Find a point x∗ ∈ C such that Ax∗ ∈ Q, where C ⊆ H and Q ⊆ H are nonempty, closed and convex sets, respectively. ([ ]) Suppose that f : Rn → R is a convex function, it is subdifferentiable everywhere and its subdifferentials are uniformly bounded on any bounded subset of Rn. we propose a new iteration method with splitting self-adaptive step size for solving the MSSFP. In order to prove that the iterative sequence {xk} is Fejér-monotone with respect to , we make the following estimations, which are based on the property of projection operator Since we found that all the three iterative methods exhibit nearly the same performance when the K -sparse signal is recovered, so we report the objective function values when the iteration process is stopped under
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