Abstract
In this paper, we present some modified relaxed CQ algorithms with different kinds of step size and perturbation to solve the Multiple-sets Split Feasibility Problem (MSSFP). Under mild assumptions, we establish weak convergence and prove the bounded perturbation resilience of the proposed algorithms in Hilbert spaces. Treating appropriate inertial terms as bounded perturbations, we construct the inertial acceleration versions of the corresponding algorithms. Finally, for the LASSO problem and three experimental examples, numerical computations are given to demonstrate the efficiency of the proposed algorithms and the validity of the inertial perturbation.
Highlights
We focus on the Multiple-sets Split Feasibility Problem (MSSFP), which is formulated as follows
Motivated by [9,18], we focus on the modified relaxed CQ algorithms to solve the MSSFP (1) in real Hilbert spaces and assert that the proposed algorithms are bounded-perturbation-resilient
Given the same number of iterations, the recovered signals generated by algorithms in this paper outperform the one generated by Yang’s algorithm; NP1 needs more CPU time and presents lower accuracy; algorithms with self-adaptive step size perform better than the algorithms with step size determined by Armijo-line search in CPU time and imposing inertial perturbation accelerates the convergence rate and accuracy of signal recovery
Summary
We focus on the Multiple-sets Split Feasibility Problem (MSSFP), which is formulated as follows. PQkj (Ax) 2, where the closed convex sets Cik and Qkj are level sets of some convex functions containing Ci and Qj, and self-adaptive step size αk = They proved that the sequence {xk} generated by algorithm (6) converges in norm to PS(μ), where S is the solution set of the MSSFP. 2, and they proved the weak convergence of the iteration sequence in Hilbert spaces The advantage of this choice of the step size lies in the fact that neither prior information about the matrix norm A nor any other conditions on Q and A are required. There are relatively fewer documents studying the algorithms of the (multiple-sets) split feasibility problem with perturbations, especially with self-adaptive step size. The latter has a bounded disturbance recovery property.
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