In this paper, we introduce a new self-adaptive CQ algorithm for solving split feasibility problems in real Hilbert spaces The algorithm is designed, such that the stepsizes are directly computed at each iteration We also consider the corresponding relaxed CQ algorithm for the proposed method Under certain mild conditions, we establish weak convergence of the proposed algorithm as well as strong convergence of its hybrid-type variant Finally, numerical examples illustrating the efficiency of our algorithm in solving the LASSO problem are presented

A new self-adaptive CQ algorithm with an application to the LASSO problem

In this paper, we use the dual variable to propose a self-adaptive iterative algorithm for solving the split common fixed point problems of averaged mappings in real Hilbert spaces. Under suitable conditions, we get the weak convergence of the proposed algorithm and give applications in the split feasibility problem and the split equality problem. Some numerical experiments are given to illustrate the efficiency of the proposed iterative algorithm. Our results improve and extend the corresponding results announced by many others.

A self-adaptive iterative algorithm for the split common fixed point problems

The split feasibility problem (SFP) is to find so that , where C and Q are non-empty closed convex subsets in Hilbert spaces and , respectively, and A is a linear bounded operator from to . Byrne proposed an iterative method called the CQ algorithm that involves the orthogonal projections onto C and Q. However, the projections onto C and Q might be hard to be implemented in general. In this paper, we propose a ball-relaxed projection method for the SFP. Instead of half spaces, we replace C and Q in the proposed algorithm by two properly chosen closed balls and . Since the projection onto the closed ball has closed form, the proposed algorithm is thus easy to be implemented. Under some mild conditions, we establish the weak convergence of the proposed algorithm to a solution of the SFP. As an application, we obtain new algorithms for solving the split equality problem. Preliminary numerical experiments show the efficiency of the proposed method.

The ball-relaxed CQ algorithms for the split feasibility problem

In this work, we suggest modifications of the self-adaptive method for solving the split feasibility problem and the fixed point problem of nonexpansive mappings in the framework of Banach spaces. ...

Iterative methods for the split feasibility problem and the fixed point problem in Banach spaces

In this paper, we study the the split common fixed point problem in Hilbert spaces. We establish a weak convergence theorem for the method recently introduced by Wang, which extends a existing result from firmly nonexpansive mappings to strictly pseudo-contractive mappings. Moreover, our condition that guarantees the weak convergence is much weaker than that of Wang’s. A strong convergence theorem is also obtained under some additional conditions. As an application, we obtain several new methods for solving various split inverse problems and split equality problems. Numerical examples are included to illustrate the applications in signal processing of the proposed algorithm.

Iterative solutions of the split common fixed point problem for strictly pseudo-contractive mappings

The split common fixed-point problem is an inverse problem that consists in finding an element in a fixed-point set such that its image under a linear transformation belongs to another fixed-point set. In this paper, we propose a new algorithm for the split common fixed-point problem that does not need any priori information of the operator norm. Under standard assumptions, we establish a weak convergence theorem of the proposed algorithm.

Fenghui Wang

Papers

A new method for split common fixed-point problem without priori knowledge of operator norms