Feasibility Problem In Hilbert Spaces Research Articles

A New Hybrid CQ Algorithm for the Split Feasibility Problem in Hilbert Spaces and Its Applications to Compressed Sensing

In this paper, we focus on studying the split feasibility problem (SFP), which has many applications in signal processing and image reconstruction. A popular technique is to employ the iterative method which is so called the relaxed CQ algorithm. However, the speed of convergence usually depends on the way of selecting the step size of such algorithms. We aim to suggest a new hybrid CQ algorithm for the SFP by using the self adaptive and the line-search techniques. There is no computation on the inverse and the spectral radius of a matrix. We then prove the weak convergence theorem under mild conditions. Numerical experiments are included to illustrate its performance in compressed sensing. Some comparisons are also given to show the efficiency with other CQ methods in the literature.

On Inertial Relaxation CQ Algorithm for Split Feasibility Problems

In this work, we introduce an inertial relaxation CQ algorithm for the split feasibility problem in Hilbert spaces. We prove weak convergence theorem under suitable conditions. Numerical examples illustrating our methods’s efficiency are presented for comparing some known methods.

Extragradient and linesearch methods for solving split feasibility problems in Hilbert spaces

Utilizing the Tikhonov regularization method and extragradient and linesearch methods, some new extragradient and linesearch algorithms have been introduced in the framework of Hilbert spaces. In the presented algorithms, the convexity of optimization subproblems is assumed, which is weaker than the strong convexity assumption that is usually supposed in the literature, and also, the auxiliary equilibrium problem is not used. Some strong convergence theorems for the sequences generated by these algorithms have been proven. It has been shown that the limit point of the generated sequences is a common element of the solution set of an equilibrium problem and the solution set of a split feasibility problem in Hilbert spaces. To illustrate the usability of our results, some numerical examples are given. Optimization subproblems in these examples have been solved by FMINCON toolbox in MATLAB.

Iterative Regularization Methods for the Multiple-Sets Split Feasibility Problem in Hilbert Spaces

In this paper, we introduce iterative regularization methods for solving the multiple-sets split feasibility problem, that is to find a point closest to a family of closed convex subsets in one space such that its image under a bounded linear mapping will be closest to another family of closed convex subsets in the image space. We consider the cases, when the families are either finite or infinite. We also give two numerical examples for illustrating our main method.

A new relaxed CQ algorithm for solving split feasibility problems in Hilbert spaces and its applications

Inspired by the works of Lopez et al. [ 21 ] and the recent paper of Dang et al. [ 15 ], we devise a new inertial relaxation of the CQ algorithm for solving Split Feasibility Problems (SFP) in real Hilbert spaces. Under mild and standard conditions we establish weak convergence of the proposed algorithm. We also propose a Mann-type variant which converges strongly. The performances and comparisons with some existing methods are presented through numerical examples in Compressed Sensing and Sparse Binary Tomography by solving the LASSO problem.

A New CQ Algorithm for Solving Split Feasibility Problems in Hilbert Spaces

In this paper, we propose a CQ-type algorithm for solving the split feasibility problem (SFP) in real Hilbert spaces. The algorithm is designed such that the step-sizes are directly computed at each iteration. We will show that the sequence generated by the proposed algorithm converges in norm to the minimum-norm solution of the SFP under appropriate conditions. In addition, we give some numerical examples to verify the implementation of our method. Our result improves and complements many known related results in the literature.

Iterative algorithms for solving the split feasibility problem in Hilbert spaces

In this paper, we introduce two iterative algorithms for the split feasibility problem in real Hilbert spaces by reformulating it as a fixed point equation. Under suitable conditions, weak and strong convergence theorems are established. As a consequence, we obtain weak and strong convergence iterative sequences for the split equality problem introduced by Moudafi. The efficiency of the proposed algorithms is illustrated by numerical experiments. Our results improve and extend the corresponding results announced by many others.

A fixed point method for solving a split feasibility problem in Hilbert spaces

In this paper, a fixed method is introduced and investigated for solving a split feasibility problem. A strong convergence theorem of solutions is established in the framework of infinite dimensional Hilbert spaces. As an application, a split equality problem is also investigated.

Viscosity approximation of solutions of a split feasibility problem in Hilbert spaces

Viscosity approximation of solutions of a split feasibility problem in Hilbert spaces

Strong convergence result for proximal split feasibility problem in Hilbert spaces

In this paper we describe and analyse new computational technique for solving proximal split feasibility problem (SFP) using a modified proximal split feasibility algorithm. The two convex and lower semi-continuous objective functions are assumed to be non-smooth. Some application to SFP are given. We demonstrate the computational efficiency of the proposed algorithm with nontrivial numerical experiments. We also compare our method with other relevant methods in the literature in terms of convergence, stability, efficiency and implementation with our illustrative numerical examples.

A regularization algorithm for a splitting feasibility problem in Hilbert spaces

A regularization algorithm for a splitting feasibility problem in Hilbert spaces

An approach for the convex feasibility problem via Monotropic Programming

In this note, we use recent zero duality results arising from Monotropic Programming problem for analyzing consistency of the convex feasibility problem in Hilbert spaces. We characterize consistency in terms of the lower semicontinuity of the infimal convolution of the associated support functions.

Linear convergence of CQ algorithms and applications in gene regulatory network inference

In the present paper, we consider the varying stepsize CQ algorithm for solving the split feasibility problem in Hilbert spaces, investigate the linear convergence issue and explore an application in systems biology. In particular, we introduce a notion of bounded linear regularity property for the split feasibility problem, and use it to establish the linear convergence property for the varying stepsize CQ algorithm when using some suitable types of stepsizes, which covers most types of stepsizes used in the literature of CQ algorithms. We also provide some mild sufficient conditions for ensuring this bounded linear regularity property, and then conclude the linear convergence rate of the varying stepsize CQ algorithm for many application cases. To the best of our knowledge, this is the first work to study the linear convergence rate of CQ algorithms. In the aspect of application, we consider the gene regulatory network inference arising in systems biology, which is formulated as a group Dantzig selector and then cast into a split feasibility problem. The numerical study on gene expression data of mouse embryonic stem cell shows that the varying stepsize CQ algorithm is applicable to gene regulatory network inference in the sense that it obtains a reliable solution matching with biological standards.

Iterative algorithms for the multiple-sets split feasibility problem in Hilbert spaces

In this paper, for the multiple-sets split feasibility problem, that is to find a point closest to a family of closed convex subsets in one space such that its image under a linear bounded mapping will be closest to another family of closed convex subsets in the image space, we study several iterative methods for finding a solution, which solves a certain variational inequality. We show that particular cases of our algorithms are some improvements for existing ones in literature. We also give two numerical examples for illustrating our algorithms.

Regularized gradient-projection methods for finding the minimum-norm solution of the constrained convex minimization problem

Let H be a real Hilbert space and C be a nonempty closed convex subset of H. Assume that g is a real-valued convex function and the gradient ∇g is frac{1}{L}-ism with L>0. Let 0<lambda <frac{2}{L+2}, 0<beta_{n}<1. We prove that the sequence {x_{n}} generated by the iterative algorithm x_{n+1}=P_{C}(I-lambda(nabla g+beta_{n}I))x_{n}, forall ngeq0 converges strongly to qin U, where q=P_{U}(0) is the minimum-norm solution of the constrained convex minimization problem, which also solves the variational inequality langle-q, p-qrangleleq0, forall pin U. Under suitable conditions, we obtain some strong convergence theorems. As an application, we apply our algorithm to solving the split feasibility problem in Hilbert spaces.

Iterative methods for generalized split feasibility problems in Banach spaces

Inspired by the recent work of Takahashi et al. [W. Takahashi, H.-K. Xu and J.-C. Yao, Iterative methods for generalized split feasibility problems in Hilbert spaces, Set-Valued Var. Anal., 23 (2015), 205–221], in this paper, we study generalized split feasibility problems (GSFPs) in the setting of Banach spaces. We propose iterative algorithms to compute the approximate solutions of such problems. The weak convergence of the sequence generated by the proposed algorithms is studied. As applications, we derive some algorithms and convergence results for some problems from nonlinear analysis, namely, split feasibility problems, equilibrium problems, etc. Our results generalize several known results in the literature including the results of Takahashi et al. [W. Takahashi, H.-K. Xu and J.-C. Yao, Iterative methods for generalized split feasibility problems in Hilbert spaces, SetValued Var. Anal., 23 (2015), 205–221].

Iterative methods of strong convergence theorems for the split feasibility problem in Hilbert spaces.

In this paper, we propose several new iterative algorithms to solve the split feasibility problem in the Hilbert spaces. By virtue of new analytical techniques, we prove that the iterative sequence generated by these iterative procedures converges to the solution of the split feasibility problem which is the best close to a given point. In particular, the minimum-norm solution can be found via our iteration method.

Splitting methods for a convex feasibility problem in Hilbert spaces

Splitting methods for a convex feasibility problem in Hilbert spaces

New strong convergence theorems for split variational inclusion problems in Hilbert spaces

The split variational inclusion problem is an important problem, and it is a generalization of the split feasibility problem. In this paper, we present a descent-conjugate gradient algorithm for the split variational inclusion problems in Hilbert spaces. Next, a strong convergence theorem of the proposed algorithm is proved under suitable conditions. As an application, we give a new strong convergence theorem for the split feasibility problem in Hilbert spaces. Finally, we give numerical results for split variational inclusion problems to demonstrate the efficiency of the proposed algorithm.

A general convergence theorem for multiple-set split feasibility problem in Hilbert spaces

We establish strong convergence result of split feasibility problem for a family of quasi-nonexpansive multi-valued mappings and a total asymptotically strict pseudo-contractive mapping in infinite dimensional Hilbert spaces.