In this paper, we propose a novel relaxed method in order to solve the split feasibility problem in Hilbert spaces. Our algorithms involve a new strategy where the projection onto the half-space is replaced by the pro...
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In this paper, we propose a novel relaxed method in order to solve the split feasibility problem in Hilbert spaces. Our algorithms involve a new strategy where the projection onto the half-space is replaced by the projection onto the intersection of two half-spaces. In addition, our algorithms do not require any prior information about the operator norm. We also establish the weak convergence and strong convergence of the proposed algorithms under standard conditions. Finally, the results of numerical experiments indicate that the proposed algorithm is effective in the LASSO problem.
In this paper, we introduce a new relaxed method for solving the split feasibility problem in Hilbert spaces. In our method, the projection to the halfspace is replaced by the one to the intersection of two halfspaces...
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In this paper, we introduce a new relaxed method for solving the split feasibility problem in Hilbert spaces. In our method, the projection to the halfspace is replaced by the one to the intersection of two halfspaces. We give convergence of the sequence generated by our method under some suitable assumptions. Finally, we give a numerical example for illustrating the efficiency and implementation of our algorithms in comparison with existing algorithms in the literature.
In this paper, by combining the hyperplane projection technique and gradient projection method with Polyak's stepsizes, we propose a novel projection algorithm to solve the split feasibility problem in Hilbert spa...
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In this paper, by combining the hyperplane projection technique and gradient projection method with Polyak's stepsizes, we propose a novel projection algorithm to solve the split feasibility problem in Hilbert spaces. Unlike the existing projection methods, our method is designed such that the next iterate is closer to the solution set than the previous one. The weak and strong convergence of new algorithms under standard assumptions are established. We examine the performance of our method on the sparse signal recovery problem. The reported numerical results demonstrate the efficiency of the proposed method.
In this paper,by combining the inertial technique and the gradient descent method with Polyak's stepsizes,we propose a novel inertial self-adaptive gradient algorithm to solve the split feasi-bility problem in Hil...
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In this paper,by combining the inertial technique and the gradient descent method with Polyak's stepsizes,we propose a novel inertial self-adaptive gradient algorithm to solve the split feasi-bility problem in Hilbert spaces and prove some strong and weak convergence theorems of our method under standard *** examine the performance of our method on the sparse recovery prob-lem beside an example in an infinite dimensional Hilbert space with synthetic data and give some numerical results to show the potential applicability of the proposed method and comparisons with related methods emphasize it further.
In this paper, we study two inertial-type algorithms with a relaxed splitting method for solving the split feasibility problem in Hilbert spaces. Weak convergence is established without assuming conditions such as Sig...
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In this paper, we study two inertial-type algorithms with a relaxed splitting method for solving the split feasibility problem in Hilbert spaces. Weak convergence is established without assuming conditions such as Sigma(infinity)(k=1) theta(k)parallel to x(k) - x(k-1)parallel to(2) < infinity and Sigma(infinity)(k=1) parallel to x(k) - x(k-1)parallel to(2) < infinity, where theta(k) is the inertial factor. < Compared with existing results where inertial factors are usually less than one, the inertial factor in the first algorithm can be taken as nonnegative numbers greater than one. The efficiency and advantage of our algorithms are illustrated by numerical experiments.
In this paper, we analyze a Thakur three-step iterative process adapted for the context of non-self-mappings. Based on this iteration, we state and prove the existence of best proximity points for the recently introdu...
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In this paper, we analyze a Thakur three-step iterative process adapted for the context of non-self-mappings. Based on this iteration, we state and prove the existence of best proximity points for the recently introduced class of (EP)-non-self-mappings. Under certain assumptions, we study the convergence of the considered Thakur process to a best proximity point for the same class of operators. Moreover, we design a cq-type algorithm which strongly converges to a best proximity point of such kind of mappings. In addition, we present the cq-variant of the proposed algorithm that strongly converges to a best proximity pair. Some examples and numerical simulations sustain the efficiency of our new algorithms. (C) 2021 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved.
In this article, we introduce a general splitting method with linearization to solve the split feasibility problem and propose a way of selecting the stepsizes such that the implementation of the method does not need ...
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In this article, we introduce a general splitting method with linearization to solve the split feasibility problem and propose a way of selecting the stepsizes such that the implementation of the method does not need any prior information about the operator norm. We present the constant and adaptive relaxation parameters, and the latter is "optimal" in theory. These ways of selecting stepsizes and relaxation parameters are also practised to the relaxed splitting method with linearization where the two closed convex sets are both level sets of convex functions. The weak convergence of two proposed methods is established under standard conditions and the linear convergence of the general splitting method with linearization is analyzed. The numerical examples are presented to illustrate the advantage of our methods by comparing with other methods.
In this article, we consider sparse signal reconstruction problems by an alternative sec-ond order self-adaptive dynamical system . By split feasibility problem of sparse signal re-construction, we introduce a new sec...
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In this article, we consider sparse signal reconstruction problems by an alternative sec-ond order self-adaptive dynamical system . By split feasibility problem of sparse signal re-construction, we introduce a new second order self-adaptive dynamical system. Then, we prove that the proposed system has a unique solution under reasonable conditions. Fur-thermore, it is shown that the corresponding orbit of the system always converges. Finally, all kinds of numerical results on synthetic data and data from practical problems verify the efficiency of the proposed approach .(c) 2023 Elsevier Inc. All rights reserved.
The split feasibility problem (SFP) is to find x is an element of C so that Ax is an element of Q, where C and Q are non-empty closed convex subsets in Hilbert spaces H-1 and H-2, respectively, and A is a linear bound...
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The split feasibility problem (SFP) is to find x is an element of C so that Ax is an element of Q, where C and Q are non-empty closed convex subsets in Hilbert spaces H-1 and H-2, respectively, and A is a linear bounded operator from H-1 to H-2. 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 C-k(b) and Q(k)(b). 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.
In this paper, we consider the varying stepsize gradient projection algorithm (GPA) for solving the split equality problem (SEP) in Hilbert spaces, and study its linear convergence. In particular, we introduce a notio...
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In this paper, we consider the varying stepsize gradient projection algorithm (GPA) for solving the split equality problem (SEP) in Hilbert spaces, and study its linear convergence. In particular, we introduce a notion of bounded linear regularity property for the SEP, and use it to establish the linear convergence property for the varying stepsize GPA. We provide some mild sufficient conditions to ensure the bounded linear regularity property, and then conclude the linear convergence rate of the varying stepsize GPA. To the best of our knowledge, this is the first work to study the linear convergence for the SEP.
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