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.
A variable Krasnosel'skii-Mann algorithm generates a sequence {x(n)} via the formula x(n+1) = (1 - alpha(n))x(n) + alpha(n)T(n)x(n), where {alpha(n)} is a sequence in [0, 1] and {T-n} is a sequence of nonexpansive...
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A variable Krasnosel'skii-Mann algorithm generates a sequence {x(n)} via the formula x(n+1) = (1 - alpha(n))x(n) + alpha(n)T(n)x(n), where {alpha(n)} is a sequence in [0, 1] and {T-n} is a sequence of nonexpansive mappings. We will show, in a fairly general Banach space, that the sequence {x(n)} generated converges weakly. This result is used to solve the split feasibility problem which is to find a point x with the property that x is an element of C and Ax is an element of Q, where C and Q are closed convex subsets of Hilbert spaces H-1 and H-2, respectively, and A is a bounded linear operator from H-1 to H-2. The multiple-set split feasibility problem recently introduced by Censor et al is stated as finding a point x is an element of boolean AND C-N(i=1)i such that Ax is an element of boolean AND(M)(j=1) Q(j), where N and M are positive integers, {C-1,..., C-N} and {Q(1),..., Q(M)} are closed convex subsets of H-1 and H-2, respectively, and A is again a linear bounded operator from H-1 to H-2. One of the purposes of this paper is to introduce more iterative algorithms that solve this problem in the framework of infinite-dimensional Hilbert spaces.
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 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.
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, we are concerned with the split feasibility problem (SFP) whenever the convex sets involved are composed of level sets. By applying Polyak's gradient method, we get a new and simple algorithm for su...
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In this paper, we are concerned with the split feasibility problem (SFP) whenever the convex sets involved are composed of level sets. By applying Polyak's gradient method, we get a new and simple algorithm for such a problem. Under standard assumptions, we prove that the whole sequence generated by the algorithm weakly converges to a solution. We also modify the proposed algorithm and state the strong convergence without regularity conditions on the sets involved. Numerical experiments are included to illustrate its applications in signal processing.
The variable stepsize methods are effective to accelerate many iteration algorithms, one aim of them is to construct an adaptive stepsize, which has more simple and efficient format. The purpose of this paper is to in...
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ISBN:
(纸本)9781467390262
The variable stepsize methods are effective to accelerate many iteration algorithms, one aim of them is to construct an adaptive stepsize, which has more simple and efficient format. The purpose of this paper is to introduce a new simpler variable stepsize for the cq (Convexes C and Q) algorithm and to reconstruct the sparse compressed sensing data from noise. In order to solve the split feasibility problem with faster cq algorithm, through analysing the former adaptive stepsizes, the paper propsoed a much more simpler stepsize format, which can avoid to compute the objective function. Then, the convegence of to the new modified cq algorihtm is proved. In the experiment of reconstruct compressed sensing data, satisfied results not only show that the proposed modified stepsize can accelerate cq algorothm better, but also give out a new method to reconstruct the sparse signal
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.
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