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.
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. ...
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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.
In this paper, we regard the cq algorithm as a fixed point algorithm for averaged mapping, and also try to study the split feasibility problem by the following hybrid steepest method;where {(n)}(0,1). It is noted that...
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In this paper, we regard the cq algorithm as a fixed point algorithm for averaged mapping, and also try to study the split feasibility problem by the following hybrid steepest method;where {(n)}(0,1). It is noted that Xu's original iterative method can conclude only weak convergence. Consequently, we obtain the sequence {x(n)} generated by our iteration method converges strongly to , where is the unique solution of the variational inequality Our result extends and improves the result of Xu, as shown in the literature, from weak to strong convergence theorems. Finally, in the last section, numerical examples for study behavior convergence analysis of this algorithm are obtained.
The purpose of this paper is to present accelerations of the Mann and cq algorithms. We first apply the Picard algorithm to the smooth convex minimization problem and point out that the Picard algorithm is the steepes...
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The purpose of this paper is to present accelerations of the Mann and cq algorithms. We first apply the Picard algorithm to the smooth convex minimization problem and point out that the Picard algorithm is the steepest descent method for solving the minimization problem. Next, we provide the accelerated Picard algorithm by using the ideas of conjugate gradient methods that accelerate the steepest descent method. Then, based on the accelerated Picard algorithm, we present accelerations of the Mann and cq algorithms. Under certain assumptions, we show that the new algorithms converge to a fixed point of a nonexpansive mapping. Finally, we show the efficiency of the accelerated Mann algorithm by numerically comparing with the Mann algorithm. A numerical example is provided to illustrate that the acceleration of the cq algorithm is ineffective.
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
The split feasibility problem(SFP) is to find a point x ∈ C such that Ax ∈ Q,where A:H1 → H2 is a bounded linear operator,and C and Q be nonempty closed convex subset of Hilbert space H1 and H2,*** this paper,we pr...
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The split feasibility problem(SFP) is to find a point x ∈ C such that Ax ∈ Q,where A:H1 → H2 is a bounded linear operator,and C and Q be nonempty closed convex subset of Hilbert space H1 and H2,*** this paper,we proposed a relaxed cq algorithm for solving split feasibility *** iterative algorithm generates a sequence {xn} as follows xn+1 =(1-αn)xn + αn PCn(xn-γA*(I-PQn)Axn),n ≥ 0,where 0 < γ < 2/ A 2,x0 ∈ H1,PCnand PQnare the nearest point projections onto Cn and Qn,*** we proved that cq algorithm converges weakly to a solution of the SFP.
In this article, we propose a new algorithm for solving the multiple-sets split feasibility problem in Banach spaces. Under some mild conditions we establish the norm convergence of the proposed algorithm. One advanta...
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In this article, we propose a new algorithm for solving the multiple-sets split feasibility problem in Banach spaces. Under some mild conditions we establish the norm convergence of the proposed algorithm. One advantage of the proposed algorithm is that its norm convergence does not need the weak-to-weak continuity of the duality mapping.
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.
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