In this article, we first introduce a modified inertial Mann algorithm and an inertial cq-algorithm by combining the accelerated Mann algorithm and the cq-algorithm with the inertial extrapolation, respectively. This ...
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In this article, we first introduce a modified inertial Mann algorithm and an inertial cq-algorithm by combining the accelerated Mann algorithm and the cq-algorithm with the inertial extrapolation, respectively. This strategy is intended to speed up the convergence of the given algorithms. Then we established the convergence theorems for two provided algorithms. For the inertial cq-algorithm, the conditions on the inertial parameters are very weak. Finally, the numerical experiments are presented to illustrate that the modified inertial Mann algorithm and inertial cq-algorithm may have a number of advantages over other methods in computing for some cases.
Let H-1, H-2 H-3 be real Hilbert spaces, let C subset of H-1, Q subset of H-2 be two nonempty closed convex level sets, let A : H-1 -> H-3, B : H-2 -> H-3 be two bounded linear operators. Our interest is in solv...
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Let H-1, H-2 H-3 be real Hilbert spaces, let C subset of H-1, Q subset of H-2 be two nonempty closed convex level sets, let A : H-1 -> H-3, B : H-2 -> H-3 be two bounded linear operators. Our interest is in solving the following new convex feasibility problem Find x is an element of C, y is an element of Q such that Ax = By, (1.1) which allows asymmetric and partial relations between the variables x and y. In this paper, we present and study the convergence of a relaxed alternating cq-algorithm (RAcqA) and show that the sequences generated by such an algorithm weakly converge to a solution of (1.1). The interest of RAcqA is that we just need projections onto half-spaces, thus making the relaxed cq-algorithm implementable. Note that, by taking B = I, in (1.1), we recover the split convex feasibility problem originally introduced in Censor and Elfving (1994) [13] and used later in intensity-modulated radiation therapy (Censor et al. (2006) [11]). We also recover the relaxed cq-algorithm introduced by Yang (2004) [8] by particularizing both B and a given parameter. (C) 2012 Elsevier Ltd. All rights reserved.
In this paper a dynamical system model is proposed for solving the split convex feasibility problem. Under mild conditions, it is shown that the proposed dynamical system globally converges to a solution of the split ...
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In this paper a dynamical system model is proposed for solving the split convex feasibility problem. Under mild conditions, it is shown that the proposed dynamical system globally converges to a solution of the split convex feasibility problem. An exponential convergence is obtained provided that the bounded linear regularity property is satisfied. The validity and transient behavior of the dynamical system is demonstrated by several numerical examples. The method proposed in this paper can be regarded as not only a continuous version but also an interior version of the known cq-method for solving the split convex feasibility problem.
A modular string averaging procedure (MSA, for short) for a finite number of operators was first introduced by Reich and Zalas in 2016. The MSA concept provides a flexible algorithmic framework for solving various fea...
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A modular string averaging procedure (MSA, for short) for a finite number of operators was first introduced by Reich and Zalas in 2016. The MSA concept provides a flexible algorithmic framework for solving various feasibility problems such as common fixed point and convex feasibility problems. In 2001 Bauschke and Combettes introduced the notion of coherence and applied it to proving weak and strong convergence of many iterative methods. In 2019 Barshad, Reich and Zalas proposed a stronger variant of coherence which provides a more convenient sufficient convergence condition for such *** this paper we combine the ideas of both modular string averaging and coherence. Focusing on extending the above MSA procedure to an infinite sequence of operators with admissible controls, we establish strong coherence of its output operators. Various applications of these concepts are presented with respect to weak and strong convergence. They also provide important generalizations of known results, where the weak convergence of sequences of operators generated by the MSA procedure with intermittent controls was considered.
We study the multiple-sets split feasibility problem that requires to find a point closest to a family of closed convex sets in one space such that its image under a linear transformation will be closest to another fa...
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We study the multiple-sets split feasibility problem that requires to find a point closest to a family of closed convex sets in one space such that its image under a linear transformation will be closest to another family of closed convex sets in the image space. By casting the problem into an equivalent problem in a suitable product space we are able to present a simultaneous subgradients projections algorithm that generates convergent sequences of iterates in the feasible case. We further derive and analyze a perturbed projection method for the multiple-sets split feasibility problem and, additionally, furnish alternative proofs to two known results. (c) 2006 Elsevier Inc. All rights reserved.
Many inverse problems can be formulated as split feasibility problems. To find feasible solutions, one has to minimize proximity functions. We show that the existence of minimizers to the proximity function for Censor...
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Many inverse problems can be formulated as split feasibility problems. To find feasible solutions, one has to minimize proximity functions. We show that the existence of minimizers to the proximity function for Censor-Elfving's split feasibility problem is equivalent to the existence of projections on appropriate convex sets and provide conditions under which such projections exist. These projections turn out to be the unique optimal solution of their Fenchel-Rockafellar duals and can be computed by the proximal point algorithm efficiently. Applications to linear equations and linear feasibility problems are given.
A split feasibility formulation for the inverse problem of intensity-modulated radiation therapy treatment planning with dose-volume constraints included in the planning algorithm is presented. It involves a new type ...
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A split feasibility formulation for the inverse problem of intensity-modulated radiation therapy treatment planning with dose-volume constraints included in the planning algorithm is presented. It involves a new type of sparsity constraint that enables the inclusion of a percentage-violation constraint in the model problem and its handling by continuous (as opposed to integer) methods. We propose an iterative algorithmic framework for solving such a problem by applying the feasibility-seeking cq-algorithm of Byrne combined with the automatic relaxation method that uses cyclic projections. Detailed implementation instructions are furnished. Functionality of the algorithm was demonstrated through the creation of an intensity-modulated proton therapy plan for a simple 2D C-shaped geometry and also for a realistic base-of-skull chordoma treatment site. Monte Carlo simulations of proton pencil beams of varying energy were conducted to obtain dose distributions for the 2D test case. A research release of the Pinnacle(3) proton treatment planning system was used to extract pencil beam doses for a clinical base-of-skull chordoma case. In both cases the beamlet doses were calculated to satisfy dose-volume constraints according to our new algorithm. Examination of the dose-volume histograms following inverse planning with our algorithm demonstrated that it performed as intended. The application of our proposed algorithm to dose-volume constraint inverse planning was successfully demonstrated. Comparison with optimized dose distributions from the research release of the Pinnacle(3) treatment planning system showed the algorithm could achieve equivalent or superior results.
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