A generalized fractional programming problem is defined as the problem of minimizing a nonlinear function, defined as the maximum of several ratios of functions on a feasible domain. In this paper, we propose new meth...
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A generalized fractional programming problem is defined as the problem of minimizing a nonlinear function, defined as the maximum of several ratios of functions on a feasible domain. In this paper, we propose new methods based on the method of centers, on the proximal point algorithm and on the idea of bundle methods, for solving such problems. First, we introduce proximal point algorithms, in which, at each iteration, an approximate prox-regularized parametric subproblem is solved inexactly to obtain an approximate solution to the original problem. Based on this approach and on the idea of bundle methods, we propose implementable proximal bundle algorithms, in which the objective function of the last mentioned prox-regularized parametric subproblem is replaced by an easier one, typically a piecewise linear function. The methods deal with nondifferentiable nonlinearly constrained convex minimax fractional problems. We prove the convergence, give the rate of convergence of the proposed procedures and present numerical tests to illustrate their behavior.
In this paper, we propose a new modified proximal point algorithm for finding a common element of the set of common minimizers of a finite family of convex and lower semi-continuous functions and the set of common fix...
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In this paper, we propose a new modified proximal point algorithm for finding a common element of the set of common minimizers of a finite family of convex and lower semi-continuous functions and the set of common fixed points of a finite family of nonexpansive mappings in complete CAT(0) spaces, and prove some convergence theorems of the proposed algorithm under suitable conditions. A numerical example is presented to illustrate the proposed method and convergence result. Our results improve and extend the corresponding results existing in the literature.
We study preconditioned algorithms of alternating direction method of multipliers type for nonsmooth optimization problems. The alternating direction method of multipliers is a popular first-order method for general c...
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We study preconditioned algorithms of alternating direction method of multipliers type for nonsmooth optimization problems. The alternating direction method of multipliers is a popular first-order method for general constrained optimization problems. However, one of its drawbacks is the need to solve implicit subproblems. In various applications, these subproblems are either easily solvable or linear, but nevertheless challenging. We derive a preconditioned version that allows for flexible and efficient preconditioning for these linear subproblems. The original and preconditioned version is written as a new kind of proximalpoint method for the primal problem, and the weak (strong) convergence in infinite (finite) dimensional Hilbert spaces is proved. Various efficient preconditioners with any number of inner iterations may be used in this preconditioned framework. Furthermore, connections between the preconditioned version and the recently introduced preconditioned Douglas-Rachford method for general nonsmooth problems involving quadratic-linear terms are established. The methods are applied to total variation denoising problems, and their benefits are shown in numerical experiments.
Recently, Xu (J Glob Optim 36:115-125 (2006)) introduced a regularized proximal point algorithm for approximating a zero of a maximal monotone operator. In this note, we shall prove the strong convergence of this algo...
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Recently, Xu (J Glob Optim 36:115-125 (2006)) introduced a regularized proximal point algorithm for approximating a zero of a maximal monotone operator. In this note, we shall prove the strong convergence of this algorithm under some weaker conditions.
It is known, by Rockafellar (SIAM J Control Optim 14:877-898, 1976), that the proximal point algorithm (PPA) converges weakly to a zero of a maximal monotone operator in a Hilbert space, but it fails to converge stron...
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It is known, by Rockafellar (SIAM J Control Optim 14:877-898, 1976), that the proximal point algorithm (PPA) converges weakly to a zero of a maximal monotone operator in a Hilbert space, but it fails to converge strongly. Lehdili and Moudafi (Optimization 37:239-252, 1996) introduced the new prox-Tikhonov regularization method for PPA to generate a strongly convergent sequence and established a convergence property for it by using the technique of variational distance in the same space setting. In this paper, the prox-Tikhonov regularization method for the proximal point algorithm of finding a zero for an accretive operator in the framework of Banach space is proposed. Conditions which guarantee the strong convergence of this algorithm to a particular element of the solution set is provided. An inexact variant of this method with error sequence is also discussed.
In this paper, we give a sufficient condition which guarantees that the sequence generated by the proximal point algorithm terminates after a finite number of iterations.
In this paper, we give a sufficient condition which guarantees that the sequence generated by the proximal point algorithm terminates after a finite number of iterations.
We present several strong convergence results for the modified, Halpern-type, proximal point algorithm x(n+1) = alpha(n)u + (1 - alpha(n)) J(beta nxn) + e(n) (n = 0, 1,...;u, x(0) is an element of H given, and J(beta ...
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We present several strong convergence results for the modified, Halpern-type, proximal point algorithm x(n+1) = alpha(n)u + (1 - alpha(n)) J(beta nxn) + e(n) (n = 0, 1,...;u, x(0) is an element of H given, and J(beta n) = (I + beta(n)A)(-1), for a maximal monotone operator A) in a real Hilbert space, under new sets of conditions on alpha(n) is an element of (0, 1) and beta(n) is an element of (0, infinity). These conditions are weaker than those known to us and our results extend and improve some recent results such as those of H. K. Xu. We also show how to apply our results to approximate minimizers of convex functionals. In addition, we give convergence rate estimates for a sequence approximating the minimum value of such a functional.
In this paper, we introduce a new modified proximal point algorithm involving fixed point iterates of nonexpansive mappings in CAT(0) spaces and prove that the sequence generated by our iterative process converges to ...
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In this paper, we introduce a new modified proximal point algorithm involving fixed point iterates of nonexpansive mappings in CAT(0) spaces and prove that the sequence generated by our iterative process converges to a minimizer of a convex function and a fixed point of mappings.
In medical imaging many conventional regularization methods, such as total variation or total generalized variation, impose strong prior assumptions which can only account for very limited classes of images. A more re...
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In medical imaging many conventional regularization methods, such as total variation or total generalized variation, impose strong prior assumptions which can only account for very limited classes of images. A more reasonable sparse representation frame for images is still badly needed. Visually understandable images contain meaningful patterns, and combinations or collections of these patterns can be utilized to form some sparse and redundant representations which promise to facilitate image reconstructions. In this work, we propose and study block matching sparsity regularization (BMSR) and devise an optimization program using BMSR for computed tomography (CT) image reconstruction for an incomplete projection set. The program is built as a constrained optimization, minimizing the L1-norm of the coefficients of the image in the transformed domain subject to data observation and positivity of the image itself. To solve the program efficiently, a practical method based on the proximal point algorithm is developed and analyzed. In order to accelerate the convergence rate, a practical strategy for tuning the BMSR parameter is proposed and applied. The experimental results for various settings, including real CT scanning, have verified the proposed reconstruction method showing promising capabilities over conventional regularization.
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