Douglas-Rachford Splitting Algorithm Research Articles

Convergence analysis of the generalized Douglas-Rachford splitting method under Hölder subregularity assumptions

<p style='text-indent:20px;'>In solving convex optimization and monotone inclusion problems, operator-splitting methods are often employed to transform optimization and inclusion problems into fixed-point equations as the equations obtained from operator-splitting methods are often easy to be solved by standard techniques. For the inclusion problem involving two maximally monotone operators, under the Hölder metric subregularity of the concerned operator, which is weaker than the strong monotonicity of the operator, we derive relationships between the convergence rate of the generalized Douglas-Rachford splitting algorithm and the order of the Hölder metric subregularity of the concerned operator. Moreover, for general multifunctions in Hilbert spaces, by proximal coderivative, we provided some dual sufficient conditions for Hölder metric subregularity.</p>

Directional asymptotics of Fejér monotone sequences

The notion of Fejér monotonicity is instrumental in unifying the convergence proofs of many iterative methods, such as the Krasnoselskii–Mann iteration, the proximal point method, the Douglas-Rachford splitting algorithm, and many others. In this paper, we present directionally asymptotical results of strongly convergent subsequences of Fejér monotone sequences. We also provide examples to show that the sets of directionally asymptotic cluster points can be large and that weak convergence is needed in infinite-dimensional spaces.

A CT reconstruction method based on constrained data fidelity range estimation.

For the CT iterative reconstruction, choosing the parameters of different regularization terms has been a difficult problem. Transforming the reconstruction problem into constrained optimization can solve this problem, but determining the constraint range and accurately solving it remains a challenge. This paper proposes a CT reconstruction method based on constrained data fidelity term, which estimates the distribution of the constraint function by Taylor expansion to determine the constraint range. We respectively use Douglas-Rachford splitting (DRS) and Projection-based primal-dual algorithm (PPD) to split the reconstruction problem and solve the data fidelity subproblem. This method can accurately estimate the constrained range of data fidelity terms to ensure reconstruction accuracy and use different regularization terms for reconstruction without parameter adjustment. Three regularization terms are used for reconstruction experiments, and simulation results show that the proposed method can converge stably, and its reconstruction quality is better than the filtered back-projection.

Convergence of an Inertial Shadow Douglas-Rachford Splitting Algorithm for Monotone Inclusions

An inertial shadow Douglas-Rachford splitting algorithm for finding zeros of the sum of monotone operators is proposed in Hilbert spaces. Moreover, a three-operator splitting algorithm for solving a class of monotone inclusion problems is also concerned. The weak convergence of the algorithms is investigated under mild assumptions. Some numerical experiments are implemented to illustrate our main convergence results.

Robust classification with feature selection using an application of the Douglas-Rachford splitting algorithm

This paper deals with supervised classification and feature selection with application in the context of high dimensional features. A classical approach leads to an optimization problem minimizing the within sum of squares in the clusters (I2 norm) with an I1 penalty in order to promote sparsity. It has been known for decades that I1 norm is more robust than I2 norm to outliers. In this paper, we deal with this issue using a new proximal splitting method for the minimization of a criterion using I2 norm both for the constraint and the loss function. Since the I1 criterion is only convex and not gradient Lipschitz, we advocate the use of a Douglas-Rachford minimization solution. We take advantage of the particular form of the cost and, using a change of variable, we provide a new efficient tailored primal Douglas-Rachford splitting algorithm which is very effective on high dimensional dataset. We also provide an efficient classifier in the projected space based on medoid modeling. Experiments on two biological datasets and a computer vision dataset show that our method significantly improves the results compared to those obtained using a quadratic loss function.

Preconditioned Douglas-Rachford type primal-dual method for solving composite monotone inclusion problems with applications

<p style='text-indent:20px;'>This paper is concerned with the monotone inclusion involving the sum of a finite number of maximally monotone operators and the parallel sum of two maximally monotone operators with bounded linear operators. To solve this monotone inclusion, we first transform it into the formulation of the sum of three maximally monotone operators in a proper product space. Then we derive two efficient iterative algorithms, which combine the partial inverse method with the preconditioned Douglas-Rachford splitting algorithm and the preconditioned proximal point algorithm. Furthermore, we develop an iterative algorithm, which relies on the preconditioned Douglas-Rachford splitting algorithm without using the partial inverse method. We carefully analyze the theoretical convergence of the proposed algorithms. Finally, in order to demonstrate the effectiveness and efficiency of these algorithms, we conduct numerical experiments on a novel image denoising model for salt-and-pepper noise removal. Numerical results show the good performance of the proposed algorithms.

Anderson Accelerated Douglas--Rachford Splitting

We consider the problem of non-smooth convex optimization with linear equality constraints, where the objective function is only accessible through its proximal operator. This problem arises in many different fields such as statistical learning, computational imaging, telecommunications, and optimal control. To solve it, we propose an Anderson accelerated Douglas-Rachford splitting (A2DR) algorithm, which we show either globally converges or provides a certificate of infeasibility/unboundedness under very mild conditions. Applied to a block separable objective, A2DR partially decouples so that its steps may be carried out in parallel, yielding an algorithm that is fast and scalable to multiple processors. We describe an open-source implementation and demonstrate its performance on a wide range of examples.

Automatic knee joint segmentation using Douglas-Rachford splitting method

In the medical field, magnetic resonance imaging (MRI) scans are widely used for conducting research in osteoarthritis and to study about the disease of a patient. Still the MRI scans provide the details of the knee joint image of a patient, it is difficult to perform quantification of bone, cartilage, and meniscus regions. A fully automatic segmentation is required to segment knee joint, cartilage images from the MRI scan is necessary to reduce manual intervention. In this work, an automatic segmentation technique based on the proximal splitting method is presented. Douglas-Rachford splitting algorithm is employed in this paper. The knee joint structures are analyzed and the cartilage region is segmented. Then the quantization of the cartilage region is performed in which several morphological measures are computed. These measures are used to find out the growth of OA and the effects of drugs on OA.

Douglas-Rachford algorithm for magnetorelaxometry imaging using random and deterministic activations

Magnetorelaxometry imaging is a novel tool for quantitative determination of the spatial distribution of magnetic nanoparticle inside an organism. The use of multiple excitation patterns has been demonstrated to significantly improve spatial resolution. However, increasing the number of excitation patterns is considerably more time consuming, because several sequential measurements have to be performed. In this paper, we use compressed sensing in combination with sparse recovery to reduce the total measurement time and to improve spatial resolution. For image reconstruction, we propose using the Douglas-Rachford splitting algorithm applied to the sparse Tikhonov functional including a positivity constraint. Our numerical experiments demonstrate that the resulting algorithm is capable to accurately recover the magnetic nanoparticle distribution from a small number of activation patterns. For example, our algorithm applied with 10 activations yields half the reconstruction error of quadratic Tikhonov regularization applied with 50 activations, for a tumor-like phantom.

Convergence Analysis of an Inexact Three-Operator Splitting Algorithm

The three-operator splitting algorithm is a new splitting algorithm for finding monotone inclusion problems of the sum of three maximally monotone operators, where one is cocoercive. As the resolvent operator is not available in a closed form in the original three-operator splitting algorithm, in this paper, we introduce an inexact three-operator splitting algorithm to solve this type of monotone inclusion problem. The theoretical convergence properties of the proposed iterative algorithm are studied in general Hilbert spaces under mild conditions on the iterative parameters. As a corollary, we obtain general convergence results of the inexact forward-backward splitting algorithm and the inexact Douglas-Rachford splitting algorithm, which extend the existing results in the literature.

Stadium Norm and Douglas-Rachford Splitting: A New Approach to Road Design Optimization

The basic optimization problem of road design is quite challenging due to an objective function that is the sum of nonsmooth functions and the presence of set constraints. In this paper, we model and solve this problem by employing the Douglas-Rachford splitting algorithm. This requires a careful study of new proximity operators related to minimizing area and to the stadium norm. We compare our algorithm to a state-of-the-art projection algorithm. Our numerical results illustrate the potential of this algorithm to significantly reduce cost in road design.

Eventual linear convergence of the Douglas-Rachford iteration for basis pursuit

We provide a simple analysis of the Douglas-Rachford splitting algorithm in the context of $\ell^1$ minimization with linear constraints, and quantify the asymptotic linear convergence rate in terms of principal angles between relevant vector spaces. In the compressed sensing setting, we show how to bound this rate in terms of the restricted isometry constant. More general iterative schemes obtained by $\ell^2$-regularization and over-relaxation including the dual split Bregman method are also treated, which answers the question how to choose the relaxation and soft-thresholding parameters to accelerate the asymptotic convergence rate. We make no attempt at characterizing the transient regime preceding the onset of linear convergence.

X-ray CT Metal Artifact Reduction Using Wavelet Domain &lt;formula formulatype="inline"&gt;&lt;tex Notation="TeX"&gt;$L_{0}$&lt;/tex&gt;&lt;/formula&gt; Sparse Regularization

X-ray computed tomography (CT) imaging of patients with metallic implants usually suffers from streaking metal artifacts. In this paper, we propose a new projection completion metal artifact reduction (MAR) algorithm by formulating the completion of missing projections as a regularized inverse problem in the wavelet domain. The Douglas-Rachford splitting (DRS) algorithm was used to iteratively solve the problem. Two types of prior information were exploited in the algorithm: 1) the sparsity of the wavelet coefficients of CT sinograms in a dictionary of translation-invariant wavelets and 2) the detail wavelet coefficients of a prior sinogram obtained from the forward projection of a segmented CT image. A pseudo- L0 synthesis prior was utilized to exploit and promote the sparsity of wavelet coefficients. The proposed L0-DRS MAR algorithm was compared with standard linear interpolation and the normalized metal artifact reduction (NMAR) approach proposed by Meyer using both simulated and clinical studies including hip prostheses, dental fillings, spine fixation and electroencephalogram electrodes in brain imaging. The qualitative and quantitative evaluations showed that our algorithm substantially suppresses streaking artifacts and can outperform both linear interpolation and NMAR algorithms.

The primal douglas-rachford splitting algorithm for a class of monotone mappings with application to the traffic equilibrium problem

We apply the Douglas-Rachford splitting algorithm to a class of multi-valued equations consisting of the sum of two monotone mappings. Compared with the dual application of the same algorithm, which is known as the alternating direction method of multipliers, the primal application yields algorithms that seem somewhat involved. However, the resulting algorithms may be applied effectively to problems with certain special structure. In particular we show that they can be used to derive decomposition algorithms for solving the variational inequality formulation of the traffic equilibrium problem.