一类单调变分不等式的非精确交替方向法

交替方向法适合于求解大规模问题.该文对于一类变分不等式提出了一种新的交替方向法.在每步迭代计算中,新方法提出了易于计算的子问题,该子问题由强单调的线性变分不等式和良态的非线性方程系统构成.基于子问题的精确求解,该文证明了算法的收敛性.进一步,又提出了一类非精确交替方向法,每步迭代计算只需非精确求解子问题.在一定的非精确条件下,算法的收敛性得以证明.     

Two-Stage Image Segmentation Scheme Based on Inexact Alternating Direction Method

Image segmentation is a fundamental problem in both image processing and computer vision with numerous applications. In this paper, we propose a two-stage image segmentation scheme based on inexact alternating direction method. Specifically, we first solve the convex variant of the Mumford-Shah model to get the smooth solution, and the segmentation is then obtained by applying the K-means clustering method to the solution. Some numerical comparisons are arranged to show the effectiveness of our proposed schemes by segmenting many kinds of images such as artificial images, natural images, and brain MRI images.     

一种求解双目标规划的非精确交替方向法

本文提出了一种求解双目标规划的直接算法---非精确交替方向方法,并证明了算法的收敛性.初步的数值实验说明了所提出的算法是有效可行的.     

An Inexact Alternating Direction Method for Structured Variational Inequalities

Recently, the alternating direction method of multipliers has attracted great attention. For a class of variational inequalities (VIs), this method is efficient, when the subproblems can be solved exactly. However, the subproblems could be too difficult or impossible to be solved exactly in many practical applications. In this paper, we propose an inexact method for structured VIs based on the projection and contraction method. Instead of solving the subproblems exactly, we use the simple projection to get a predictor and correct it to approximate the subproblems’ real solutions. The convergence of the proposed method is proved under mild assumptions and its efficiency is also verified by some numerical experiments.     

求解一类非对称单调变分不等式的非精确自适应交替方向法

本文研究了一类单调非对称变分不等式的非精确自适应交替方向法,证明了方法的收敛性.     

A partial inexact alternating direction method for structured variational inequalities

In this article, a new method is proposed for solving a class of structured variational inequalities (SVIs). The proposed method is referred to as the partial inexact proximal alternating direction (piPAD) method. In the method, two subproblems are solved independently. One is handled by an inexact proximal point method and the other is solved directly. This feature is the major difference between the proposed method and some existing alternating direction-like methods. The convergence of the piPAD method is proved. Two examples of the modern convex optimization problem arising from engineering and information sciences, which can be reformulated into the encountered SVIs, are presented to demonstrate the applicability of the piPAD method. Also, some preliminary numerical results are reported to validate the feasibility and efficiency of the piPAD method.     

An inexact alternating direction method of multipliers for the solution of linear complementarity problems arising from free boundary problems

A large number of free boundary problems can be formulated as linear-complementarity problems. In this paper, we propose an inexact alternating direction method of multipliers for solving linear complementarity problem arising from free boundary problems by using the special structure of these problems. The convergence of our proposed method is proved. Numerical results show that the proposed method is feasible and effective, and it is significantly faster than modified alternating direction implicit algorithm and many other methods, especially when dimension of the problem being solved is large.     

Symmetric Alternating Direction Method with Indefinite Proximal Regularization for Linearly Constrained Convex Optimization

The proximal alternating direction method of multipliers is a popular and useful method for linearly constrained, separable convex problems, especially for the linearized case. In the literature, convergence of the proximal alternating direction method has been established under the assumption that the proximal regularization matrix is positive semi-definite. Recently, it was shown that the regularizing proximal term in the proximal alternating direction method of multipliers does not necessarily have to be positive semi-definite, without any additional assumptions. However, it remains unknown as to whether the indefinite setting is valid for the proximal version of the symmetric alternating direction method of multipliers. In this paper, we confirm that the symmetric alternating direction method of multipliers can also be regularized with an indefinite proximal term. We theoretically prove the global convergence of the indefinite method and establish its worst-case convergence rate in an ergodic sense. In addition, the generalized alternating direction method of multipliers proposed by Eckstein and Bertsekas is a special case in our discussion. Finally, we demonstrate the performance improvements achieved when using the indefinite proximal term through experimental results.     

Alternating oblique projections for coupled linear systems

In this work we propose the use of alternating oblique projections (AOP) for the solution of the saddle points systems resulting from the discretization of domain decomposition problems. These systems are called coupled linear systems. The AOP method is a descent method in which the descent direction is defined by using alternating oblique projections onto the search subspaces. We prove that this method is a preconditioned simple gradient (Uzawa) method with a particular preconditioner. Finally, a preconditioned conjugate gradient based version of AOP is proposed. AMS subject classification 65F10, 65N22, 65Y05     

An Extragradient-Based Alternating Direction Method for Convex Minimization

In this paper, we consider the problem of minimizing the sum of two convex functions subject to linear linking constraints. The classical alternating direction type methods usually assume that the two convex functions have relatively easy proximal mappings. However, many problems arising from statistics, image processing and other fields have the structure that while one of the two functions has an easy proximal mapping, the other function is smoothly convex but does not have an easy proximal mapping. Therefore, the classical alternating direction methods cannot be applied. To deal with the difficulty, we propose in this paper an alternating direction method based on extragradients. Under the assumption that the smooth function has a Lipschitz continuous gradient, we prove that the proposed method returns an \(\epsilon \)-optimal solution within \(O(1/\epsilon )\) iterations. We apply the proposed method to solve a new statistical model called fused logistic regression. Our numerical experiments show that the proposed method performs very well when solving the test problems. We also test the performance of the proposed method through solving the lasso problem arising from statistics and compare the result with several existing efficient solvers for this problem; the results are very encouraging.