Shrinking gradient descent algorithms for total variation regularized image denoising

Total variation regularization introduced by Rudin, Osher, and Fatemi (ROF) is widely used in image denoising problems for its capability to preserve repetitive textures and details of images. Many efforts have been devoted to obtain efficient gradient descent schemes for dual minimization of ROF model, such as Chambolle’s algorithm or gradient projection (GP) algorithm. In this paper, we propose a general gradient descent algorithm with a shrinking factor. Both Chambolle’s and GP algorithm can be regarded as the special cases of the proposed methods with special parameters. Global convergence analysis of the new algorithms with various step lengths and shrinking factors are present. Numerical results demonstrate their competitiveness in computational efficiency and reconstruction quality with some existing classic algorithms on a set of gray scale images.     

中等稀疏条件下基因交互作用的两步Bayes方法

植物遗传与基因组学研究表明许多重要的农艺性状有影响的基因位点不是稀疏的,受到大量微效基因的影响,并且还存在基因交互项的影响.本文基于重要油料作物油菜的花期数据,研究中等稀疏条件下的基因选择问题,提出了一种两步Bayes模型选择方法.考虑基因间的交互作用,模型的维数急剧增长,加上数据结构特别,通常的变量选择方法效果不好....     

Learning the coordinate gradients

In this paper we study the problem of learning the gradient function with application to variable selection and determining variable covariation. Firstly, we propose a novel unifying framework for coordinate gradient learning from the perspective of multi-task learning. Various variable selection methods can be regarded as special instances of this framework. Secondly, we formulate the dual problems of gradient learning with general loss functions. This enables the direct application of standard optimization toolboxes to the case of gradient learning. For instance, gradient learning with SVM loss can be solved by quadratic programming (QP) routines. Thirdly, we propose a novel gradient learning formulation which can be cast as a learning the kernel matrix problem. Its relation with sparse regularization is highlighted. A semi-infinite linear programming (SILP) approach and an iterative optimization approach are proposed to efficiently solve this problem. Finally, we validate our proposed approaches on both synthetic and real datasets.     

互补约束规划问题的一个广义梯度投影算法

本文研究了一类均衡约束最优化问题.利用广义梯度投影法,结合罚函数思想,得到了一个初始点可以任意的广义梯度投影算法.在较弱的条件下,证明了算法的全局收敛性.     

基于Huber正则化的红外与可见光图像融合

图像融合通常是指从多源信道采集同一目标图像,将互补的多焦点、多模态、多时相和/或多视点图像集成在一起,形成新图像的过程.在本文中,我们采用基于Huber正则化的红外与可见光图像的融合模型.该模型通过约束融合图像与红外图像相似的像素强度保持热辐射信息,以及约束融合图像与可见光图像相似的灰度梯度和像素强度保持图像的边缘和纹理等外观信息,同时能够改善图像灰度梯度相对较小区域的阶梯效应.为了最小化这种变分模型,我们结合增广拉格朗日方法(ALM)和量身定做有限点方法(TFPM)的思想设计数值算法,并给出了算法的收敛性分析.最后,我们将所提模型和算法与其他七种图像融合方法进行定性和定量的比较,分析了本文所提模型的特点和所提数值算法的有效性.     

Spectral DY-Type Projection Method for Nonlinear Monotone Systems of Equations

In this paper, we propose a spectral DY-type projection method for nonlinear monotone systems of equations, which is a reasonable combination of DY conjugate gradient method, the spectral gradient method and the projection technique. Without the differentiability assumption on the system of equations, we establish the global convergence of the proposed method, which does not rely on any merit function. Furthermore, this method is derivative-free and so is very suitable to solve large-scale nonlinear monotone systems. The preliminary numerical results show the feasibility and effectiveness of the proposed method.     

Efficient projected gradient methods for cardinality constrained optimization

Sparse optimization has attracted increasing attention in numerous areas such as compressed sensing, financial optimization and image processing. In this paper, we first consider a special class of cardinality constrained optimization problems, which involves box constraints and a singly linear constraint. An efficient approach is provided for calculating the projection over the feasibility set after a careful analysis on the projection subproblem. Then we present several types of projected gradient methods for a general class of cardinality constrained optimization problems. Global convergence of the methods is established under suitable assumptions. Finally, we illustrate some applications of the proposed methods for signal recovery and index tracking.Especially for index tracking, we propose a new model subject to an adaptive upper bound on the sparse portfolio weights. The computational results demonstrate that the proposed projected gradient methods are efficient in terms of solution quality.     

A conjugate Rosen’s gradient projection method with global line search for piecewise linear concave optimization

The Kelley cutting plane method is one of the methods commonly used to optimize the dual function in the Lagrangian relaxation scheme. Usually the Kelley cutting plane method uses the simplex method as the optimization engine. It is well known that the simplex method leaves the current vertex, follows an ascending edge and stops at the nearest vertex. What would happen if one would continue the line search up to the best point instead? As a possible answer, we propose the face simplex method, which freely explores the polyhedral surface by following the Rosen’s gradient projection combined with a global line search on the whole surface. Furthermore, to avoid the zig-zagging of the gradient projection, we propose a conjugate gradient version of the face simplex method. For our preliminary numerical tests we have implemented this method in Matlab. This implementation clearly outperforms basic Matlab implementations of the simplex method. In the case of state-of-the-art simplex implementations in C or similar, our Matlab implementation is only competitive for the case of many cutting planes.     

Linear convergence analysis of the use of gradient projection methods on total variation problems

Optimization problems using total variation frequently appear in image analysis models, in which the sharp edges of images are preserved. Direct gradient descent methods usually yield very slow convergence when used for such optimization problems. Recently, many duality-based gradient projection methods have been proposed to accelerate the speed of convergence. In this dual formulation, the cost function of the optimization problem is singular, and the constraint set is not a polyhedral set. In this paper, we establish two inequalities related to projected gradients and show that, under some non-degeneracy conditions, the rate of convergence is linear.     

谱HS投影算法求解非线性单调方程组

借助谱梯度法和HS共轭梯度法的结构, 建立一种求解非线性单调方程组问题的谱HS投影算法. 该算法继承了谱梯度法和共轭梯度法储存量小和计算简单的特征, 且不需要任何导数信息, 因此它适应于求解大规模非光滑的非线性单调方程组问题. 在适当的条件下, 证明了该算法的收敛性, 并通过数值实验表明了该算法的有效性.     

Error Analysis of SAV Finite Element Method to Phase Field Crystal Model

In this paper, we construct and analyze an energy stable scheme by combining the latest developed scalar auxiliary variable (SAV) approach and linear finite element method (FEM) for phase field crystal (PFC) model, and show rigorously that the scheme is first-order in time and second-order in space for the $L^2$ and $H^{-1}$ gradient flow equations. To reduce efficiently computational cost and capture accurately the phase interface, we give a simple adaptive strategy, equipped with a posteriori gradient estimator, i.e., $L^2$ norm of the recovered gradient. Extensive numerical experiments are presented to verify our theoretical results and to demonstrate the effectiveness and accuracy of our proposed method.     

Duality-based algorithms for total-variation-regularized image restoration

Image restoration models based on total variation (TV) have become popular since their introduction by Rudin, Osher, and Fatemi (ROF) in 1992. The dual formulation of this model has a quadratic objective with separable constraints, making projections onto the feasible set easy to compute. This paper proposes application of gradient projection (GP) algorithms to the dual formulation. We test variants of GP with different step length selection and line search strategies, including techniques based on the Barzilai-Borwein method. Global convergence can in some cases be proved by appealing to existing theory. We also propose a sequential quadratic programming (SQP) approach that takes account of the curvature of the boundary of the dual feasible set. Computational experiments show that the proposed approaches perform well in a wide range of applications and that some are significantly faster than previously proposed methods, particularly when only modest accuracy in the solution is required.     

A projection method based on the splitting Bregman iteration for the image denoising

By analyzing the connection between the projection operator and the shrink operator, we propose a projection method based on the splitting Bregman iteration for image denoising problem in this paper. Compared with the splitting Bregman method, the proposed method has a more compact form so that it is more fast and efficient. Following from the operator theory, the convergence of the proposed method is proved. Some numerical comparisons between the proposed method and the splitting Bregman method are arranged for solving two basic image denoising models.     

A primal-dual active-set algorithm for bilaterally constrained total variation deblurring and piecewise constant Mumford-Shah segmentation problems

In this paper, we propose a fast primal-dual algorithm for solving bilaterally constrained total variation minimization problems which subsume the bilaterally constrained total variation image deblurring model and the two-phase piecewise constant Mumford-Shah image segmentation model. The presence of the bilateral constraints makes the optimality conditions of the primal-dual problem semi-smooth which can be solved by a semi-smooth Newton’s method superlinearly. But the linear system to solve at each iteration is very large and difficult to precondition. Using a primal-dual active-set strategy, we reduce the linear system to a much smaller and better structured one so that it can be solved efficiently by conjugate gradient with an approximate inverse preconditioner. Locally superlinear convergence results are derived for the proposed algorithm. Numerical experiments are also provided for both deblurring and segmentation problems. In particular, for the deblurring problem, we show that the addition of the bilateral constraints to the total variation model improves the quality of the solutions.     

求解低秩矩阵恢复问题的非单调交替梯度方向法

低秩矩阵恢复问题作为一类在图像处理和信号数据分析等领域中都十分重要的问题已被广泛研究.本文在交替方向算法的框架下,应用非单调技术,提出一种求解低秩矩阵恢复问题的新算法.该算法在每一步迭代过程中,首先利用一步带有变步长梯度算法同时更新低秩部分的两块变量,然后采用非单调技术更新稀疏部分的变量.在一定的假设条件下,本文证明了...     

A modified quasi‐Newton diagonal update algorithm for total variation denoising problems and nonlinear monotone equations with applications in compressive sensing

In this paper, we present a new algorithm to accelerate the Chambolle gradient projection method for total variation image restoration. The new proposed method considers an approximation of the Hessian based on the secant equation. Combined with the quasi‐Cauchy equations and diagonal updating, we can obtain a positive definite diagonal matrix. In the proposed minimization method model, we use the positive definite diagonal matrix instead of the constant time stepsize in Chambolle's method. The global convergence of the proposed scheme is proved. Some numerical results illustrate the efficiency of this method. Moreover, we also extend the quasi‐Newton diagonal updating method to solve nonlinear systems of monotone equations. Performance comparisons show that the proposed method is efficient. A practical application of the monotone equations is shown and tested on sparse signal reconstruction in compressed sensing. Copyright © 2015 John Wiley & Sons, Ltd.     

最小化三个凸函数之和的一个简单原始-对偶算法

提出一个简单的原始-对偶算法求解三个凸函数之和的最小化问题, 其中目标函数包含有梯度李普希兹连续的光滑函数, 非光滑函数和含有复合算子的非光滑函数. 在新方法中, 对偶变量迭代使用预估-矫正的方案. 分析了算法的收敛性和收敛速率. 最后, 数值实验说明了算法的有效性.     

Limited Memory Space Dilation and Reduction Algorithms

In this paper, we present variants of Shor and Zhurbenko's r-algorithm, motivated by the memoryless and limited memory updates for differentiable quasi-Newton methods. This well known r-algorithm, which employs a space dilation strategy in the direction of the difference between two successive subgradients, is recognized as being one of the most effective procedures for solving nondifferentiable optimization problems. However, the method needs to store the space dilation matrix and update it at every iteration, resulting in a substantial computational burden for large-sized problems. To circumvent this difficulty, we first propose a memoryless update scheme, which under a suitable choice of parameters, yields a direction of motion that turns out to be a convex combination of two successive anti-subgradients. Moreover, in the space transformation sense, the new update scheme can be viewed as a combination of space dilation and reduction operations. We prove convergence of this new method, and demonstrate how it can be used in conjunction with a variable target value method that allows a practical, convergent implementation of the method. We also examine a memoryless variant that uses a fixed dilation parameter instead of varying degrees of dilation and/or reduction as in the former algorithm, as well as another variant that examines a two-step limited memory update. These variants are tested along with Shor's r-algorithm and also a modified version of a related algorithm due to Polyak that employs a projection onto a pair of Kelley's cutting planes. We use a set of standard test problems from the literature as well as randomly generated dual transportation and assignment problems in our computational experiments. The results exhibit that the proposed space dilation and reduction method and the modification of Polyak's method are competitive, and offer a substantial advantage over the r-algorithm and over the other tested limited memory variants with respect to accuracy as well as effort.     

Sparsity-promoting distributed charging control for plug-in electric vehicles over distribution networks

Uncoordinated charging of plug-in electric vehicles brings a new challenge on the operation of a power system as it causes power flow fluctuations and even unacceptable load peaks. To ensure the stability of power network, plug-in charging needs to be scheduled against the base load properly. In this paper, we propose a sparsity-promoting charging control model to address this issue. In the model, the satisfaction of customers is improved through sparsity-promoting charging where the numbers of charging time slots are optimized. Dynamic feeder overload constraints are imposed in the model to avoid any unacceptable load peaks, and thus ensure the network stability. Then, a distributed solution strategy is developed to solve the problem based on the alternating direction method of multipliers (ADMM) since most of power networks are managed typically in a distributed manner. During solving process, Lagrangian duality is used to transform the original problem into an equivalent dual problem, which can be decomposed into a set of homogeneous small-scaled sub-problems. Particularly, each sub-problem either has a closed-form solution or can be solved locally by an accelerated dual gradient method. The global convergence of the proposed algorithm is also established. Finally, numerical simulations are presented to illustrate our proposed method. In contrast to traditional charging models, our sparsity-promoting charging model not only ensures the stability of power network, but also improves the satisfaction of customers.     

Adaptive limited memory bundle method for bound constrained large-scale nonsmooth optimization

Typically, practical optimization problems involve nonsmooth functions of hundreds or thousands of variables. As a rule, the variables in such problems are restricted to certain meaningful intervals. In this article, we propose an efficient adaptive limited memory bundle method for large-scale nonsmooth, possibly nonconvex, bound constrained optimization. The method combines the nonsmooth variable metric bundle method and the smooth limited memory variable metric method, while the constraint handling is based on the projected gradient method and the dual subspace minimization. The preliminary numerical experiments to be presented confirm the usability of the method.