一种基于交替方向乘子法改进的模型预测控制方法

模型预测控制能够有效处理实际应用中的扰动、多控制变量和复杂约束,因此被广泛应用于各种大规模控制问题.然而,在处理基于复杂线性时不变系统的模型预测控制问题时,传统的二次规划算法存在计算负载过大,实时性差的缺点.为此,本文引入交替方向乘子法对模型预测控制问题进行分布式地求解.仿真实验结果表明,基于交替方向乘子法的模型预测控制相较传统的方法,计算效率明显提高,更适于求解大规模优化控制问题.     

图像修复TV模型的快速算法研究

关于图像修复的全变分（ TV）模型的求解有很多方法。在图像修复的全变分（ TV）模型中,文中针对含有非光滑项的凸优化问题提出了一种基于交替方向乘子法（ ADMM）的快速求解算法。 ADMM方法对迭代公式中具体的子问题求解过程一般采用Gauss-Seidel方法,文中通过分析TV修复模型的性质,对ADMM算法进行了相应的改进,使得具体的数值求解可以用快速傅里叶变换方法,并证明了该算法的收敛性。实验结果表明,文中所提出的新算法与采用Gauss-Seidel迭代的方法相比较,不但修复效果更好,而且修复速度更快。     

基于交替方向乘子法的电力系统完全分布式动态经济调度方法

本文基于交替方向乘子法(alternating direction multiplier method,ADMM)提出了一种完全分布式的跨区域电力系统动态经济调度方法.其中的经济调度模型以整个系统的运行成本最小为目标,并满足各种系统运行约束.为了实现模型的分布式求解,本文利用交替方向乘子法将各区域之间的联系解耦,将整个系统的大型优化问题分解为各个区域内部的子优化问题,通过迭代求解每个区域的子问题即可得到整个系统的最优解.进一步地,本文算法取消了负责乘子更新的数据中心,实现了完全分布式的调度策略.同时,为了兼顾电力系统中时间断面之间的紧密联系,本文的经济调度模型采用了多时段优化方法.最后,本文对基于IEEE标准测试系统的3区域互联系统算例进行了分析,验证了本文的调度策略的有效性.     

麦克风阵列鲁棒频率不变波束形成算法

在实际应用中,频率不变波束形成器通常受到麦克风阵列失配误差的影响,因此提高频率不变波束形成器的鲁棒性具有重要意义。针对上述问题提出了一种约束优化模型,可以在保持频率不变波束形成的同时提高阵列的鲁棒性。首先设计目标波束图,考虑到差分麦克风阵列本身具有频率不变的波束图,选用传统二阶超心型差分麦克风波束图做为目标波束图。上述模型以麦克风阵列权矢量的二范数作为目标函数来最大化鲁棒性,在无失真约束,目标波束主瓣逼近约束以及旁瓣增益精准控制约束下实现频率不变。然后在交替方向乘子法算法框架下,将优化问题分解为多个优化子问题求解,然后对每个优化子问题分别求解,通过仿真验证了在交替方向乘子法算法下上述模型的可行性与有效性,最终达到了麦克风阵列鲁棒频率不变波束响应的效果。     

一种用于可形变对象操作的动力学模型隐式求解方法

机器人操作可形变对象在多类场景中具有重要的应用需求。然而,对象复杂的高维动力学模型导致难以准确、快速地计算其形变。本文建立了一种基于交替方向乘子法的对象形变动力学模型快速、准确、隐式优化求解方法。该方法以通用对象几何模型为输入创建有限元模型,并分别构建材料本构势能函数和操作及碰撞交互单元的位置约束能量函数。随后,采用动力学投影方法构建基于交替方向乘子法的两项优化求解问题,最终快速、准确地计算对象在机器人操作下的形变。数值实验表明,所提出的方法能够在保证相对形变计算误差低于5%的条件下,实现高于24帧/秒的物理形态更新。针对实际应用场景,对所提出的方法开展了从形变仿真预测到在线操作执行的量化评估以及全局约束环境下的离线规划和仿真应用验证。     

分裂可行性问题的外推加速线性交替方向乘子法及其全局收敛性

针对在图像重建以及语言处理系统等领域有着广泛应用的分裂可行性问题(SFP)的最优化求解，提出了外推加速线性交替方向乘子法。首先将SFP描述为一个具有线性约束的可分离凸极小化问题；然后引进外推线性交替方向乘子法，利用问题的可分离结构，产生了具有闭式解的子问题，并在适当条件下证明了该算法的全局收敛性；最后，通过数值实验验证了该算法的可行性和有效性。     

基于牛顿法的并行优化算法

针对非线性数值优化问题,提出一种在分布式环境下的基于牛顿法的并行算法。引入松弛变量,将不等式约束转换为等式约束,利用广义拉格朗日乘子将约束优化问题转换为无约束子优化问题。为了并行地求解这些子优化问题,将Newton迭代法中的Hessian矩阵进行适当的分裂,采用简单迭代法求解Newton法中的线性方程组。在理论上对该算法进行了收敛性分析。在HP rx2600集群上进行的数值实验结果表明并行效率达90%以上。     

图像修补的优化变换方法

采用整体变分(TV)模型修补图像,提出一种图像修补的优化变换方法.引入一个辅助变量,利用优化变换,将TV模型中单变量函数的优化问题转化为等效双变量函数的优化问题,并利用交替迭代最小化算法和Chambolle's投影算法求解模型.实验结果表明,与采用梯度下降法的TV模型算法相比,该方法的图像修补效率和修补效果较优.     

纹理图像分割的非局部Mumford-Shah-TV模型及其ADMM算法

为了提高纹理图像分割的准确率,解决纹理图像中纹理图像成分及纹理区域边界难以描述的问题.基于总变差(total variation, TV)规则项可得到纹理图像区域隐藏的图像结构、非局部算子可以描述纹理图像特征的特点,综合TV模型、非局部Mumford-Shah模型,并用二值标记函数划分区域,提出纹理图像分割的非局部Mumford-Shah-TV变分模型;为了提高计算效率,对所提出的模型设计了相应的交替方向乘子算法,将原问题分解为一系列优化子问题求解.数值实验结果表明,该模型计算的纹理图像区域边界较好,并具有较高的准确率.     

有等式约束优化问题的粒子群优化算法

目前大多数粒子群优化算法针对无约束优化问题或不等式约束优化问题,求解有等式约束优化问题的方法是把每个等式约束变成两个不等式约束,这种方法的缺点是在进化过程中粒子位置很难满足等式约束条件,影响了收敛速度和解的精度。提出了求解有等式约束优化问题的两种新粒子群优化算法,数值试验结果表明,算法是有效的。     

Adaptive algorithm for sparse signal recovery

The development of compressive sensing in recent years has given much attention to sparse signal recovery. In sparse signal recovery, spike and slab priors are playing a key role in inducing sparsity. The use of such priors, however, results in non-convex and mixed integer programming problems. Most of the existing algorithms to solve non-convex and mixed integer programming problems involve either simplifying assumptions, relaxations or high computational expenses. In this paper, we propose a new adaptive alternating direction method of multipliers (AADMM) algorithm to directly solve the suggested non-convex and mixed integer programming problem. The algorithm is based on the one-to-one mapping property of the support and non-zero element of the signal. At each step of the algorithm, we update the support by either adding an index to it or removing an index from it and use the alternating direction method of multipliers to recover the signal corresponding to the updated support. Moreover, as opposed to the competing “adaptive sparsity matching pursuit” and “alternating direction method of multipliers” methods our algorithm can solve non-convex problems directly. Experiments on synthetic data and real-world images demonstrated that the proposed AADMM algorithm provides superior performance and is computationally cheaper than the recently developed iterative convex refinement (ICR) and adaptive matching pursuit (AMP) algorithms.     

HSAC-ALADMM: an asynchronous lazy ADMM algorithm based on hierarchical sparse allreduce communication

The Journal of Supercomputing - The distributed alternating direction method of multipliers (ADMM) is an effective algorithm for solving large-scale optimization problems. However, its high...     

Alternating direction method of multipliers as a simple effective heuristic for mixed-integer nonlinear optimization

Structural and Multidisciplinary Optimization - In this paper we propose to utilize a variation of the alternating direction method of multipliers (ADMM) as a simple heuristic for mixed-integer...     

Improving Medical CT Image Blind Restoration Algorithm Based on Dictionary Learning by Alternating Direction Method of Multipliers

In this paper, the medical CT image blind restoration is translated into two sub problems, namely, image estimation based on dictionary learning and point spread function estimation. A blind restoration algorithm optimized by the alternating direction method of multipliers for medical CT images was proposed. At present, the existing methods of blind image restoration based on dictionary learning have the problem of low efficiency and precision. This paper aims to improve the effectiveness and accuracy of the algorithm and to improve the robustness of the algorithm. The local CT images are selected as training samples, and the K-SVD algorithm is used to construct the dictionary by iterative optimization, which is beneficial to improve the efficiency of the algorithm. Then, the orthogonal matching pursuit algorithm is employed to implement the dictionary update. Dictionary learning is accomplished by sparse representation of medical CT images. The alternating direction method of multipliers (ADMM) is used to solve the objective function and realize the local image restoration, so as to eliminate the influence of point spread function. Secondly, the local restoration image is used to estimate the point spread function, and the convex quadratic optimization method is used to solve the point spread function sub problems. Finally, the optimal estimation of point spread function is obtained by iterative method, and the global sharp image is obtained by the alternating direction method of multipliers. Experimental results show that, compared with the traditional adaptive dictionary restoration algorithm, the new algorithm improves the objective image quality metrics, such as peak signal to noise ratio, structural similarity, and universal image quality index. The new algorithm optimizes the restoration effect, improves the robustness of noise immunity and improves the computing efficiency.     

A classification of methods for distributed system optimization based on formulation structure

Augmented Lagrangian coordination (ALC) is a provably convergent coordination method for multidisciplinary design optimization (MDO) that is able to treat both linking variables and linking functions (i.e. system-wide objectives and constraints). Contrary to quasi-separable problems with only linking variables, the presence of linking functions may hinder the parallel solution of subproblems and the use of the efficient alternating directions method of multipliers. We show that this unfortunate situation is not the case for MDO problems with block-separable linking constraints. We derive a centralized formulation of ALC for block-separable constraints, which does allow parallel solution of subproblems. Similarly, we derive a distributed coordination variant for which subproblems cannot be solved in parallel, but that still enables the use of the alternating direction method of multipliers. The approach can also be used for other existing MDO coordination strategies such that they can include block-separable linking constraints. This work is funded by MicroNed, grant number 10005898.     

Block-separable linking constraints in augmented Lagrangian coordination

Augmented Lagrangian coordination (ALC) is a provably convergent coordination method for multidisciplinary design optimization (MDO) that is able to treat both linking variables and linking functions (i.e. system-wide objectives and constraints). Contrary to quasi-separable problems with only linking variables, the presence of linking functions may hinder the parallel solution of subproblems and the use of the efficient alternating directions method of multipliers. We show that this unfortunate situation is not the case for MDO problems with block-separable linking constraints. We derive a centralized formulation of ALC for block-separable constraints, which does allow parallel solution of subproblems. Similarly, we derive a distributed coordination variant for which subproblems cannot be solved in parallel, but that still enables the use of the alternating direction method of multipliers. The approach can also be used for other existing MDO coordination strategies such that they can include block-separable linking constraints.     

A Dantzig–Wolfe decomposition algorithm for linear economic model predictive control of dynamically decoupled subsystems

This paper presents a warm-started Dantzig–Wolfe decomposition algorithm tailored to economic model predictive control of dynamically decoupled subsystems. We formulate the constrained optimal control problem solved at each sampling instant as a linear program with state space constraints, input limits, input rate limits, and soft output limits. The objective function of the linear program is related directly to the cost of operating the subsystems, and the cost of violating the soft output constraints. Simulations for large-scale economic power dispatch problems show that the proposed algorithm is significantly faster than both state-of-the-art linear programming solvers, and a structure exploiting implementation of the alternating direction method of multipliers. It is also demonstrated that the control strategy presented in this paper can be tuned using a weighted ℓ₁-regularization term. In the presence of process and measurement noise, such a regularization term is critical for achieving a well-behaved closed-loop performance.     

A faster stochastic alternating direction method for large scale convex composite problems

Inspired by the fact that certain randomization schemes incorporated into the stochastic (proximal) gradient methods allow for a large reduction in computational time, we incorporate such a scheme into stochastic alternating direction method of multipliers (ADMM), yielding a faster stochastic alternating direction method (FSADM) for solving a class of large scale convex composite problems. In the numerical experiments, we observe a reduction of this method in computational time compared to previous methods. More importantly, we unify the stochastic ADMM for solving general convex and strongly convex composite problems (i.e., the iterative scheme does not change when the the problem goes from strongly convex to general convex). In addition, we establish the convergence rates of FSADM for these two cases.

     

A new fuzzy c-means method with total variation regularization for segmentation of images with noisy and incomplete data

The objective function of the original (fuzzy) c-mean method is modified by a regularizing functional in the form of total variation (TV) with regard to gradient sparsity, and a regularization parameter is used to balance clustering and smoothing. An alternating direction method of multipliers in conjunction with the fast discrete cosine transform is used to solve the TV-regularized optimization problem. The new algorithm is tested on both synthetic and real data, and is demonstrated to be effective and robust in treating images with noise and missing data (incomplete data).     

Convergence of alternating direction method for minimizing sum of two nonconvex functions with linear constraints

The efficiency of the classic alternating direction method of multipliers has been exhibited by various applications for large-scale separable optimization problems, both for convex objective functions and for nonconvex objective functions. While there are a lot of convergence analysis for the convex case, the nonconvex case is still an open problem and the research for this case is in its infancy. In this paper, we give a partial answer on this problem. Specially, under the assumption that the associated function satisfies the Kurdyka–?ojasiewicz inequality, we prove that the iterative sequence generated by the alternating direction method converges to a critical point of the problem, provided that the penalty parameter is greater than 2L, where L is the Lipschitz constant of the gradient of one of the involved functions. Under some further conditions on the problem's data, we also analyse the convergence rate of the algorithm.