基于序列线性组合的原始–对偶算法

双线性鞍点问题及其对应的原问题和对偶问题在信号图像处理、机器学习、统计和高维数据处理等领域具有重要的应用,原始对偶算法是求解该类问题的有效算法。利用序列的线性组合技术,改进了Chambolle-Pock原始对偶算法子问题的求解,提出了一种求解双线性鞍点问题的新原始对偶算法。该算法也是Arrow-Hurwicz算法的修正,在子问题求解中将线性组合和经典的外插技术进行结合,得到了更一般的收敛性。利用变分分析证明了算法的收敛性和遍历■(1/N)收敛率,获得了保证算法收敛的步长和组合参数取值范围,求解非负最小二乘和Lasso问题的数值实验验证了算法的有效性。     

求解互补问题的原始对偶起作用集算法

在本文中我们得到了求解带T-单调算子的互补问题的原始对偶活跃集算法的收敛结果.当原始对偶活跃集算法求解此类互补问题时,此算法可以作为一类特殊的半光滑牛顿法.收敛结果和数值试验说明了此算法的迭代次数不超过问题未知数的个数.最终,计算结果表明此算法的可行性.     

基于原始对偶分裂方法求解一类约束可分离凸优化问题及其应用

本文研究一类具有代表性的约束可分离凸优化模型,其目标函数中的数据误差项满足可微性条件,许多图像恢复和图像重建等问题都可以归结为该模型的求解.为克服现有求解该模型方法的不足,文中首先借助指示函数,将原模型转化为无约束的凸优化模型;然后基于原始对偶分裂方法思想,提出一种新的迭代算法,该算法具有结构简单和参数选取容易的特点,同时证明所提算法的收敛性.最后,为验证算法的有效性,我们将其应用于CT图像重建问题,数值实验结果表明所提出的算法在重建时间和重建图像质量上优于现有的其他算法.     

求解鞍点问题的多项式加速超松弛方法

为了快速有效地求解大型稀疏鞍点问题,在广义逐次超松弛(GSOR)迭代算法的基础上,结合Chebyshev多项式加速技术,本文构造了一种多项式加速超松弛迭代算法,并研究了该算法的收敛性.通过讨论加速后迭代矩阵的收敛性证明了新方法比加速前的迭代法具有快的收敛速度.数值例子也表明新方法提高了GSOR算法的收敛效率.     

非线性指标-3微分-代数系统的波形松弛算法的实现

本文探讨非线性指标-3微分-代数系统的波形松弛算法所涉及的理论模型和具体算例的求解。对于收敛性问题,我们利用谱半径条件证明该理论模型的收敛性,所得结论的收敛性条件相对较弱,并利用算例测试了收敛定理的正确性。结果显示:在求解非线性微分-代数系统时,波形松弛类算法是具有内在并行性的有效算法。     

求解对称锥互补问题的一种非精确光滑牛顿方法(英文)

本文给出了一种求解对称锥互补问题的非精确光滑牛顿方法,所采用的互补函数是含一个参数且以FB和CHKS为特例的光滑函数.新方法的每步迭代中,都采用非精确牛顿方法求解由原问题产生的子问题.在一定条件下,新算法具有全局收敛和局部超线性收敛的性质.数值试验表明算法对于求解大规模对称锥互补问题是非常有效的.     

关于两个拟非扩张映像的混杂投影算法及其数值实验（英文）

在Hilbert空间中,构造了一种新的混杂投影算法用于逼近两个拟非扩张映像的公共不动点,并利用拟非扩张映像的定义和投影算子等技巧证明了其公共不动点的强收敛定理.最后,给出具体的数值实验描述了所提出算法的有效性.     

分裂可行问题的两种强收敛CQ算法(英文)

为保证Hilbert空间中求解分裂可行问题迭代算法的强收敛性,本文首先通过引入三个参数序列提出了求解分裂可行问题的改进CQ算法,并在较弱的条件下证明了算法的强收敛性.然后改进算法中的一个算子,即选择另外一个参数序列嵌入到一个算子里,得到了一种新的算法.在参数序列满足一定条件下也证明了算法的强收敛性.本文拓展了现已有的相关研究成果.     

非线性混合整数规划问题的改进差分进化算法

针对非线性混合整数规划问题,本文采用非固定多段映射罚函数法处理约束条件、用混合整数编码技术处理连续变量和整数变量,并在基本差分进化算法中加入一种新型的凸组合变异算子和一种指数递增交叉算子,由此构造出了一种求解非线性混合整数规划问题的改进差分进化算法。实验表明,所提出的算法全局收敛速度快,精度高,鲁棒性强。     

一维变分不等式问题的自适应有限元分析新探

结构工程中的弹性薄膜接触和杆件弹塑性扭转等问题是典型的变分不等式问题,对其高效精确求解,特别是满足给定精度要求下的自适应求解,是挑战性课题。该文作者新近成功实现了一维变分不等式问题的自适应有限元分析,该文对此进展作一报道。对于变分不等式的有限元求解,该文提出区域二分法和C检验技术,极大提升了松弛迭代的收敛速度,一般4次~5次线性解即可得到收敛的有限元解答,进而采用作者提出的EEP(单元能量投影)超收敛公式计算超收敛解答,用其检验误差并指导网格细分,逐步得到堪称为数值精确解的解答,亦即得到按照最大模度量逐点满足精度要求的解答。该文给出的数值算例表明所提出的算法具有高效、可靠、精确的优良特性。     

框式凸二次规划的原始-对偶不可行内点算法

对框式凸二次规划提出了一种原始一对偶不可行内点算法,在初始点取在中心路径的领域N时,证明了算法的全局收敛性。     

Implementation of an exact finite reduction scheme for steady‐state reaction‐diffusion equations

In this paper we propose the numerical solution of a steady‐state reaction‐diffusion problem by means of application of a non‐local Lyapunov–Schmidt type reduction originally devised for field theory. A numerical algorithm is developed on the basis of the discretization of the differential operator by means of simple finite differences. The eigendecomposition of the resulting matrix is used to implement a discrete version of the reduction process. By the new algorithm the problem is decomposed into two coupled subproblems of different dimensions. A large subproblem is solved by means of a fixed point iteration completely controlled by the features of the original equation, and a second problem, with dimensions that can be made much smaller than the former, which inherits most of the non‐linear difficulties of the original system. The advantage of this approach is that sophisticated linearization strategies can be used to solve this small non‐linear system, at the expense of a partial eigendecomposition of the discretized linear differential operator. The proposed scheme is used for the solution of a simple non‐linear one‐dimensional problem. The applicability of the procedure is tested and experimental convergence estimates are consolidated. Numerical results are used to show the performance of the new algorithm. Copyright © 2006 John Wiley & Sons, Ltd.     

A Primal-Dual Interior-Point Method to Solve the Optimal Power Flow Dispatching Problem

This paper presents a primal-dual path-following interior-point method for the solution of the optimal power flow dispatching (OPFD) problem. The underlying idea of most path-following algorithms is relatively similar: starting from the Fiacco-McCormick barrier function, define the central path and loosely follow it to the optimum solution. Several primal-dual methods for OPF have been suggested, all of which are essentially direct extensions of primal-dual methods for linear programming. Nevertheless, there are substantial variations in some crucial details which include the formulation of the non-linear problem, the associated linear system, the linear algebraic procedure to solve this system, the line search, strategies for adjusting the centring parameter, estimating higher order correction terms for the homotopy path, and the treatment of indefiniteness. This paper discusses some of the approaches that were undertaken in implementing a specific primal-dual method for OPFD. A comparison is carried out with previous research on interior-point methods for OPF. Numerical tests on standard IEEE systems and on a realistic network are very encouraging and show that the new algorithm converges where other algorithms fail.     

A Convex and Exact Approach to Discrete Constrained TV-L1 Image Approximation

We study the TV-L1 image approximation model from primal and dual perspective, based on a proposed equivalent convex formulations. More specifically, we apply a convex TV-L1 based approach to globally solve the discrete constrained optimization problem of image approximation, where the unknown image function $u(x)∈\{f_1 ,... , f_n\}$, $∀x ∈ Ω$. We show that the TV-L1 formulation does provide an exact convex relaxation model to the non-convex optimization problem considered. This result greatly extends recent studies of Chan et al., from the simplest binary constrained case to the general gray-value constrained case, through the proposed rounding scheme. In addition, we construct a fast multiplier-based algorithm based on the proposed primal-dual model, which properly avoids variability of the concerning TV-L1 energy function. Numerical experiments validate the theoretical results and show that the proposed algorithm is reliable and effective.     

A new fixed‐point algorithm for hardening plasticity based on non‐linear mixed variational inequalities

Moving from the seminal papers of Han and Reddy, we propose a fixed‐point algorithm for the solution of hardening plasticity two‐dimensional problems. The continuous problem may be classified as a mixed non‐linear non‐differentiable variational inequality of the second type and is discretized by means of a truly mixed finite‐element scheme. One of the main peculiarities of our approach is the use of the composite triangular element of Johnson and Mercier for the approximation of the stress field. The non‐differentiability is coped with via regularization whereas the non‐linearity is approached with a fixed‐point iterative strategy. Numerical results are proposed that investigate the sensitivity of the approach with respect to the mesh size and the regularization parameter ε. The simplicity of the proposed fixed‐point scheme with respect to more classical return mapping approaches seems to represent one of the key features of our algorithm. Copyright © 2003 John Wiley & Sons, Ltd.     

一种自适应EPF算法及其在光纤水听器中的应用

基于椭圆拟合的相位生成载波(Phase Generated Carrier,PGC)解调方法是消除非线性因素对光纤水听器PGC解调结果影响的一种有效手段,椭圆曲线参数的最优估计问题是实现该方法的关键。扩展卡尔曼粒子滤波(Extended Kalman Particle Filter,EPF)是解决此类非线性估计问题的一种常用的最优估计算法。但传统的EPF算法在用于常参数过程方程的参数或状态估计问题时,过程噪声的方差通常设置为一个常量,这使得算法难以兼顾收敛速度和估计精度,一定程度上限制了算法的整体性能。为了解决这个问题,文章对现有的EPF进行了改进,提出了一种自适应扩展卡尔曼粒子滤波(Adaptive Extended Kalman Particle Filter,AEPF)算法。模拟仿真和实验结果表明,文中所提出的AEPF算法能根据基于椭圆拟合的PGC解调方法有效地解调出待测声信号,相比EKF算法和EPF算法,AEPF算法的收敛速度和估计精度都得到了提升。此外,文章所提出的AEPF算法也适用于其他具有常参数过程方程的参数或状态估计问题,具有一定的通用性。     

Second-order cone programming with warm start for elastoplastic analysis with von Mises yield criterion

The incremental problem for quasistatic elastoplastic analysis with the von?Mises yield criterion is discussed within the framework of the second-order cone programming (SOCP). We show that the associated flow rule under the von?Mises yield criterion with the linear isotropic/kinematic hardening is equivalently rewritten as a second-order cone complementarity problem. The minimization problems of the potential energy and the complementary energy for incremental analysis are then formulated as the primal-dual pair of SOCP problems, which can be solved with a primal-dual interior-point method. To enhance numerical performance of tracing an equilibrium path, we propose a warm-start strategy for a primal-dual interior-point method based on the primal-dual penalty method. In this warm-start strategy, we solve a penalized SOCP problem to find the equilibrium solution at the current loading step. An advanced initial point for solving this penalized SOCP problem is defined by using information of the solution at the previous loading step.     

基于SCAD的压缩感知阈值迭代算法的收敛性分析

基于SCAD罚函数的压缩感知在有噪声稀疏信号重建中具有优良的理论及应用效果,开展其快速重建算法研究有着重要的意义,阈值迭代算法是解决压缩传感问题最有效的算法之一.本文研究了基于SCAD罚函数的压缩感知阈值迭代算法的收敛性问题,给出了算法收敛到稀疏解的充分条件,并证明了迭代估计值以指数阶速率收敛于最优值.进一步,本文给出了基于AMP改进的SCAD阈值迭代算法的收敛性分析.     

Linearized Alternating Direction Method of Multipliers for Constrained Linear Least-Squares Problem

The alternating direction method of multipliers (ADMM) is applied to a constrained linear least-squares problem, where the objective function is a sum of two least-squares terms and there are box constraints. The original problem is decomposed into two easier least-squares subproblems at each iteration, and to speed up the inner iteration we linearize the relevant subproblem whenever it has no known closed-form solution. We prove the convergence of the resulting algorithm, and apply it to solve some image deblurring problems. Its efficiency is demonstrated, in comparison with Newton-type methods.