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.     

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.     

一类非对称单调变分不等式的交替方向法

对一类非对称变分不等式问题提出了交替方向法。推广了交替方向仅适用于等式约束或不等约束的情形，得出了迭代序列的一些性质及收敛性．     

交替方向法求解带线性约束的变分不等式

１引言变分不等式是一个有广泛应用的数学问题,它的一般形式是：确定一个向量,使其满足这里ｆ是一个从到自身的一个映射,Ｓ是Ｒ中的一个闭凸集．在许多实际问题中集合Ｓ往往具有如下结构其中ＡｂＫ是中的一个简单闭凸集．例如一个正卦限,一个框形约束结构,或者一个球简言之,Ｓ是Ｒ中的一个超平面与一个简单闭凸集的交．求解问题（１）－（２）,往往是通过对线性约束Ａ引人Ｌａｇｒａｎｇｅ乘子,将原问题化为如下的变分不等式：确定使得我们记问题（３）－（４）为ＶＩ（Ｆ）．熟知［３］,ＶＩ（,Ｆ）等价于投影方程其中凡（·）表…     

一种求解变分不等式的新的交替方向法

对于一类特殊的变分不等式,提出一种新的交替方向方法.与通常的交替方向方法相比,该方法计算量更小.在函数强迫单调的条件下证明了算法的全局收敛性.     

解可分离结构变分不等式的一种新的交替方向法

交替方向法是求解可分离结构变分不等式问题的经典方法之一, 它将一个大型的变分不等式问题分解成若干个小规模的变分不等式问题进行迭代求解. 但每步迭代过程中求解的子问题仍然摆脱不了求解变分不等式子问题的瓶颈. 从数值计算上来说, 求解一个变分不等式并不是一件容易的事情.因此, 本文提出一种新的交替方向法, 每步迭代只需要求解一个变分不等式子问题和一个强单调的非线性方程组子问题. 相对变分不等式问题而言, 我们更容易、且有更多的有效算法求解一个非线性方程组问题. 在与经典的交替方向法相同的假设条件下, 我们证明了新算法的全局收敛性. 进一步的数值试验也验证了新算法的有效性.     

一种求解单调变分不等式的部分并行分裂LQP交替方向法

LQP交替方向法是求解可分离结构型单调变分不等式问题的一种非常有效的方法.它不仅可以充分地利用目标函数的可分结构,将原问题分解为多个更易求解的子问题,还更适合求解大规模问题.对于带有三个可分离算子的单调变分不等式问题,结合增广拉格朗日算法和LQP交替方向法提出了一种部分并行分裂LQP交替方向法,构造了新算法的两个下降方向,结合这两个下降方向得到了一个新的下降方向,沿着这个新的下降方向给出了最优步长.并在较弱的假设条件下,证明了新算法的全局收敛性.     

求解一类结构型变分不等式的加速随机方法

基于Glowinski的交替方向法和何炳生教授的改善步长的收缩算法,提出一个求解结构型变分不等式的加速随机方法.新方法的优势在于利用独立同分布的随机数来扩张步长,克服了传统的交替方向法中固定步长因子的缺点,证明了新方法的下降方向是可行的.在适当的假设条件下,给出新方法的性质,并证明新方法依概率收敛.通过对来自于金融和统计中问题的一系列数值试验,验证新方法的可行性和有效性.     

曲率障碍下四阶变分不等式的交替方向乘子法北大核心CSCD

对于重调和算子和曲率障碍表示的变分不等式,提出了自适应交替方向乘子数值解法(SADMM).对问题引入一个辅助变量表示曲率函数的增广Lagrange函数,导出一个约束极小值问题,并且该问题等价于一个鞍点问题.然后采用交替方向乘子法(ADMM)求解这个鞍点问题.通过采用平衡原理和迭代函数,得到了自动调整罚参数的自适应法则,从而提高了计算效率.证明了该方法的收敛性,并给出了利用迭代函数近似罚参数的具体方法.最后,用数值计算结果验证了该方法的有效性.     

关于一类随机变分不等式和随机拟变分不等式问题

本文对单值和多值情形的随机变分不等式和随机拟变分不等式得出可测解的存在性条件。另外还利用KKM-技巧及著名的Ky Fan定理对一类确定型的广义拟变分不等式讨论了解的存在性问题。本文的结果改进和发展了[10,11,12]中的重要结果。     

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.     

New inexact implicit method for general mixed quasi variational inequalities

In this paper, we suggest and analyze a new self-adaptive inexact implicit method with a variable parameter for general mixed quasi variational inequalities, where the skew-symmetry of the nonlinear bifunction plays a crucial part in the convergence analysis of this method. We use a self-adaptive technique to adjust parameter ρ at each iteration. The global convergence of the proposed method is proved under some mild conditions. Preliminary numerical results indicate that the self-adaptive adjustment rule is necessary in practice. Muhammad Aslam Noor is supported by the Higher Education Commission, Pakistan, through research grant No: 1-28/HEC/HRD/2005/90.     

Convergence analysis of a modified inexact implicit method for general mixed monotone variational inequalities

We consider a useful modification of the inexact implicit method with a variable parameter in Wang et al. J Optim Theory 111: 431–443 (2001) for generalized mixed monotone variational inequalities. One of the contributions of the proposed method in this paper is that the restrictions imposed on the variable parameter are weaker than the ones in Wang et al. J Optim Theory 111: 431–443 (2001). Another contribution is that we establish a sufficient and necessary condition for the convergence of the proposed method to a solution of the general mixed monotone variational inequality.     

A new inexact alternating directions method for monotone variational inequalities

The alternating directions method (ADM) is an effective method for solving a class of variational inequalities (VI) when the proximal and penalty parameters in sub-VI problems are properly selected. In this paper, we propose a new ADM method which needs to solve two strongly monotone sub-VI problems in each iteration approximately and allows the parameters to vary from iteration to iteration. The convergence of the proposed ADM method is proved under quite mild assumptions and flexible parameter conditions. Received: January 4, 2000 / Accepted: October 2001?Published online February 14, 2002     

An additive Schwarz method for variational inequalities

This paper proposes an additive Schwarz method for variational inequalities and their approximations by finite element methods. The Schwarz domain decomposition method is proved to converge with a geometric rate depending on the decomposition of the domain. The result is based on an abstract framework of convergence analysis established for general variational inequalities in Hilbert spaces.

     

An inexact logarithmic-quadratic proximal augmented Lagrangian method for a class of constrained variational inequalities

The augmented Lagrangian method is attractive in constraint optimizations. When it is applied to a class of constrained variational inequalities, the sub-problem in each iteration is a nonlinear complementarity problem (NCP). By introducing a logarithmic-quadratic proximal term, the sub-NCP becomes a system of nonlinear equations, which we call the LQP system. Solving a system of nonlinear equations is easier than the related NCP, because the solution of the NCP has combinatorial properties. In this paper, we present an inexact logarithmic-quadratic proximal augmented Lagrangian method for a class of constrained variational inequalities, in which the LQP system is solved approximately under a rather relaxed inexactness criterion. The generated sequence is Fejér monotone and the global convergence is proved. Finally, some numerical test results for traffic equilibrium problems are presented to demonstrate the efficiency of the method.      

An extended descent framework for variational inequalities

In this paper, we develop a very general descent framework for solving asymmetric, monotone variational inequalities. We introduce two classes of differentiable merit functions and the associated global convergence frameworks which include, as special instances, the projection, Newton, quasi-Newton, linear Jacobi, and nonlinear methods. The generic algorithm is very flexible and consequently well suited for exploiting any particular structure of the problem.This research was supported by the National Science and Engineering Research Council of Canada, Grant A5789, and by the Department of National Defence of Canada, Grant FUHBP.     

An alternating direction method for general variational inequalities

In this paper, we present an alternating direction method for structured general variational inequalities. This method only needs functional values for given variables in the solution process and does not require the estimate of the co-coercive modulus. All the computing process are easily implemented and the global convergence is also presented under mild assumptions. Some preliminary computational results are given.