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.     

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．     

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.     

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.     

关于一类变分不等式的LQP算法

本文研究一类ξ-单调的变分不等式问题.利用KKT条件将原问题转换为非线性互补问题(nonlinear complementarity problem,NCP)的方法,获得了基于logarithmic-quadratic proximal(LQP)的算法及其改进形式,推广了LQP算法的适用范围.     

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.     

解一类结构变分不等式问题的非精确并行交替方向法

带线性约束的具有两分块结构的单调变分不等式问题, 出现在许多现代应用中, 如交通和经济问题等. 基于该问题良好的可分结构, 分裂型算法被广泛研究用于其求解. 提出新的带回代的非精确并行交替方向法解该类问题, 在每一步迭代中,首先以并行模式通过投影得到预测点, 然后对其校正得到下一步的迭代点. 在压缩型算法的理论框架下, 在适当条件下证明了所提算法的全局收敛性. 数值结果表明了算法的有效性. 此外, 该算法可推广到求解具有多分块结构的问题.     

求解单调变分不等式的两类迭代算法

给出了求解单调变分不等式的两类迭代算法．通过解强单调变分不等式子问题,产生两个迭代点列,都弱收敛到变分不等式的解．最后,给出了这两类新算法的收敛性分析．     

A new LQP alternating direction method for solving variational inequality problems with separable structure

We presented a new logarithmic-quadratic proximal alternating direction scheme for the separable constrained convex programming problem. The predictor is obtained by solving series of related systems of non-linear equations in a parallel wise. The new iterate is obtained by searching the optimal step size along a new descent direction. The new direction is obtained by the linear combination of two descent directions. Global convergence of the proposed method is proved under certain assumptions. We show the O(1 / t) convergence rate for the parallel LQP alternating direction method.     

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.     

Application of the alternating direction method of multipliers to separable convex programming problems

This paper presents a decomposition algorithm for solving convex programming problems with separable structure. The algorithm is obtained through application of the alternating direction method of multipliers to the dual of the convex programming problem to be solved. In particular, the algorithm reduces to the ordinary method of multipliers when the problem is regarded as nonseparable. Under the assumption that both primal and dual problems have at least one solution and the solution set of the primal problem is bounded, global convergence of the algorithm is established.     

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 approximate proximal-extragradient type method for monotone variational inequalities

Proximal point algorithms (PPA) are attractive methods for monotone variational inequalities. The approximate versions of PPA are more applicable in practice. A modified approximate proximal point algorithm (APPA) presented by Solodov and Svaiter [Math. Programming, Ser. B 88 (2000) 371–389] relaxes the inexactness criterion significantly. This paper presents an extended version of Solodov–Svaiter's APPA. Building the direction from current iterate to the new iterate obtained by Solodov–Svaiter's APPA, the proposed method improves the profit at each iteration by choosing the optimal step length along this direction. In addition, the inexactness restriction is relaxed further. Numerical example indicates the improvement of the proposed method.     

一种求解合作博弈最公平核心的非精确平行分裂算法

针对合作博弈核心和Shapley值的特点, 将最公平核心问题转化为带有两个变 量的可分离凸优化问题, 引入结构变分不等式的算子分裂方法框架, 提出了求解最公平核心的一种非精确平行分裂算法. 而且, 该算法充分利用了所求解问题的可行域的简单闭凸性, 子问题的非精确求解是容易的. 最后, 简单算例的数值实验表明了算法的收敛性和有效性.     

A modified inexact operator splitting method for monotone variational inequalities

The Douglas–Peaceman–Rachford–Varga operator splitting methods (DPRV methods) are attractive methods for monotone variational inequalities. He et al. [Numer. Math. 94, 715–737 (2003)] proposed an inexact self-adaptive operator splitting method based on DPRV. This paper relaxes the inexactness restriction further. And numerical experiments indicate the improvement of this relaxation.      

Parallel splitting augmented Lagrangian methods for monotone structured variational inequalities

The typical structured variational inequalities can be interpreted as a system of equilibrium problems with a leader and two cooperative followers. Assume that, based on the instruction given by the leader, each follower can solve the individual equilibrium sub-problems in his own way. The responsibility of the leader is to give a more reasonable instruction for the next iteration loop based on the feedback information from the followers. This consideration leads us to present a parallel splitting augmented Lagrangian method (abbreviated to PSALM). The proposed method can be extended to solve the system of equilibrium problems with three separable operators. Finally, it is explained why we cannot use the same technique to develop similar methods for problems with more than three separable operators.