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Bingsheng He Xiang Wang Junfeng Yang 《计算数学(英文版)》2009,(4):459-473
It is interesting to compare the efficiency of two methods when their computational loads in each iteration are equal. In this paper, two classes of contraction methods for monotone variational inequalities are studied in a unified framework. The methods of both classes can be viewed as prediction-correction methods, which generate the same test vector in the prediction step and adopt the same step-size rule in the correction step. The only difference is that they use different search directions. The computational loads of each iteration of the different classes are equal. Our analysis explains theoretically why one class of the contraction methods usually outperforms the other class. It is demonstrated that many known methods belong to these two classes of methods. Finally, the presented numerical results demonstrate the validity of our analysis. 相似文献
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In this paper we study the proximal point algorithm (PPA) based prediction-correction (PC) methods for monotone variational inequalities. Each iteration of these methods consists of a prediction and a correction. The predictors are produced by inexact PPA steps. The new iterates are then updated by a correction using the PPA formula. We present two profit functions which serve two purposes: First we show that the profit functions are tight lower bounds of the improvements obtained in each iteration. Based on this conclusion we obtain the convergence inexactness restrictions for the prediction step. Second we show that the profit functions are quadratically dependent upon the step lengths, thus the optimal step lengths are obtained in the correction step. In the last part of the paper we compare the strengths of different methods based on their inexactness restrictions. 相似文献
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1引言设R~n表示n维欧式空间,‖·‖和<,>分别表示R~n中的范数和内积,K为R~n中的非空闭凸集,(?)是R~n到R∪{ ∞)的算子.对于给定的非线性算子T,g:R~n→R~n,考虑下面的广义混合变分不等式,记为GMVI:求u∈R~n满足(Tu)~T(g(v)-g(tu)) (?)(g(v))-(?)(g(u))(?)0,(?)g(v)∈R~n.(1)假如(?)是R~n中非空闭凸集K的指标集,即,(?)(u)≡I_k(u)=(?).此时GMVI等价于下面的广义变分不等式:求u∈R~n,g(u)∈K满足 相似文献
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一类求解单调变分不等式的隐式方法 总被引:6,自引:0,他引:6
1.引言变分不等式是一个非常有趣。非常困难的数学问题["].它具有广泛的应用(例如,数学规划中的许多基本问题都可以归结为一个变分不等式问题),因而得到深入的研究并有了不少算法[1,2,5-8,17-21].对线性单调变分不等式,我们最近提出了一系列投影收缩算法Ig-13].本文考虑求解单调变分不等式其中0CW是一闭凸集,F是从正p到自身的一个单调算子,一即有我们用比(·)表示到0上的投影.求解单调变分不等式的一个简单方法是基本投影法[1,6],它的迭代式为然而,如果F不是仿射函数,只有当F一致强单调且LIPSChitZ连续… 相似文献
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ALTERNATING PROJECTION BASED PREDICTION-CORRECTION METHODS FOR STRUCTURED VARIATIONAL INEQUALITIES 总被引:4,自引:0,他引:4
Bing-sheng He Li-zhi Liao Mai-jian Qian 《计算数学(英文版)》2006,24(6):693-710
The monotone variational inequalities Ⅵ(Ω,F)have vast applications, including opti-mal controls and convex programming.In this paper we focus on the Ⅵ problems thathave a particular splitting structure and in which the mapping F does not have an explicitform,therefore only its function values can be employed in the numerical methods for solv-ing such problems.We study a set of numerical methods that are easily implementable.Each iteration of the proposed methods consists of two procedures.The first(prediction)procedure utilizes alternating projections to produce a predictor.The second(correction)procedure generates the new iterate via some minor computations.Convergence of theproposed methods is proved under mild conditions.Preliminary numerical experiments forsome traffic equilibrium problems illustrate the effectiveness of the proposed methods. 相似文献
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蒋美群 《高等学校计算数学学报(英文版)》1994,(1)
In this paper the Schwarz alternating method for a fourth-order elliptic variational inequality problem is considered by way of the equivalent form, and the geometric convergence is obtained on two subdomains. 相似文献
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In this paper, we first give a smoothing approximation function of nonsmooth system based on box constrained variational inequalities and then present a new smoothing approximation algorithm. Under suitable conditions,we show that the method is globally and superlinearly convergent. A few numerical results are also reported in the paper. 相似文献
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1 IntroductionWe consider tlie variational inequality problelll, deuoted by VIP(X, F), wliicli is to find avector x* E X such thatF(X*)"(X -- X-) 2 0, VX E X, (1)where F: R" - R" is any vector-valued f11uction and X is a uonelllpty subset of R'.This problem has important applicatiolls. in equilibriun1 modeIs arising in fields such asecououtics, transportatioll scieuce alld operations research. See [1]. There exist mauy lllethodsfor solviug tlie variational li1equality problem VIP(X. … 相似文献
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The general mixed quasi variational inequality containing a nonlinear term φ is a useful and an important generalization of variational inequalities. The projection method can not be applied to solve this problem due to the presence of nonlinear term. It is well known that the variational inequalities involving the nonlinear term φ are equivalent to the fixed point problems and resolvent equations. In this article, the authors use these alternative equivalent formulations to suggest and analyze a new self-adaptive iterative method for solving general mixed quasi variational inequalities. Global convergence of the new method is proved. An example is given to illustrate the efficiency of the proposed method. 相似文献
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求解单调变分不等式问题的一个连续型迭代方法 总被引:1,自引:1,他引:0
本文给出一个求解单调变分不等式问题的连续型迭代方法,对任意单调趋于零的正数序列和任意初始点,方法产生的迭代点列均收敛到所求变分不等式问题的一个解,且在适当条件下方法具有Q-超线性收敛率.数值试验结果进一步表明了所给方法的稳定性和有效性. 相似文献
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求单调变分不等式隐式方法的一个单调下降性质 总被引:1,自引:0,他引:1
孙秀真 《高等学校计算数学学报》2002,24(1):75-80
1 引 言变分不等式在数学规划中起着很重要的作用 ,许多研究者 [3 ]讨论了这一问题 .对于单调线性变分不等式问题 ,文 [4 -7]提出了几种投影收缩算法 ( PC) .最近文 [7]中研究了如下的一类变分不等式问题( VI) u∈Ω , ( v -u) TF( u)≥ 0 , v∈Ω . ( 1 )其中Ω Rn 是一个闭凸集 ,F是 Rn到自身的连续单调映射 ,即F( u) -F( v) T( u -v)≥ 0 , u,v∈ Rn. ( 2 )由 [1 ]知 ,对于任意的 β>0 ,变分不等式 ( 1 )等价于投影方程于是求解 ( 1 )即是寻求e( u,β)∶ =u -PΩ[u -βF( u) ]的零点 .本文中 Ω*表示 ( … 相似文献
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N.H.Sweilam L.F.Abd-Elal 《计算数学(英文版)》2003,21(6):815-824
Iterative techniques for solving optimal control systems governed by parabolic variational inequalities are presented. The techniques we use are based on linear finite elements method to approximate the state equations and nonlinear conjugate gradient methods to solve the discrete optimal control problem. Convergence results and numerical experiments are presented. 相似文献
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P0-函数箱约束变分不等式的正则半光滑牛顿法 总被引:8,自引:0,他引:8
1引言设X C R~n,F:R~n→R~n,变分不等式Ⅵ(X,F)是指:求x∈X,使F(x)~T(y-x)≥0,(?)_y∈X.(1)记i∈N={1,2,…,n},当X=[a,b]:={x∈(?)~n|a_i≤x_i≤b_i,i∈N}时,称Ⅵ(X,F)为箱约束变分不等式(也有些文献称为混合互补问题),记为Ⅵ(a,b,F).若a_i=0,b_i= ∞,i∈N,即X=(?)_ ~n:={x∈(?)~n|x≥0}时,Ⅵ(a,b,F)化为非线性互补问题NCP(F):求x∈(?)_ ~n,使x≥0,F(x)≥0,x~TF(x)=0.(2) 相似文献
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关于外梯度法的步长规则 总被引:1,自引:0,他引:1
1.引言 设为Rn中的一个非空闭凸集,F(x)为Rn Rn中的一个连续向量函数.变分不等式问题(F,)就是:找一向量x 使得当 =R时,(1.1)退化成非线性互补问题。在这篇文章中总假定:(H1) ,这里表示(1.1)的解集;(H2)F(x)是单调的,即对,(x-y)(F(x)-F(x)-F(y)). 这类问题出现在工程物理、经济管理等领域,有着极为广泛的应用.因此,其数值解近年来受到重视,提出许多有效算法,见综述[1, 2].在现有的算法中, Korpelevich的外梯度法[3](何炳生称它为投影… 相似文献
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关于单调变分不等式的不精确邻近点算法的收敛性分析 总被引:7,自引:0,他引:7
王治华 《高等学校计算数学学报》2003,25(4):336-343
We consider a proximal point algorithm(PPA) for solving monotone variational inequalities. PPA generates a sequence by solving a sequence of strongly monotone subproblems .However,solving the subproblems is either expensive or impossible. Some inexact proximal point algorithms(IPPA) have been developed in many literatures. In this paper, we present a criterion for approximately solving subproblems. It only needs one simple additional work on the basis of original algorithm, and the convergence criterion becomes milder. We show that this method converges globally under new criterion provided that the solution set of the problem is nonempty. 相似文献
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本文给出了求解无单调性集值变分不等式的一个新的投影算法,该算法所产生的迭代序列在Minty变分不等式解集非空且映射满足一定的连续性条件下收敛到解.对比文献[10]中的算法,本文中的算法使用了不同的线性搜索和半空间,在计算本文所引的两个数值例子时,该算法比文献[10]中的算法所需迭代步更少. 相似文献
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1.引 言 设C Rn为非空闭凸集,为连续映射.变分不等式问题,记为VI(F,C),是求满足上述条件的向量x∈C变分不等式问题在工程力学,交通运输,经济运筹等方面具有广泛的应用并越来越受到人们的重视 [2,3] 求解变分不等式问题有很多解法,其中最简单的是投影 相似文献