求解结构矩阵低秩逼近的交替投影方法

结构矩阵低秩逼近在图像压缩、计算机代数和语音编码中有广泛应用.首先给出了几类结构矩阵的投影公式,再利用交替投影方法计算结构矩阵低秩逼近问题.数值试验表明新方法是可行的.     

一族超线性收敛的投影拟牛顿算法

本文将梯度投影与拟牛顿法相结合，给出了求解一般线性约束非线性规划问题含两组参数的算法族．在一定的条件下证明了算法族的全局收敛性与它的子族的超线性收敛速度，并给出了投影Ｄ．Ｆ．Ｐ方法、投影ＢＦＧＳ方法等一些特例．     

一般单调变分不等式的一个改进的预估-校正算法

最近何炳生等提出了解大规模单调变分不等式的一种预估-校正算法,然而,这个方法在计算每一个试验点时需要一次投影运算,因而计算量较大．为了克服这个缺点,我们提出了一个解一般大规模g-单调变分不等式的新的预估-校正算法,该方法使用了一个非常有效的预估步长准则,每个步长的选取只需要计算一次投影,这将大大减少计算量．数值试验说明我们的算法比最新文献中出现的投影类方法有效．     

最优投影策略下解病态积分方程的快速迭代算法

基于最优的投影方法,构造了求解病态积分方程的截断快速Tikhonov迭代算法,与传统投影方法相比得到了相同的最优收敛率,但内积的计算个数少于传统投影方法.同时,给出了后验参数选择办法.算例证实了算法的有效性.     

一般单调变分不等式的近似邻近外梯度算法

近似邻近点算法是求解单调变分不等式的一个有效方法,该算法通过解决一系列强单调子问题,产生近似邻近点序列来逼近变分不等式的解,而外梯度算法则通过每次迭代中增加一个投影来克服一般投影算法限制太强的缺点,但它们均未能改变迭代步骤中不规则闭凸区域上投影难计算的问题.于是,本文结合外梯度算法的迭代格式,构造包含原投影区域的半空间,将投影建立在半空间上,简化了投影的求解过程,并对新的邻近点序列作相应限制,使得改进的算法具有较好的收敛性.     

核范数和谱范数下广义Sylvester方程最小二乘问题的有效算法

本文从数值角度讨论Schatten q-范数下的广义Sylvester方程约束最小二乘问题min x∈s‖N∑i=1A_iXB_i—C‖_q,其中S为闭凸约束集合,Schatten q-范数定义为‖M‖_q~q=∑_(i=1)~nσ_i~q(M),其中σ_i(M)为M∈R~(n×n)的奇异值.该问题的几类特殊情形在图像处理、控制论等领域有广泛的应用.q=2即Frobenius范数下该问题已被充分研究,故本文着重讨论q=1,+∞,即核范数和谱范数下该问题的数值求解.采用的数值方法是非精确标准容易执行的部分非精确交替方向法,并结合奇异值阈值算法,Moreau-Yosida正则化算法,谱投影算法和LSQR算法等求解相应子问题.给出算法的收敛性证明,并用数值算例验证其高效可行性.     

一个解带线性或非线性约束最优化问题的梯度投影方法

§1 引言 Rosen在[1,2]中利用梯度投影建立了带约束非线性规划问题的可行方向算法,称为梯度投影方法.由于此方法简单易行,计算的每一步都是显式迭代,而不必去解复杂的线性规划或二次规划问题,因此人们颇为注意.现在梯度投影方法已成为非线性规划算法     

对称线性互补问题的乘性Schwarz算法

本文提出了求解对称性互补问题的乘性Schwarz算法,其中子问题用投影迭代方法求解.利用投影迭代算子的性质及投影迭代的收敛性,证明了算法产生的迭代点列的聚点为原互补问题的解,并在一定条件下,证明算法产生的迭代点列的聚点存在.     

求解非单调变分不等式的一种二次投影算法北大核心CSCD

投影算法是求解变分不等式问题的主要方法之一.目前,有关投影算法的研究通常需要假设映射是单调且Lipschitz连续的,然而在实际问题中,往往不满足这些假设条件.该文利用线搜索方法,提出了一种新的求解非单调变分不等式问题的二次投影算法.在一致连续假设下,证明了算法产生的迭代序列强收敛到变分不等式问题的解.数值实验结果表明了该文所提算法的有效性和优越性.     

求解单侧障碍问题的隐式投影方法

利用不动点原理,得到了求解一类障碍问题的隐式投影算法.采用中心差分格式将障碍问题离散为一个线性互补问题,从而得到了基于投影形式的隐式算法.该方法的每一步迭代只需要求解一个线性方程组.用投影性质很容易证明算法收敛性.给出了具体的算法过程,数值算例结果和理论分析是一致的.     

The iterative methods for monotone generalized variational inequalities

A new class of iterative methods are presented for monotone generalized variational inequality problems. These methods, which base on an equivalent formulation of the original problem, can be viewed as the extension of the symmetric projection rnethod for monotone variational inequalities. The global convergence of the methods is estab-lished under the monotonicity assumption on the functions associated the problem.Specialization of the proposed algorithms and related results to several special cases are also discussed. Moreover, two combination methods are presented for affine monotone problems. and their global and Q-linear convergence are also established     

Incomplete oblique projections for solving large inconsistent linear systems

The authors introduced in previously published papers acceleration schemes for Projected Aggregation Methods (PAM), aiming at solving consistent linear systems of equalities and inequalities. They have used the basic idea of forcing each iterate to belong to the aggregate hyperplane generated in the previous iteration. That scheme has been applied to a variety of projection algorithms for solving systems of linear equalities or inequalities, proving that the acceleration technique can be successfully used for consistent problems. The aim of this paper is to extend the applicability of those schemes to the inconsistent case, employing incomplete projections onto the set of solutions of the augmented system Ax − r = b. These extended algorithms converge to the least squares solution. For that purpose, oblique projections are used and, in particular, variable oblique incomplete projections are introduced. They are defined by means of matrices that penalize the norm of the residuals very strongly in the first iterations, decreasing their influence with the iteration counter in order to fulfill the convergence conditions. The theoretical properties of the new algorithms are analyzed, and numerical experiences are presented comparing their performance with several well-known projection methods. Dedicated to Clovis Gonzaga on the occassion of his 60th birthday.     

An iterative scheme for variational inequalities

In this paper we introduce and study a general iterative scheme for the numerical solution of finite dimensional variational inequalities. This iterative scheme not only contains, as special cases the projection, linear approximation and relaxation methods but also induces new algorithms. Then, we show that under appropriate assumptions the proposed iterative scheme converges by establishing contraction estimates involving a sequence of norms in Eⁿ induced by symmetric positive definite matrices G_m. Thus, in contrast to the above mentioned methods, this technique allows the possibility of adjusting the norm at each step of the algorithm. This flexibility will generally yield convergence under weaker assumptions.     

An assessment of two approaches to variable metric methods

Two recent suggestions in the field of variable metric methods for function minimization are reviewed: the self-scaling method, first introduced by Oren and Luenberger, and the method of Biggs. The two proposals are considered both from a theoretical and computational aspect. They are compared with methods which use correction formulae from the Broyden one-parameter family, in particular the BFGS formula and the Fletcher switching strategy.     

求解约束优化问题的记忆梯度Goldstein-Lavintin-Polyak投影算法

给求解无约束规划问题的记忆梯度算法中的参数一个特殊取法,得到目标函数的记忆梯度G o ldste in-L av in tin-Po lyak投影下降方向,从而对凸约束的非线性规划问题构造了一个记忆梯度G o ldste in-L av in tin-Po lyak投影算法,并在一维精确步长搜索和去掉迭代点列有界的条件下,分析了算法的全局收敛性,得到了一些较为深刻的收敛性结果.同时给出了结合FR,PR,HS共轭梯度算法的记忆梯度G o ldste in-L av in tin-Po lyak投影算法,从而将经典共轭梯度算法推广用于求解凸约束的非线性规划问题.数值例子表明新算法比梯度投影算法有效.     

Projection methods for stochastic differential equations with conserved quantities

In this paper, we consider the numerical methods preserving single or multiple conserved quantities, and these methods are able to reach high order of strong convergence simultaneously based on some kinds of projection methods. The mean-square convergence orders of these methods under certain conditions are given, which can reach order 1.5 or even 2 according to the supporting methods embedded in the projection step. Finally, three numerical experiments are taken into account to show the superiority of the projection methods.     

Variable metric methods for minimizing a class of nondifferentiable functions

We develop a class of methods for minimizing a nondifferentiable function which is the maximum of a finite number of smooth functions. The methods proceed by solving iteratively quadratic programming problems to generate search directions. For efficiency the matrices in the quadratic programming problems are suggested to be updated in a variable metric way. By doing so, the methods possess many attractive features of variable metric methods and can be viewed as their natural extension to the nondifferentiable case. To avoid the difficulties of an exact line search, a practical stepsize procedure is also introduced. Under mild assumptions the resulting method converge globally.Research supported by National Science Foundation under grant number ENG 7903881.     

求解非线性等式和不等式约束优化问题的三项记忆梯度广义投影算法

利用广义投影矩阵，对求解无约束规划的三项记忆梯度算法中的参数给一条件，确定它们的取值范围，以保证得到目标函数的三项记忆梯度广义投影下降方向，建立了求解非线性等式和不等式约束优化问题的三项记忆梯度广义投影算法，并证明了算法的收敛性．同时给出了结合FR，PR，HS共轭梯度参数的三项记忆梯度广义投影算法，从而将经典的共轭梯度算法推广用于求解约束规划问题．数值例子表明算法是有效的．     

信赖域方法最优曲线性质分析

本文描述了信赖域方法最优曲线在二维子空间内投影的几个性质,分析了几种信赖域折线法与该投影的关系,为推导理邹的求解信赖域子问题的折线近似提供理论依据。     

初始点任意的解非线性不等式约束优化问题的结合共轭梯度参数的超记忆梯度广义投影算法

本文利用广义投影矩阵，对求解无约束规划的超记忆梯度算法中的参数给出一种新的取值范围以保证得到目标函数的超记忆梯度广义投影下降方向，并与处理任意初始点的方法技巧结合建立求解非线性不等式约束优化问题的一个初始点任意的超记忆梯度广义投影算法，在较弱条件下证明了算法的收敛性．同时给出结合FR，PR，HS共轭梯度参数的超记忆梯度广义投影算法，从而将经典的共轭梯度法推广用于求解约束规划问题．数值例子表明算法是有效的．