一般线性或非线性约束下的共轭投影变尺度方法

梯度投影法是一类有效的约束最优化算法，在最优化领域中占有重要的地位．但是，梯度投影法所采用的投影是正交投影，不包含目标函数和约束函数的二阶导数信息·因而；收敛速度不太令人满意．本文介绍一种共轭投影概念，利用共轭投影构造了一般线性或非线性约束下的共轭投影变尺度算法，并证明了算法在一定条件下具有全局收敛性．由于算法中的共轭投影恰当地包含了目标函数和约束函数的二阶导数信息，因而收敛速度有希望加快．数值试验的结果表明算法是有效的．     

Projektive Newton-Verfahren und Anwendungen auf nichtlineare Randwertaufgaben

Summary In this paper we consider the following Newton-like methods for the solution of nonlinear equations. In each step of the Newton method the linear equations are solved approximatively by a projection method. We call this a Projective Newton method. For a fixed projection method the approximations often are the same as those of the Newton method applied to a nonlinear projection method. But the efficiency can be increased by adapting the accuracy of the projection method to the convergence of the approximations. We investigate the convergence and the order of convergence for these methods. The results are applied to some Projective Newton methods for nonlinear two point boundary value problems. Some numerical results indicate the efficiency of these methods.

     

一个线性约束优化问题的广义梯度投影法

本文对线性约束优化问题提出了一个新的广义梯度投影法，该算法采用了非精确线性搜索，并在每次迭代运算中结合了广义投影矩阵和变尺度方法的思想确定其搜索方向．在通常的假设条件下，证明了该算法的整体收敛性和超线性收敛速度．     

关于两个非线性算子的一个分裂方法(英文)

<正>This paper generalizes a class of projection type methods for monotone variational inequalities to general monotone inclusion.It is shown that when the normal cone operator in projection is replaced by any maximal monotone operator,the resulting method inherits all attractive convergence properties of projection type methods,and allows an adjusting step size rule.Weaker convergence assumption entails an extra projection at each iteration.Moreover,this paper also addresses applications of the resulting method to convex programs and monotone variational inequalities.     

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．     

在一个新步长规则下梯度投影算法的全局收敛性

考虑约束最优化问题：minx∈Ωf(x)其中:f:R^n→R是连续可微函数，Ω是一闭凸集。本文研究了解决此问题的梯度投影方法，在步长的选取时采用了一种新的策略，在较弱的条件下，证明了梯度投影响方法的全局收敛性。     

凸约束非光滑方程组一个新的谱梯度投影算法

基于寻找分离超平面的三种经典线搜索技术,本文提出了一种自适应线搜索技术.结合谱梯度投影法,提出了凸约束非光滑单调方程组的一个谱梯度投影算法.该算法不需要计算和存储任何矩阵,因而适合求解大规模非光滑的非线性单调方程组.在较弱的条件下,证明了方法的全局收敛性,并分析了算法的收敛率.数值试验结果表明算法是有效的和鲁棒的.     

平板几何迁移算子的特征值问题

研究L^p(1相似文献   

广义非凸变分不等式解的存在性与投影算法

本文运用Banach压缩映象原理和投影技巧研究一类新的广义非凸变分不等式问题解的存在唯一性,并在非凸集上建立一个逼近广义非凸变分不等式解的三步投影算法,在一定条件下证明了该投影算法所产生的迭代序列的收敛性.     

一个新的共轭投影梯度算法及其超线性收敛性

利用共轭投影梯度技巧，结合SQP算法的思想，建立了一个具有显示搜索方向的新算法，在适当的条件下，证明算法是全局收敛和强收敛的，且具有超线性收敛性，最后数值实验表明算法是有效的。     

On the O(1/t) convergence rate of the projection and contraction methods for variational inequalities with Lipschitz continuous monotone operators

Nemirovski’s analysis (SIAM J. Optim. 15:229–251, 2005) indicates that the extragradient method has the O(1/t) convergence rate for variational inequalities with Lipschitz continuous monotone operators. For the same problems, in the last decades, a class of Fejér monotone projection and contraction methods is developed. Until now, only convergence results are available to these projection and contraction methods, though the numerical experiments indicate that they always outperform the extragradient method. The reason is that the former benefits from the ‘optimal’ step size in the contraction sense. In this paper, we prove the convergence rate under a unified conceptual framework, which includes the projection and contraction methods as special cases and thus perfects the theory of the existing projection and contraction methods. Preliminary numerical results demonstrate that the projection and contraction methods converge twice faster than the extragradient method.     

A derivative-free three-term projection algorithm involving spectral quotient for solving nonlinear monotone equations

ABSTRACT

In this paper, we develop a three-term conjugate gradient method involving spectral quotient, which always satisfies the famous Dai-Liao conjugacy condition and quasi-Newton secant equation, independently of any line search. This new three-term conjugate gradient method can be regarded as a variant of the memoryless Broyden-Fletcher-Goldfarb-Shanno quasi-Newton method with regard to spectral quotient. By combining this method with the projection technique proposed by Solodov and Svaiter in 1998, we establish a derivative-free three-term projection algorithm for dealing with large-scale nonlinear monotone system of equations. We prove the global convergence of the algorithm and obtain the R-linear convergence rate under some mild conditions. Numerical results show that our projection algorithm is effective and robust, and is more competitive with the TTDFP algorithm proposed Liu and Li [A three-term derivative-free projection method for nonlinear monotone system of equations. Calcolo. 2016;53:427–450].     

Optimal Rates of Linear Convergence of Relaxed Alternating Projections and Generalized Douglas-Rachford Methods for Two Subspaces

We systematically study the optimal linear convergence rates for several relaxed alternating projection methods and the generalized Douglas-Rachford splitting methods for finding the projection on the intersection of two subspaces. Our analysis is based on a study on the linear convergence rates of the powers of matrices. We show that the optimal linear convergence rate of powers of matrices is attained if and only if all subdominant eigenvalues of the matrix are semisimple. For the convenience of computation, a nonlinear approach to the partially relaxed alternating projection method with at least the same optimal convergence rate is also provided. Numerical experiments validate our convergence analysis     

Radius of analyticity and exponential convergence for spectral projections of the generalized KdV equation

In this paper an exponential convergence rate for a spectral projection of the periodic initial-value problem for the generalized KdV equation is proved. Based on this convergence result, a method for determining the radius of analyticity of solutions of the generalized KdV equation is derived. Results from the new method and a similar method are compared.     

互补约束规划问题的一个广义梯度投影算法

本文研究了一类均衡约束最优化问题.利用广义梯度投影法,结合罚函数思想,得到了一个初始点可以任意的广义梯度投影算法.在较弱的条件下,证明了算法的全局收敛性.     

Extragradient method for solving quasivariational inequalities

We study methods for solving a class of the quasivariational inequalities in Hilbert space when the changeable set is described by translation of a fixed, closed and convex set. We consider one variant of the gradient-type projection method and an extragradient method. The possibilities of the choice of parameters of the gradient projection method in this case are wider than in the general case of a changeable set. The extragradient method on each iteration makes one trial step along the gradient, and the value of the gradient at the obtained point is used at the first point as the iteration direction. In the paper, we establish sufficient conditions for the convergence of the proposed methods and derive a new estimate of the rates of the convergence. The main result of this paper is contained in the convergence analysis of the extragradient method.     

解带线性或非线性约束最优化问题的三项记忆梯度Rosen投影算法

利用Rosen投影矩阵，建立求解带线性或非线性不等式约束优化问题的三项记忆梯度Rosen投影下降算法，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Rosen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。     

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．     

Strong Convergence Theorems for a Finite Family of Sequences of Nearly Nonexpansive Mappings in Hilbert Spaces

In this article, by using the hybrid projection method or the shrinking projection method, we introduce two strong convergence theorems for finding a common fixed point of a finite family of sequences of nearly nonexpansive mappings in Hilbert spaces.     

A CLASS OF TRUST REGION METHODS FOR LINEAR INEQUALITY CONSTRAINED OPTIMIZATION AND ITS THEORY ANALYSIS：I．ALGORITHM AND GLOBAL CONVERGENCE

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