一种新的自适应非单调牛顿算法（英文）

本文针对无约束优化问题,提出一种新的自适应非单调线搜索技术.基于新的非单调线搜索技术,提出一种自适应非单调牛顿算法.在适当的假设下,证明了新的算法的全局收敛性.数值结果表明了该算法的可行性和有效性.     

一类基于Wolfe线搜索下的谱共轭梯度法

在已有文献β■的基础上得到了一个新的谱共轭参数,从而构造了一个新的谱共轭梯度法.并且新方法的搜索方向不需要任何线性搜索条件而自动下降.利用标准Wolfe线搜索,在一般假设条件下,验证了该方法是全局收敛的.     

一个带重启步的改进PRP型谱共轭梯度法

Polak-Ribière-Polak (PRP)方法是经典共轭梯度法中数值表现较好的方法之一.结合Wolfe非精确线搜索准则对PRP公式进行改进,从而产生新的共轭参数,并基于新共轭参数设计新的谱参数,引入重启条件并构造新的重启方向,进而建立一个带重启步的谱共轭梯度算法.在常规假设及强Wolfe非精确线搜索步长准则下,...     

一类新的共轭梯度法的全局收敛性

本文提出了一类与HS方法相关的新的共轭梯度法.在强Wolfe线搜索的条件下,该方法能够保证搜索方向的充分下降性,并且在不需要假设目标函数为凸的情况下,证明了该方法的全局收敛性.同时,给出了这类新共轭梯度法的一种特殊形式,通过调整参数ρ,验证了它对给定测试函数的有效性.     

两类下降的非线性共轭梯度法的全局收敛性

本文在文献[1]中提出了一类新共轭梯度法的基础上,给出求解无约束优化问题的两类新的非线性下降共轭梯度法,此两类方法在无任何线搜索下,能够保证在每次迭代中产生下降方向.对一般非凸函数,我们在Wolfe线搜索条件下证明了两类新方法的全局收敛性.     

解非线性互补问题的非单调可行SQP方法

非线性互补问题可以转化成非线性约束优化问题.提出一种非单调线搜索的可行SQP方法.利用QP子问题的K-T点得到一个可行下降方向,通过引入一个高阶校正步以克服Maratos效应.同时,算法采用非单调线搜索技巧获得搜索步长.证明全局收敛性时不需要严格互补条件,最后给出数值试验.     

一类新的非单调记忆梯度法及其全局收敛性

在非单调Armijo线搜索的基础上提出一种新的非单调线搜索,研究了一类在该线搜索下的记忆梯度法,在较弱条件下证明了其全局收敛性。与非单调Armijo线搜索相比,新的非单调线搜索在每次迭代时可以产生更大的步长,从而使目标函数值充分下降,降低算法的计算量。     

一类带线搜索的非单调信赖域新算法

结合非单调信赖域方法,和非单调线搜索技术,提出了一种新的无约束优化算法.信赖域方法的每一步采用线搜索,使得迭代每一步都充分下降加快了迭代速度.在一定条件下,证明了算法具有全局收敛性和局部超线性.收敛速度.数值试验表明算法是十分有效的.     

Wolfe线搜索下新的共轭梯度法的全局收敛性

共轭梯度法是求解无约束优化问题的一种重要的方法.本文提出一族新的共轭梯度法,证明了其在推广的Wolfe非精确线搜索条件下具有全局收敛性.最后对算法进行了数值实验,实验结果验证了该算法的有效性.     

一类非单调保守BFGS算法研究

基于非单调线搜索在寻求优化问题最优解中的优越性,提出了一类新的非单调保守BFGS算法.同已有方法不同,该算法中用来控制非单调性程度的算法参数不是取固定值,而是利用已有目标函数和梯度函数的信息自动调整其取值,以改善算法的数值表现.在合适的假设条件下,建立了新的非单调保守BFGS算法的全局收敛性.用基准测试优化问题测试了算...     

基于凸组合共轭梯度法的ARIMA模型参数估计

提出了一种凸组合共轭梯度算法,并将其算法应用到ARIMA模型参数估计中.新算法由改进的谱共轭梯度算法与共轭梯度算法作凸组合构造而成,具有下述特性:1)具备共轭性条件;2)自动满足充分下降性.证明了在标准Wolfe线搜索下新算法具备完全收敛性,最后数值实验表明通过调节凸组合参数,新算法更加快速有效,通过具体实例证实了模型的显著拟合效果.     

An Improved Spectral Conjugate Gradient Algorithm for Nonconvex Unconstrained Optimization Problems

In this paper, an improved spectral conjugate gradient algorithm is developed for solving nonconvex unconstrained optimization problems. Different from the existent methods, the spectral and conjugate parameters are chosen such that the obtained search direction is always sufficiently descent as well as being close to the quasi-Newton direction. With these suitable choices, the additional assumption in the method proposed by Andrei on the boundedness of the spectral parameter is removed. Under some mild conditions, global convergence is established. Numerical experiments are employed to demonstrate the efficiency of the algorithm for solving large-scale benchmark test problems, particularly in comparison with the existent state-of-the-art algorithms available in the literature.     

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.     

Scaled conjugate gradient algorithms for unconstrained optimization

In this work we present and analyze a new scaled conjugate gradient algorithm and its implementation, based on an interpretation of the secant equation and on the inexact Wolfe line search conditions. The best spectral conjugate gradient algorithm SCG by Birgin and Martínez (2001), which is mainly a scaled variant of Perry’s (1977), is modified in such a manner to overcome the lack of positive definiteness of the matrix defining the search direction. This modification is based on the quasi-Newton BFGS updating formula. The computational scheme is embedded in the restart philosophy of Beale–Powell. The parameter scaling the gradient is selected as spectral gradient or in an anticipative manner by means of a formula using the function values in two successive points. In very mild conditions it is shown that, for strongly convex functions, the algorithm is global convergent. Preliminary computational results, for a set consisting of 500 unconstrained optimization test problems, show that this new scaled conjugate gradient algorithm substantially outperforms the spectral conjugate gradient SCG algorithm. The author was awarded the Romanian Academy Grant 168/2003.     

Solving nonlinear equations with the convex combination of two positive spectral coefficients

We present a spectral algorithm based on the convex combination of two modified spectral coefficients for solving systems of nonlinear equations. The proposed algorithm does not require the exact or approximated directional derivative for its implementation. By employing a derivative-free line search, the global convergence of the sequence generated by the algorithm is supported. Numerical experiments are given to demonstrate the performance of the algorithm compared with a similar algorithm in the literature for solving nonlinear equations problems.     

求解无约束优化问题的新谱共轭梯度法及其收敛性

强Wolfe条件不能保证标准CD共轭梯度法全局收敛．本文通过建立新的共轭参数,提出无约束优化问题的一个新谱共轭梯度法,该方法在精确线搜索下与标准CD共轭梯度法等价,在标准wolfe线搜索下具有下降性和全局收敛性．初步的数值实验结果表明新方法是有效的,适合于求解非线性无约束优化问题．     

A new regularized quasi-Newton method for unconstrained optimization

In this paper, we propose a new regularized quasi-Newton method for unconstrained optimization. At each iteration, a regularized quasi-Newton equation is solved to obtain the search direction. The step size is determined by a non-monotone Armijo backtracking line search. An adaptive regularized parameter, which is updated according to the step size of the line search, is employed to compute the next search direction. The presented method is proved to be globally convergent. Numerical experiments show that the proposed method is effective for unconstrained optimizations and outperforms the existing regularized Newton method.     

A modified Newton’s method for best rank-one approximation to tensors

In this paper, a modified Newton’s method for the best rank-one approximation problem to tensor is proposed. We combine the iterative matrix of Jacobi-Gauss-Newton (JGN) algorithm or Alternating Least Squares (ALS) algorithm with the iterative matrix of GRQ-Newton method, and present a modified version of GRQ-Newton algorithm. A line search along the projective direction is employed to obtain the global convergence. Preliminary numerical experiments and numerical comparison show that our algorithm is efficient.     

Foundations of a Multi-way Spectral Clustering Framework for Hybrid Linear Modeling

The problem of Hybrid Linear Modeling (HLM) is to model and segment data using a mixture of affine subspaces. Different strategies have been proposed to solve this problem, however, rigorous analysis justifying their performance is missing. This paper suggests the Theoretical Spectral Curvature Clustering (TSCC) algorithm for solving the HLM problem and provides careful analysis to justify it. The TSCC algorithm is practically a combination of Govindu’s multi-way spectral clustering framework (CVPR 2005) and Ng et al.’s spectral clustering algorithm (NIPS 2001). The main result of this paper states that if the given data is sampled from a mixture of distributions concentrated around affine subspaces, then with high sampling probability the TSCC algorithm segments well the different underlying clusters. The goodness of clustering depends on the within-cluster errors, the between-clusters interaction, and a tuning parameter applied by TSCC. The proof also provides new insights for the analysis of Ng et al. (NIPS 2001). This work was supported by NSF grant #0612608.