Optimal Conditioning and Convergence in Rank One Quasi-Newton Updates

1988 ◽  
Vol 25 (1) ◽  
pp. 206-221 ◽  
Author(s):  
Chi Ming Ip ◽  
Michael J. Todd
Keyword(s):  
Optimal Conditioning ◽  
Rank One ◽  
Quasi Newton

scholarly journals Convergence of Symmetric Rank-One method based on Modified Quasi-Newton equation

2010 ◽  
Vol 2 (3) ◽  
Author(s):  
Farzin Modarres Khiyabani ◽  
Malik Abu Hassan ◽  
Wah June Leong
Keyword(s):  
Newton Equation ◽  
Rank One ◽  
Symmetric Rank One ◽  
Quasi Newton

Descent Symmetrization of the Dai–Liao Conjugate Gradient Method

2016 ◽  
Vol 33 (02) ◽  
pp. 1650008 ◽  
Author(s):  
Saman Babaie-Kafaki ◽  
Reza Ghanbari
Keyword(s):  
Conjugate Gradient ◽  
Search Direction ◽  
Test Problems ◽  
Uniformly Convex ◽  
Objective Functions ◽  
Descent Property ◽  
Rank One ◽  
Nonlinear Conjugate Gradient ◽  
Well Conditioning ◽  
Quasi Newton

Symmetrizing the Dai–Liao (DL) search direction matrix by a rank-one modification, we propose a one-parameter class of nonlinear conjugate gradient (CG) methods which includes the memoryless Broyden–Fletcher–Goldfarb–Shanno (MLBFGS) quasi-Newton updating formula. Then, conducting an eigenvalue analysis, we suggest two choices for the parameter of the proposed class of CG methods which simultaneously guarantee the descent property and well-conditioning of the search direction matrix. A global convergence analysis is made for uniformly convex objective functions. Computational experiments are done on a set of unconstrained optimization test problems of the CUTEr collection. Results of numerical comparisons made by the Dolan–Moré performance profile show that proper choices for the mentioned parameter may lead to promising computational performances.

scholarly journals A Symmetric Rank-One Quasi-Newton Method for Nonnegative Matrix Factorization

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Shu-Zhen Lai ◽  
Hou-Biao Li ◽  
Zu-Tao Zhang
Keyword(s):  
Matrix Factorization ◽  
Nonnegative Matrix Factorization ◽  
Hessian Matrix ◽  
Synthetic Data ◽  
Nonnegative Matrix ◽  
Matrix Approximation ◽  
Active Set ◽  
Rank One ◽  
Symmetric Rank One ◽  
Quasi Newton

As is well known, the nonnegative matrix factorization (NMF) is a dimension reduction method that has been widely used in image processing, text compressing, signal processing, and so forth. In this paper, an algorithm on nonnegative matrix approximation is proposed. This method is mainly based on a relaxed active set and the quasi-Newton type algorithm, by using the symmetric rank-one and negative curvature direction technologies to approximate the Hessian matrix. The method improves some recent results. In addition, some numerical experiments are presented in the synthetic data, imaging processing, and text clustering. By comparing with the other six nonnegative matrix approximation methods, this method is more robust in almost all cases.

scholarly journals Optimal conditioning of quasi-Newton methods

1970 ◽  
Vol 24 (111) ◽  
pp. 657-657 ◽  
Author(s):  
D. F. Shanno ◽  
P. C. Kettler
Keyword(s):  
Newton Methods ◽  
Optimal Conditioning ◽  
Quasi Newton

Convergence of quasi-Newton matrices generated by the symmetric rank one update

1991 ◽  
Vol 50 (1-3) ◽  
pp. 177-195 ◽  
Author(s):  
A. R. Conn ◽  
N. I. M. Gould ◽  
Ph. L. Toint
Keyword(s):  
Rank One ◽  
Symmetric Rank One ◽  
Quasi Newton
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