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

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.

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

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

Optimization With Discrete Variables Via Recursive Quadratic Programming: Part 2—Algorithm and Results

1989 ◽  
Vol 111 (1) ◽  
pp. 130-136 ◽  
Author(s):  
J. Z. Cha ◽  
R. W. Mayne
Keyword(s):  
Quadratic Programming ◽  
Performance Comparison ◽  
Programming Algorithm ◽  
Discrete Variables ◽  
Contour Analysis ◽  
Recursive Quadratic Programming ◽  
Analysis Technique ◽  
Rank One ◽  
Symmetric Rank One ◽  
Monotonicity Analysis

A discrete recursive quadratic programming algorithm is developed for a class of mixed discrete constrained nonlinear programming (MDCNP) problems. The symmetric rank one (SR1) Hessian update formula is used to generate second order information. Also, strategies, such as the watchdog technique (WT), the monotonicity analysis technique (MA), the contour analysis technique (CA), and the restoration of feasibility have been considered. Heuristic aspects of handling discrete variables are treated via the concepts and convergence discussions of Part I. This paper summarizes the details of the algorithm and its implementation. Test results for 25 different problems are presented to allow evaluation of the approach and provide a basis for performance comparison. The results show that the suggested method is a promising one, efficient and robust for the MDCNP problem.

A new three-term spectral conjugate gradient algorithm with higher numerical performance for solving large scale optimization problems based on Quasi-Newton equation

2021 ◽  
pp. 2150053
Author(s):  
Jie Guo ◽  
Zhong Wan
Keyword(s):  
Conjugate Gradient ◽  
Large Scale ◽  
Line Search ◽  
Optimization Problems ◽  
Gradient Algorithm ◽  
Large Scale Optimization ◽  
Newton Equation ◽  
Numerical Performance ◽  
Scale Optimization ◽  
Quasi Newton

A new spectral three-term conjugate gradient algorithm in virtue of the Quasi-Newton equation is developed for solving large-scale unconstrained optimization problems. It is proved that the search directions in this algorithm always satisfy a sufficiently descent condition independent of any line search. Global convergence is established for general objective functions if the strong Wolfe line search is used. Numerical experiments are employed to show its high numerical performance in solving large-scale optimization problems. Particularly, the developed algorithm is implemented to solve the 100 benchmark test problems from CUTE with different sizes from 1000 to 10,000, in comparison with some similar ones in the literature. The numerical results demonstrate that our algorithm outperforms the state-of-the-art ones in terms of less CPU time, less number of iteration or less number of function evaluation.

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