scholarly journals Convergence analysis of sparse quasi-Newton updates with positive definite matrix completion for two-dimensional functions

2011 ◽  
Vol 1 (1) ◽  
pp. 61-69 ◽  
Author(s):  
Yuhong Dai ◽  
◽  
Nobuo Yamashita ◽  
Keyword(s):  
Convergence Analysis ◽  
Matrix Completion ◽  
Positive Definite Matrix ◽  
Positive Definite ◽  
Two Dimensional ◽  
Quasi Newton

scholarly journals A New Sparse Quasi-Newton Update Method

2012 ◽  
Vol 16 ◽  
pp. 30
Author(s):  
Minghou Cheng ◽  
Yu-Hong Dai ◽  
Rui Diao
Keyword(s):  
Optimization Problems ◽  
Matrix Completion ◽  
Positive Definite Matrix ◽  
Positive Definite ◽  
Test Problems ◽  
Unconstrained Optimization Problems ◽  
Secant Condition ◽  
Sparsity Structure ◽  
Quasi Newton ◽  
Local And Superlinear Convergence

Based on the idea of maximum determinant positive definite matrix completion, Yamashita proposed a sparse quasi-Newton update, called MCQN, for unconstrained optimization problems with sparse Hessian structures. Such an MCQN update keeps the sparsity structure of the Hessian while relaxing the secant condition. In this paper, we propose an alternative to the MCQN update, in which the quasi-Newton matrix satisfies the secant condition, but does not have the same sparsity structure as the Hessian in general. Our numerical results demonstrate the usefulness of the new MCQN update with the BFGS formula for a collection of test problems. A local and superlinear convergence analysis is also provided for the new MCQN update with the DFP formula.  

scholarly journals A self-correcting variable-metric algorithm framework for nonsmooth optimization

2019 ◽  
Vol 40 (2) ◽  
pp. 1154-1187 ◽  
Author(s):  
Frank E Curtis ◽  
Daniel P Robinson ◽  
Baoyu Zhou
Keyword(s):  
Positive Definite Matrix ◽  
Positive Definite ◽  
Symmetric Positive Definite Matrix ◽  
Variable Metric ◽  
Symmetric Positive Definite ◽  
Nonsmooth Functions ◽  
Variable Metric Algorithms ◽  
Algorithm Framework ◽  
Newton Scheme ◽  
Quasi Newton

Abstract An algorithm framework is proposed for minimizing nonsmooth functions. The framework is variable metric in that, in each iteration, a step is computed using a symmetric positive-definite matrix whose value is updated as in a quasi-Newton scheme. However, unlike previously proposed variable-metric algorithms for minimizing nonsmooth functions, the framework exploits self-correcting properties made possible through Broyden–Fletcher–Goldfarb–Shanno-type updating. In so doing, the framework does not overly restrict the manner in which the step computation matrices are updated, yet the scheme is controlled well enough that global convergence guarantees can be established. The results of numerical experiments for a few algorithms are presented to demonstrate the self-correcting behaviours that are guaranteed by the framework.

scholarly journals Active Positive-Definite Matrix Completion

2017 ◽  
pp. 264-272 ◽  
Author(s):  
Charalampos Mavroforakis ◽  
Dóra Erdös ◽  
Mark Crovella ◽  
Evimaria Terzi
Keyword(s):  
Matrix Completion ◽  
Positive Definite Matrix ◽  
Positive Definite
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