Nonstandard Scaling Matrices for Trust Region Gauss–Newton Methods

1989 ◽  
Vol 10 (4) ◽  
pp. 654-670 ◽  
Author(s):  
Hubert Schwetlick ◽  
Volker Tiller
Keyword(s):  
Trust Region ◽  
Newton Methods

scholarly journals An inexact regularized Newton framework with a worst-case iteration complexity of $ {\mathscr O}(\varepsilon^{-3/2}) $ for nonconvex optimization

2018 ◽  
Vol 39 (3) ◽  
pp. 1296-1327 ◽  
Author(s):  
Frank E Curtis ◽  
Daniel P Robinson ◽  
Mohammadreza Samadi
Keyword(s):  
Nonconvex Optimization ◽  
Optimization Problems ◽  
Trust Region ◽  
Newton Methods ◽  
Positive Real ◽  
Worst Case ◽  
Nonconvex Optimization Problems ◽  
Trust Region Algorithm ◽  
Wide Range ◽  
Regularized Newton Methods

Abstract An algorithm for solving smooth nonconvex optimization problems is proposed that, in the worst-case, takes ${\mathscr O}(\varepsilon ^{-3/2})$ iterations to drive the norm of the gradient of the objective function below a prescribed positive real number $\varepsilon $ and can take ${\mathscr O}(\varepsilon ^{-3})$ iterations to drive the leftmost eigenvalue of the Hessian of the objective above $-\varepsilon $. The proposed algorithm is a general framework that covers a wide range of techniques including quadratically and cubically regularized Newton methods, such as the Adaptive Regularization using Cubics (arc) method and the recently proposed Trust-Region Algorithm with Contractions and Expansions (trace). The generality of our method is achieved through the introduction of generic conditions that each trial step is required to satisfy, which in particular allows for inexact regularized Newton steps to be used. These conditions center around a new subproblem that can be approximately solved to obtain trial steps that satisfy the conditions. A new instance of the framework, distinct from arc and trace, is described that may be viewed as a hybrid between quadratically and cubically regularized Newton methods. Numerical results demonstrate that our hybrid algorithm outperforms a cubically regularized Newton method.

Relations Between Several Path Following Algorithms and Local and Global Newton Methods

SIAM Review ◽  
1980 ◽  
Vol 22 (3) ◽  
pp. 263-274 ◽  
Author(s):  
C. B. Garcia ◽  
F. J. Gould
Keyword(s):  
Path Following ◽  
Newton Methods ◽  
Path Following Algorithms
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