scholarly journals A Broyden Class of Quasi-Newton Methods for Riemannian Optimization

2015 ◽  
Vol 25 (3) ◽  
pp. 1660-1685 ◽  
Author(s):  
Wen Huang ◽  
K. A. Gallivan ◽  
P.-A. Absil
Keyword(s):  
Newton Methods ◽  
Riemannian Optimization ◽  
Broyden Class ◽  
Quasi Newton

scholarly journals Exact linesearch limited-memory quasi-Newton methods for minimizing a quadratic function

2021 ◽  
Author(s):  
David Ek ◽  
Anders Forsgren
Keyword(s):  
Optimization Problems ◽  
Linear Equations ◽  
Quadratic Function ◽  
Compact Representation ◽  
Test Collection ◽  
Newton Methods ◽  
Limited Memory ◽  
Broyden Class ◽  
Systems Of Linear Equations ◽  
Quasi Newton

AbstractThe main focus in this paper is exact linesearch methods for minimizing a quadratic function whose Hessian is positive definite. We give a class of limited-memory quasi-Newton Hessian approximations which generate search directions parallel to those of the BFGS method, or equivalently, to those of the method of preconditioned conjugate gradients. In the setting of reduced Hessians, the class provides a dynamical framework for the construction of limited-memory quasi-Newton methods. These methods attain finite termination on quadratic optimization problems in exact arithmetic. We show performance of the methods within this framework in finite precision arithmetic by numerical simulations on sequences of related systems of linear equations, which originate from the CUTEst test collection. In addition, we give a compact representation of the Hessian approximations in the full Broyden class for the general unconstrained optimization problem. This representation consists of explicit matrices and gradients only as vector components.

scholarly journals Rates of superlinear convergence for classical quasi-Newton methods

2021 ◽  
Author(s):  
Anton Rodomanov ◽  
Yurii Nesterov
Keyword(s):  
Nonlinear Optimization ◽  
Superlinear Convergence ◽  
Local Convergence ◽  
Lipschitz Constant ◽  
Rates Of Convergence ◽  
Newton Methods ◽  
Long Time ◽  
Broyden Class ◽  
Quasi Newton ◽  
Convexity Parameter

AbstractWe study the local convergence of classical quasi-Newton methods for nonlinear optimization. Although it was well established a long time ago that asymptotically these methods converge superlinearly, the corresponding rates of convergence still remain unknown. In this paper, we address this problem. We obtain first explicit non-asymptotic rates of superlinear convergence for the standard quasi-Newton methods, which are based on the updating formulas from the convex Broyden class. In particular, for the well-known DFP and BFGS methods, we obtain the rates of the form $$(\frac{n L^2}{\mu ^2 k})^{k/2}$$ ( n L 2 μ 2 k ) k / 2 and $$(\frac{n L}{\mu k})^{k/2}$$ ( nL μ k ) k / 2 respectively, where k is the iteration counter, n is the dimension of the problem, $$\mu $$ μ is the strong convexity parameter, and L is the Lipschitz constant of the gradient.

Quasi-Newton methods for saddlepoints

1985 ◽  
Vol 47 (4) ◽  
pp. 393-399 ◽  
Author(s):  
F. Biegler-König
Keyword(s):  
Newton Methods ◽  
Quasi Newton

Inexact Newton and quasi-Newton methods for the output feedback pole assignment problem

2013 ◽  
Vol 33 (3) ◽  
pp. 517-542 ◽  
Author(s):  
El-Sayed M. E. Mostafa ◽  
Mohamed A. Tawhid ◽  
Eman R. Elwan
Keyword(s):  
Output Feedback ◽  
Assignment Problem ◽  
Pole Assignment ◽  
Newton Methods ◽  
Quasi Newton

scholarly journals Multi-step quasi-Newton methods for optimization

1994 ◽  
Vol 50 (1-3) ◽  
pp. 305-323 ◽  
Author(s):  
J.A. Ford ◽  
I.A. Moghrabi
Keyword(s):  
Newton Methods ◽  
Quasi Newton

Phase equilibrium calculations with quasi-Newton methods

2015 ◽  
Vol 406 ◽  
pp. 194-208 ◽  
Author(s):  
Dan Vladimir Nichita ◽  
Martin Petitfrere
Keyword(s):  
Phase Equilibrium ◽  
Newton Methods ◽  
Quasi Newton ◽  
Equilibrium Calculations

Maximum entropy derivation of quasi-Newton methods

2016 ◽  
Author(s):  
Steven H. Waldrip ◽  
Robert K. Niven
Keyword(s):  
Maximum Entropy ◽  
Newton Methods ◽  
Quasi Newton
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