An inexact regularized Newton framework with a worst-case iteration complexity of $$ {\mathscr O}(\varepsilon^{-3/2}) $$ for nonconvex optimization
Abstract
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.
- Authors:
-
- Department of Industrial and Systems Engineering, Lehigh University
- Department of Applied Mathematics and Statistics, Johns Hopkins University
- Publication Date:
- Sponsoring Org.:
- USDOE
- OSTI Identifier:
- 1436375
- Grant/Contract Number:
- DE–SC0010615
- Resource Type:
- Published Article
- Journal Name:
- IMA Journal of Numerical Analysis
- Additional Journal Information:
- Journal Name: IMA Journal of Numerical Analysis Journal Volume: 39 Journal Issue: 3; Journal ID: ISSN 0272-4979
- Publisher:
- Oxford University Press
- Country of Publication:
- United Kingdom
- Language:
- English
Citation Formats
Curtis, Frank E., Robinson, Daniel P., and Samadi, Mohammadreza. An inexact regularized Newton framework with a worst-case iteration complexity of $ {\mathscr O}(\varepsilon^{-3/2}) $ for nonconvex optimization. United Kingdom: N. p., 2018.
Web. doi:10.1093/imanum/dry022.
Curtis, Frank E., Robinson, Daniel P., & Samadi, Mohammadreza. An inexact regularized Newton framework with a worst-case iteration complexity of $ {\mathscr O}(\varepsilon^{-3/2}) $ for nonconvex optimization. United Kingdom. https://doi.org/10.1093/imanum/dry022
Curtis, Frank E., Robinson, Daniel P., and Samadi, Mohammadreza. Tue .
"An inexact regularized Newton framework with a worst-case iteration complexity of $ {\mathscr O}(\varepsilon^{-3/2}) $ for nonconvex optimization". United Kingdom. https://doi.org/10.1093/imanum/dry022.
@article{osti_1436375,
title = {An inexact regularized Newton framework with a worst-case iteration complexity of $ {\mathscr O}(\varepsilon^{-3/2}) $ for nonconvex optimization},
author = {Curtis, Frank E. and Robinson, Daniel P. and Samadi, Mohammadreza},
abstractNote = {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.},
doi = {10.1093/imanum/dry022},
journal = {IMA Journal of Numerical Analysis},
number = 3,
volume = 39,
place = {United Kingdom},
year = {Tue May 08 00:00:00 EDT 2018},
month = {Tue May 08 00:00:00 EDT 2018}
}
https://doi.org/10.1093/imanum/dry022
Web of Science
Works referenced in this record:
GALAHAD, a library of thread-safe Fortran 90 packages for large-scale nonlinear optimization
journal, December 2003
- Gould, Nicholas I. M.; Orban, Dominique; Toint, Philippe L.
- ACM Transactions on Mathematical Software, Vol. 29, Issue 4
Trust Region Methods
book, January 2000
- Conn, Andrew R.; Gould, Nicholas I. M.; Toint, Philippe L.
- MOS-SIAM Series on Optimization
A trust region algorithm with a worst-case iteration complexity of $$\mathcal{O}(\epsilon ^{-3/2})$$ O ( ϵ - 3 / 2 ) for nonconvex optimization
journal, May 2016
- Curtis, Frank E.; Robinson, Daniel P.; Samadi, Mohammadreza
- Mathematical Programming, Vol. 162, Issue 1-2
Cubic regularization of Newton method and its global performance
journal, April 2006
- Nesterov, Yurii; Polyak, B. T.
- Mathematical Programming, Vol. 108, Issue 1
Worst-case evaluation complexity for unconstrained nonlinear optimization using high-order regularized models
journal, August 2016
- Birgin, E. G.; Gardenghi, J. L.; Martínez, J. M.
- Mathematical Programming, Vol. 163, Issue 1-2
ARC q : a new adaptive regularization by cubics
journal, May 2017
- Dussault, Jean-Pierre
- Optimization Methods and Software, Vol. 33, Issue 2
On solving trust-region and other regularised subproblems in optimization
journal, February 2010
- Gould, Nicholas I. M.; Robinson, Daniel P.; Thorne, H. Sue
- Mathematical Programming Computation, Vol. 2, Issue 1
Introductory Lectures on Convex Optimization
book, January 2004
- Nesterov, Yurii
- Applied Optimization
Adaptive cubic regularisation methods for unconstrained optimization. Part I: motivation, convergence and numerical results
journal, May 2009
- Cartis, Coralia; Gould, Nicholas I. M.; Toint, Philippe L.
- Mathematical Programming, Vol. 127, Issue 2
The Use of Quadratic Regularization with a Cubic Descent Condition for Unconstrained Optimization
journal, January 2017
- Birgin, E. G.; Martínez, J. M.
- SIAM Journal on Optimization, Vol. 27, Issue 2
On the Complexity of Steepest Descent, Newton's and Regularized Newton's Methods for Nonconvex Unconstrained Optimization Problems
journal, January 2010
- Cartis, C.; Gould, N. I. M.; Toint, Ph. L.
- SIAM Journal on Optimization, Vol. 20, Issue 6
Updating the regularization parameter in the adaptive cubic regularization algorithm
journal, December 2011
- Gould, N. I. M.; Porcelli, M.; Toint, P. L.
- Computational Optimization and Applications, Vol. 53, Issue 1
Benchmarking optimization software with performance profiles
journal, January 2002
- Dolan, Elizabeth D.; Moré, Jorge J.
- Mathematical Programming, Vol. 91, Issue 2
Adaptive cubic regularisation methods for unconstrained optimization. Part II: worst-case function- and derivative-evaluation complexity
journal, January 2010
- Cartis, Coralia; Gould, Nicholas I. M.; Toint, Philippe L.
- Mathematical Programming, Vol. 130, Issue 2