An inexact regularized Newton framework with a worstcase 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 worstcase, 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 TrustRegion 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 02724979
 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 worstcase 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 worstcase iteration complexity of $ {\mathscr O}(\varepsilon^{3/2}) $ for nonconvex optimization. United Kingdom. doi:10.1093/imanum/dry022.
Curtis, Frank E., Robinson, Daniel P., and Samadi, Mohammadreza. Tue .
"An inexact regularized Newton framework with a worstcase iteration complexity of $ {\mathscr O}(\varepsilon^{3/2}) $ for nonconvex optimization". United Kingdom. doi:10.1093/imanum/dry022.
@article{osti_1436375,
title = {An inexact regularized Newton framework with a worstcase 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 worstcase, 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 TrustRegion 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 = {2018},
month = {5}
}
DOI: 10.1093/imanum/dry022
Works referenced in this record:
GALAHAD, a library of threadsafe Fortran 90 packages for largescale nonlinear optimization
journal, December 2003
 Gould, Nicholas I. M.; Orban, Dominique; Toint, Philippe L.
 ACM Transactions on Mathematical Software, Vol. 29, Issue 4
A trust region algorithm with a worstcase 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 12
Cubic regularization of Newton method and its global performance
journal, April 2006
 Nesterov, Yurii; Polyak, B. T.
 Mathematical Programming, Vol. 108, Issue 1
Worstcase evaluation complexity for unconstrained nonlinear optimization using highorder regularized models
journal, August 2016
 Birgin, E. G.; Gardenghi, J. L.; Martínez, J. M.
 Mathematical Programming, Vol. 163, Issue 12
ARC _{q} : a new adaptive regularization by cubics
journal, May 2017
 Dussault, JeanPierre
 Optimization Methods and Software, Vol. 33, Issue 2
On solving trustregion 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
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: worstcase function and derivativeevaluation complexity
journal, January 2010
 Cartis, Coralia; Gould, Nicholas I. M.; Toint, Philippe L.
 Mathematical Programming, Vol. 130, Issue 2