Stochastic derivativefree optimization using a trust region framework
Abstract
This study presents a trust region algorithm to minimize a function f when one has access only to noisecorrupted function values f¯. The modelbased algorithm dynamically adjusts its step length, taking larger steps when the model and function agree and smaller steps when the model is less accurate. The method does not require the user to specify a fixed pattern of points used to build local models and does not repeatedly sample points. If f is sufficiently smooth and the noise is independent and identically distributed with mean zero and finite variance, we prove that our algorithm produces iterates such that the corresponding function gradients converge in probability to zero. As a result, we present a prototype of our algorithm that, while simplistic in its management of previously evaluated points, solves benchmark problems in fewer function evaluations than do existing stochastic approximation methods.
 Authors:

 Argonne National Lab. (ANL), Lemont, IL (United States)
 Univ. of Colorado, Denver, CO (United States)
 Publication Date:
 Research Org.:
 Argonne National Lab. (ANL), Argonne, IL (United States)
 Sponsoring Org.:
 USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)
 OSTI Identifier:
 1366470
 Grant/Contract Number:
 AC0206CH11357
 Resource Type:
 Accepted Manuscript
 Journal Name:
 Computational Optimization and Applications
 Additional Journal Information:
 Journal Volume: 64; Journal Issue: 3; Journal ID: ISSN 09266003
 Publisher:
 Springer
 Country of Publication:
 United States
 Language:
 English
 Subject:
 97 MATHEMATICS AND COMPUTING; derivativefree optimization; modelbased trust region methods; stochastic optimization
Citation Formats
Larson, Jeffrey, and Billups, Stephen C. Stochastic derivativefree optimization using a trust region framework. United States: N. p., 2016.
Web. doi:10.1007/s105890169827z.
Larson, Jeffrey, & Billups, Stephen C. Stochastic derivativefree optimization using a trust region framework. United States. doi:10.1007/s105890169827z.
Larson, Jeffrey, and Billups, Stephen C. Wed .
"Stochastic derivativefree optimization using a trust region framework". United States. doi:10.1007/s105890169827z. https://www.osti.gov/servlets/purl/1366470.
@article{osti_1366470,
title = {Stochastic derivativefree optimization using a trust region framework},
author = {Larson, Jeffrey and Billups, Stephen C.},
abstractNote = {This study presents a trust region algorithm to minimize a function f when one has access only to noisecorrupted function values f¯. The modelbased algorithm dynamically adjusts its step length, taking larger steps when the model and function agree and smaller steps when the model is less accurate. The method does not require the user to specify a fixed pattern of points used to build local models and does not repeatedly sample points. If f is sufficiently smooth and the noise is independent and identically distributed with mean zero and finite variance, we prove that our algorithm produces iterates such that the corresponding function gradients converge in probability to zero. As a result, we present a prototype of our algorithm that, while simplistic in its management of previously evaluated points, solves benchmark problems in fewer function evaluations than do existing stochastic approximation methods.},
doi = {10.1007/s105890169827z},
journal = {Computational Optimization and Applications},
number = 3,
volume = 64,
place = {United States},
year = {2016},
month = {2}
}
Web of Science
Works referenced in this record:
Estimating Derivatives of Noisy Simulations
journal, April 2012
 Moré, Jorge J.; Wild, Stefan M.
 ACM Transactions on Mathematical Software, Vol. 38, Issue 3
UOBYQA: unconstrained optimization by quadratic approximation
journal, May 2002
 Powell, M. J. D.
 Mathematical Programming, Vol. 92, Issue 3
Stochastic TrustRegion ResponseSurface Method (STRONG)—A New ResponseSurface Framework for Simulation Optimization
journal, May 2013
 Chang, KuoHao; Hong, L. Jeff; Wan, Hong
 INFORMS Journal on Computing, Vol. 25, Issue 2
Introduction to DerivativeFree Optimization
book, January 2009
 Conn, Andrew R.; Scheinberg, Katya; Vicente, Luis N.
Lipschitzian optimization without the Lipschitz constant
journal, October 1993
 Jones, D. R.; Perttunen, C. D.; Stuckman, B. E.
 Journal of Optimization Theory and Applications, Vol. 79, Issue 1
An adaptive Monte Carlo algorithm for computing mixed logit estimators
journal, January 2006
 Bastin, Fabian; Cirillo, Cinzia; Toint, Philippe L.
 Computational Management Science, Vol. 3, Issue 1
DerivativeFree Optimization of Expensive Functions with Computational Error Using Weighted Regression
journal, January 2013
 Billups, Stephen C.; Larson, Jeffrey; Graf, Peter
 SIAM Journal on Optimization, Vol. 23, Issue 1
Extension of the direct optimization algorithm for noisy functions
conference, December 2007
 Deng, Geng; Ferris, Michael C.
 2007 Winter Simulation Conference
A Simplex Method for Function Minimization
journal, January 1965
 Nelder, J. A.; Mead, R.
 The Computer Journal, Vol. 7, Issue 4
Multivariate stochastic approximation using a simultaneous perturbation gradient approximation
journal, March 1992
 Spall, J. C.
 IEEE Transactions on Automatic Control, Vol. 37, Issue 3
Estimating Computational Noise
journal, January 2011
 Moré, Jorge J.; Wild, Stefan M.
 SIAM Journal on Scientific Computing, Vol. 33, Issue 3
The effect of deterministic noise in subgradient methods
journal, January 2009
 Nedić, Angelia; Bertsekas, Dimitri P.
 Mathematical Programming, Vol. 125, Issue 1
Adaptation of the Uobyqa Algorithm for Noisy Functions
conference, December 2006
 Deng, Geng; Ferris, Michael
 Proceedings of the 2006 Winter Simulation Conference
On sampling controlled stochastic approximation
journal, January 1991
 Dupuis, P.; Simha, R.
 IEEE Transactions on Automatic Control, Vol. 36, Issue 8
Benchmarking DerivativeFree Optimization Algorithms
journal, January 2009
 Moré, Jorge J.; Wild, Stefan M.
 SIAM Journal on Optimization, Vol. 20, Issue 1
Stochastic Estimation of the Maximum of a Regression Function
journal, September 1952
 Kiefer, J.; Wolfowitz, J.
 The Annals of Mathematical Statistics, Vol. 23, Issue 3
Benchmarking optimization software with performance profiles
journal, January 2002
 Dolan, Elizabeth D.; Moré, Jorge J.
 Mathematical Programming, Vol. 91, Issue 2
Convergence of TrustRegion Methods Based on Probabilistic Models
journal, January 2014
 Bandeira, A. S.; Scheinberg, K.; Vicente, L. N.
 SIAM Journal on Optimization, Vol. 24, Issue 3
Sample size selection for improved NelderMead performance
conference, January 1995
 Tomick, J. J.; Arnold, S. F.; Barton, R. R.
 1995 Winter Simulation Conference, Winter Simulation Conference Proceedings, 1995.
Response Surface Methodology: A Retrospective and Literature Survey
journal, January 2004
 Myers, Raymond H.; Montgomery, Douglas C.; Vining, G. Geoffrey
 Journal of Quality Technology, Vol. 36, Issue 1
Works referencing / citing this record:
Newtontype methods for nonconvex optimization under inexact Hessian information
journal, May 2019
 Xu, Peng; Roosta, Fred; Mahoney, Michael W.
 Mathematical Programming, Vol. 184, Issue 12
Robust optimization of noisy blackbox problems using the Mesh Adaptive Direct Search algorithm
journal, January 2018
 Audet, Charles; Ihaddadene, Amina; Le Digabel, Sébastien
 Optimization Letters, Vol. 12, Issue 4