A derivative-free trust-region algorithm for unconstrained optimization with controlled error

Previous Article
A nonconvergent example for the iterative water-filling algorithm
NACO Home
This Issue
Next Article
Filter-based genetic algorithm for mixed variable programming

2011, 1(1): 117-145. doi: 10.3934/naco.2011.1.117

A derivative-free trust-region algorithm for unconstrained optimization with controlled error

Jun Takaki ^1, and Nobuo Yamashita ^2,

1.	Graduate School of Informatics, Kyoto University, Yoshida-honmachi, Sakyo-ku, Kyoto, 606-8501,, Japan
2.	Graduate School of Informatics, Kyoto University, Yoshida-honmachi, Sakyo-ku, Kyoto, 606-8501

Received September 2010 Revised December 2010 Published February 2011

Abstract
Related Papers

In this paper, we consider the unconstrained optimization problem under the following conditions: (S1) The objective function is evaluated with a certain bounded error, (S2) the error is controllable, that is, the objective function can be evaluated to any desired accuracy, and (S3) more accurate evaluation requires a greater computation time. This situation arises in many fields such as engineering and financial problems, where objective function values are obtained from considerable numerical calculation or a simulation. Under (S2) and (S3), it seems reasonable to set the accuracy of the evaluation to be low at points far from a solution, and high at points in the neighborhood of a solution. In this paper, we propose a derivative-free trust-region algorithm based on this idea. For this purpose, we consider (i) how to construct a quadratic model function by exploiting pointwise errors and (ii) how to control the accuracy of function evaluations to reduce the total computation time of the algorithm. For (i), we propose a method based on support vector regression. For (ii), we present an updating formula of the accuracy of which is related to the trust-region radius. We present numerical results for several test problems taken from CUTEr and a financial problem of estimating implied volatilities from option prices.

Keywords: controlled error., trust region algorithm, Derivative-free method.

Mathematics Subject Classification: Primary: 90C56; Secondary: 90C3.

Citation: Jun Takaki, Nobuo Yamashita. A derivative-free trust-region algorithm for unconstrained optimization with controlled error. Numerical Algebra, Control and Optimization, 2011, 1 (1) : 117-145. doi: 10.3934/naco.2011.1.117

References:

[1]	I. Bongartz, A. R. Conn, N. I. M. Gould and Ph. L. Toint, CUTE: Constrained and unconstrained testing environment, ACM Transactions on Mathematical Software, 21 (1995), 123-160. doi: 10.1145/200979.201043.
[2]	A. J. Booker, J. E. Dennis, P. D. Frank, D. B. Serafini, V. Torczon and M. W. Trosset, A rigorous framework for optimization of expensive functions by surrogates, Structural and Multidisciplinary Optimization, 17 (1999), 1-13.
[3]	T. D. Choi and C. T. Kelley, Superlinear Convergence and Implicit Filtering, SIAM Journal on Optimization, 10 (2000), 1149-1162. doi: 10.1137/S1052623499354096.
[4]	B. Colson, P. Marcotte and G. Savard, Bilevel programming: A survey, 4OR: A Quarterly Journal of Operations Research, 3 (2005), 87-107.
[5]	A. R. Conn, N. I. M. Gould and Ph. L. Toint, "Trust-Region Methods,'' SIAM, Philadelphia, 2000. doi: 10.1137/1.9780898719857.
[6]	A. R. Conn and Ph. L. Toint, An algorithm using quadratic interpolation for unconstrained derivative free optimization, in "Nonlinear Optimization and Applications,'' Plenum Publishing, (1996), 27-47.
[7]	A. R. Conn, K. Scheinberg and Ph. L. Toint, A derivative free optimization algorithm in practice, the American Institute of Aeronautics and Astronautics Conference, St Louis, 1998.
[8]	A. R. Conn, K. Scheinberg and Ph. L. Toint, A derivative free optimization method via support vector machines, 1999. Available from: www.cas.mcmaster.ca/ oplab/seminar/conn/conn_deriv_free.ps
[9]	A. R. Conn, K. Scheinberg and Ph. L. Toint, On the convergence of derivative-free methods for unconstrained optimization, in "Approximation Theory and Optimization,'' Cambridge University Press, (1997), 83-108.
[10]	A. R. Conn, K. Scheinberg and L. N. Vicente, Geometry of interpolation sets in derivative free optimization, Mathematical Programming, 111 (2008), 141-172. doi: 10.1007/s10107-006-0073-5.
[11]	A. R. Conn, K. Scheinberg and L. N. Vicente, Geometry of sample sets in derivative free optimization: Polynomial regression and underdetermined interpolation, IMA Journal of Numerical Analysis, 28 (2008), 721-748. doi: 10.1093/imanum/drn046.
[12]	C. Cox and M. Rubinstein, "Option Markets,'' Prentice-Hall, 1985.
[13]	N. Cristianini and J. Shawe-Taylor, "An Introduction to Support Vector Machines and Other Kernel-based Methods,'' Cambridge University Press, Cambridge, UK, 2000.
[14]	J. E. Dennis and V. Torczon, Direct search methods on parallel machines, SIAM Journal on Optimization, 1 (1991), 448-474. doi: 10.1137/0801027.
[15]	R. A. Fisher, "The Design of Experiments,'' Oliver and Boyd Ltd., 1951.
[16]	P. Gilmore and C. T. Kelley, An implicit filtering algorithm for optimization of functions with many local minima, SIAM Journal on Optimization, 5 (1995), 269-285. doi: 10.1137/0805015.
[17]	B. Karasözen, Survey of trust-region derivative free optimization methods, Journal of Industrial and Management Optimization, 3 (2007), 321-334.
[18]	T. G. Kolda, R. M. Lewis and V. Torzcon, Optimization by direct search: new perspectives of some classical and modern methods, SIAM Review, 45 (2003), 385-482. doi: 10.1137/S003614450242889.
[19]	J. A. Nelder and R. Mead, A simplex method for function minimization, Computer Journal, 7 (1965), 308-313.
[20]	J. Nocedal and S. J. Wright, "Numerical Optimization,'' Springer-Verlag, New York, 1999. doi: 10.1007/b98874.
[21]	M. J. D. Powell, Trust region methods that employ quadratic interpolation to the objective function, Presentation at the 5th SIAM Conference on Optimization, 1996.
[22]	M. J. D. Powell, UOBYQA: unconstrained optimization by quadratic approximation, Mathematical Programming, 92 (2002), 555-582. doi: 10.1007/s101070100290.
[23]	J. A. Tilley, Valuing American options in a path simulation model, Transactions of the Society of Actuaries, 45 (1993), 83-104.
[24]	V. Torczon, On the convergence of the multidirectional search algorithm, SIAM Journal on Optimization, 1 (1991), 123-145. doi: 10.1137/0801010.
[25]	D. Winfield, "Function and Functional Optimization by Interpolation in Data Tables,'' PhD thesis, Harvard University in Cambridge, 1969.
[26]	D. Winfield, Functional minimization by interpolation in a data table, Journal of the Institute of Mathematics and its Applications, 12 (1973), 339-347. doi: 10.1093/imamat/12.3.339.

show all references

References:

[1]	I. Bongartz, A. R. Conn, N. I. M. Gould and Ph. L. Toint, CUTE: Constrained and unconstrained testing environment, ACM Transactions on Mathematical Software, 21 (1995), 123-160. doi: 10.1145/200979.201043.
[2]	A. J. Booker, J. E. Dennis, P. D. Frank, D. B. Serafini, V. Torczon and M. W. Trosset, A rigorous framework for optimization of expensive functions by surrogates, Structural and Multidisciplinary Optimization, 17 (1999), 1-13.
[3]	T. D. Choi and C. T. Kelley, Superlinear Convergence and Implicit Filtering, SIAM Journal on Optimization, 10 (2000), 1149-1162. doi: 10.1137/S1052623499354096.
[4]	B. Colson, P. Marcotte and G. Savard, Bilevel programming: A survey, 4OR: A Quarterly Journal of Operations Research, 3 (2005), 87-107.
[5]	A. R. Conn, N. I. M. Gould and Ph. L. Toint, "Trust-Region Methods,'' SIAM, Philadelphia, 2000. doi: 10.1137/1.9780898719857.
[6]	A. R. Conn and Ph. L. Toint, An algorithm using quadratic interpolation for unconstrained derivative free optimization, in "Nonlinear Optimization and Applications,'' Plenum Publishing, (1996), 27-47.
[7]	A. R. Conn, K. Scheinberg and Ph. L. Toint, A derivative free optimization algorithm in practice, the American Institute of Aeronautics and Astronautics Conference, St Louis, 1998.
[8]	A. R. Conn, K. Scheinberg and Ph. L. Toint, A derivative free optimization method via support vector machines, 1999. Available from: www.cas.mcmaster.ca/ oplab/seminar/conn/conn_deriv_free.ps
[9]	A. R. Conn, K. Scheinberg and Ph. L. Toint, On the convergence of derivative-free methods for unconstrained optimization, in "Approximation Theory and Optimization,'' Cambridge University Press, (1997), 83-108.
[10]	A. R. Conn, K. Scheinberg and L. N. Vicente, Geometry of interpolation sets in derivative free optimization, Mathematical Programming, 111 (2008), 141-172. doi: 10.1007/s10107-006-0073-5.
[11]	A. R. Conn, K. Scheinberg and L. N. Vicente, Geometry of sample sets in derivative free optimization: Polynomial regression and underdetermined interpolation, IMA Journal of Numerical Analysis, 28 (2008), 721-748. doi: 10.1093/imanum/drn046.
[12]	C. Cox and M. Rubinstein, "Option Markets,'' Prentice-Hall, 1985.
[13]	N. Cristianini and J. Shawe-Taylor, "An Introduction to Support Vector Machines and Other Kernel-based Methods,'' Cambridge University Press, Cambridge, UK, 2000.
[14]	J. E. Dennis and V. Torczon, Direct search methods on parallel machines, SIAM Journal on Optimization, 1 (1991), 448-474. doi: 10.1137/0801027.
[15]	R. A. Fisher, "The Design of Experiments,'' Oliver and Boyd Ltd., 1951.
[16]	P. Gilmore and C. T. Kelley, An implicit filtering algorithm for optimization of functions with many local minima, SIAM Journal on Optimization, 5 (1995), 269-285. doi: 10.1137/0805015.
[17]	B. Karasözen, Survey of trust-region derivative free optimization methods, Journal of Industrial and Management Optimization, 3 (2007), 321-334.
[18]	T. G. Kolda, R. M. Lewis and V. Torzcon, Optimization by direct search: new perspectives of some classical and modern methods, SIAM Review, 45 (2003), 385-482. doi: 10.1137/S003614450242889.
[19]	J. A. Nelder and R. Mead, A simplex method for function minimization, Computer Journal, 7 (1965), 308-313.
[20]	J. Nocedal and S. J. Wright, "Numerical Optimization,'' Springer-Verlag, New York, 1999. doi: 10.1007/b98874.
[21]	M. J. D. Powell, Trust region methods that employ quadratic interpolation to the objective function, Presentation at the 5th SIAM Conference on Optimization, 1996.
[22]	M. J. D. Powell, UOBYQA: unconstrained optimization by quadratic approximation, Mathematical Programming, 92 (2002), 555-582. doi: 10.1007/s101070100290.
[23]	J. A. Tilley, Valuing American options in a path simulation model, Transactions of the Society of Actuaries, 45 (1993), 83-104.
[24]	V. Torczon, On the convergence of the multidirectional search algorithm, SIAM Journal on Optimization, 1 (1991), 123-145. doi: 10.1137/0801010.
[25]	D. Winfield, "Function and Functional Optimization by Interpolation in Data Tables,'' PhD thesis, Harvard University in Cambridge, 1969.
[26]	D. Winfield, Functional minimization by interpolation in a data table, Journal of the Institute of Mathematics and its Applications, 12 (1973), 339-347. doi: 10.1093/imamat/12.3.339.

[1]	Liang Zhang, Wenyu Sun, Raimundo J. B. de Sampaio, Jinyun Yuan. A wedge trust region method with self-correcting geometry for derivative-free optimization. Numerical Algebra, Control and Optimization, 2015, 5 (2) : 169-184. doi: 10.3934/naco.2015.5.169
[2]	Bülent Karasözen. Survey of trust-region derivative free optimization methods. Journal of Industrial and Management Optimization, 2007, 3 (2) : 321-334. doi: 10.3934/jimo.2007.3.321
[3]	A. M. Bagirov, Moumita Ghosh, Dean Webb. A derivative-free method for linearly constrained nonsmooth optimization. Journal of Industrial and Management Optimization, 2006, 2 (3) : 319-338. doi: 10.3934/jimo.2006.2.319
[4]	Wei-Zhe Gu, Li-Yong Lu. The linear convergence of a derivative-free descent method for nonlinear complementarity problems. Journal of Industrial and Management Optimization, 2017, 13 (2) : 531-548. doi: 10.3934/jimo.2016030
[5]	Min Xi, Wenyu Sun, Jun Chen. Survey of derivative-free optimization. Numerical Algebra, Control and Optimization, 2020, 10 (4) : 537-555. doi: 10.3934/naco.2020050
[6]	Jirui Ma, Jinyan Fan. On convergence properties of the modified trust region method under Hölderian error bound condition. Journal of Industrial and Management Optimization, 2021 doi: 10.3934/jimo.2021222
[7]	Yannan Chen, Jingya Chang. A trust region algorithm for computing extreme eigenvalues of tensors. Numerical Algebra, Control and Optimization, 2020, 10 (4) : 475-485. doi: 10.3934/naco.2020046
[8]	Gaohang Yu. A derivative-free method for solving large-scale nonlinear systems of equations. Journal of Industrial and Management Optimization, 2010, 6 (1) : 149-160. doi: 10.3934/jimo.2010.6.149
[9]	Dong-Hui Li, Xiao-Lin Wang. A modified Fletcher-Reeves-Type derivative-free method for symmetric nonlinear equations. Numerical Algebra, Control and Optimization, 2011, 1 (1) : 71-82. doi: 10.3934/naco.2011.1.71
[10]	Yigui Ou, Wenjie Xu. A unified derivative-free projection method model for large-scale nonlinear equations with convex constraints. Journal of Industrial and Management Optimization, 2022, 18 (5) : 3539-3560. doi: 10.3934/jimo.2021125
[11]	Nobuko Sagara, Masao Fukushima. trust region method for nonsmooth convex optimization. Journal of Industrial and Management Optimization, 2005, 1 (2) : 171-180. doi: 10.3934/jimo.2005.1.171
[12]	Xin Zhang, Jie Wen, Qin Ni. Subspace trust-region algorithm with conic model for unconstrained optimization. Numerical Algebra, Control and Optimization, 2013, 3 (2) : 223-234. doi: 10.3934/naco.2013.3.223
[13]	Honglan Zhu, Qin Ni, Meilan Zeng. A quasi-Newton trust region method based on a new fractional model. Numerical Algebra, Control and Optimization, 2015, 5 (3) : 237-249. doi: 10.3934/naco.2015.5.237
[14]	Jun Chen, Wenyu Sun, Zhenghao Yang. A non-monotone retrospective trust-region method for unconstrained optimization. Journal of Industrial and Management Optimization, 2013, 9 (4) : 919-944. doi: 10.3934/jimo.2013.9.919
[15]	Lijuan Zhao, Wenyu Sun. Nonmonotone retrospective conic trust region method for unconstrained optimization. Numerical Algebra, Control and Optimization, 2013, 3 (2) : 309-325. doi: 10.3934/naco.2013.3.309
[16]	Dan Xue, Wenyu Sun, Hongjin He. A structured trust region method for nonconvex programming with separable structure. Numerical Algebra, Control and Optimization, 2013, 3 (2) : 283-293. doi: 10.3934/naco.2013.3.283
[17]	Zhou Sheng, Gonglin Yuan, Zengru Cui, Xiabin Duan, Xiaoliang Wang. An adaptive trust region algorithm for large-residual nonsmooth least squares problems. Journal of Industrial and Management Optimization, 2018, 14 (2) : 707-718. doi: 10.3934/jimo.2017070
[18]	Chunlin Hao, Xinwei Liu. A trust-region filter-SQP method for mathematical programs with linear complementarity constraints. Journal of Industrial and Management Optimization, 2011, 7 (4) : 1041-1055. doi: 10.3934/jimo.2011.7.1041
[19]	Jing Zhou, Cheng Lu, Ye Tian, Xiaoying Tang. A SOCP relaxation based branch-and-bound method for generalized trust-region subproblem. Journal of Industrial and Management Optimization, 2021, 17 (1) : 151-168. doi: 10.3934/jimo.2019104
[20]	Hatim Tayeq, Amal Bergam, Anouar El Harrak, Kenza Khomsi. Self-adaptive algorithm based on a posteriori analysis of the error applied to air quality forecasting using the finite volume method. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2557-2570. doi: 10.3934/dcdss.2020400

Impact Factor:

Tools

Metrics

Other articles
by authors

on AIMS
Jun Takaki Nobuo Yamashita
on Google Scholar
Jun Takaki Nobuo Yamashita

[Back to Top]