American Journal of Applied Mathematics
Volume 6, Issue 2, April 2018, Pages: 71-77
Received: Jun. 25, 2018;
Published: Jun. 26, 2018
Views 886 Downloads 51
Min Xi, School of Finance, Guangdong University of Foreign Studies, Guangzhou, China
Ailing Xiao, School of Finance, Guangdong University of Foreign Studies, Guangzhou, China
A class of nonsmooth composite minimization problems are considered in this paper. In practice, many problems in computational science, engineering and other enormous areas fall into this class of problems. Two obvious applications of this class of problems are least square problems with nonsmooth data and the problem of solving a system of nonsmooth equations. Because of its practical importance, this class of problem has received wide attention from the mathematical optimization community. In particular, Sampaio, Yuan and Sun has proposed a trust region method for this class of problems and also provided the convergence analysis to support their algorithm. Motivated partly by their work, in this paper a nonmonotone trust region algorithm for this class of nonsmooth composite minimization problems is presented. Different from most existing monotone line search and trust region methods, this method combines the nonmonotone technique to improve the efficiency of the trust region method. After a brief introduction of the class of problems in the first section, some fundamental concepts and properties which will be used in this paper are presented. Then, the new nonmonotone trust region algorithm for the class of problem is described followed by the global convergence analysis of the new algorithm. A simple application of this algorithm is discussed in the last part of this paper.
A Nonmonotone Trust Region Method for Nonsmooth Composite Programming Problems, American Journal of Applied Mathematics.
Vol. 6, No. 2,
2018, pp. 71-77.
R. M. Chamberlain, M. J. D. Powell, C. Lemarechal and H. C. Pedersen, The watchdog technique for forcing convergence in algorithms for constrained optimization, Mathematical Programming Studies, 1982, vol. 16: pp. 1-17.
F. H. Clarke, Optimization and Nonsmooth Analysis, John Wiley, New York, 1983.
A. R. Conn, N. I. M. Gould and P. L. Toint, Trust Region Methods. Philadelphia: SIAM, 2000.
N. Y. Deng, Y. Xiao and F. J. Zhou, Nonmonotonic trust region algorithm, Journal of Optimization Theory and Applications, 1993, vol. 76, pp. 259-285.
R. Fletcher, Practical Methods of Optimization, Volume 2, John Wiley and Sons, New York, 1981.
L. Grippo, F. Lampariello and S Lucidi, A nonmonotone line search technique for Newton’s method 1986, vol. 23: pp. 88.
M. Kimiaei, A new class of nonmonotone adaptive trust-region methods for nonlinear equations with box constraints, Calcolo, 2016, vol. 54: pp. 1-44
J. Nocedal and S. J. Wright, Numerical Optimization, 2nd edition, Springer Series in Operations Research and Financial Engineering, Springer, New York, 2006.
J. S. Pang, S. P. Han, and N. Rangaraj, Minimization of locally Lipschitzian function, SIAM Journal on Optimization, 1991, vol. 1, pp. 57-82.
L. Qi and J. Sun, A trust region algorithm for minimization of locally Lipschitzian functions, Mathematical Programming, 1994, vol. 66: pp. 25-43.
S. Rezaee and S. Babaie-Kafaki, A modified nonmonotone trust region line search method, Journal of Applied Mathematics and Computing, 2017, https://doi.org/10.1007/s12190-017-1113-4.
R. J. de Sampaio, J. Yuan and W. Sun, Trust region algorithm for nonsmooth optimization, Applied Mathematics and Computation, 1997, vol. 85: pp. 109-116.
Ph. L. Toint, An assessment of nonmonotone line search techniques for unconstrained optimization, SIAM Journal on Scientific Computing, 1996, vol. 17: pp. 725-739.
Ph. L. Toint, A non-monotone trust-region algorithms for nonlinear optimization subject to convex constraints, Mathematical Programming, 1997, vol. 17: pp. 69-94.
X. Wang and Y. Yuan, An augmented Lagrangian trust region method for equality constrained optimization, Optimization Methods and Software, 2015, vol. 30: pp. 559-582.
Y. Yuan, Conditions for convergence of trust region algorithms for nonsmooth optimization, Mathematical Programming, 1985, vol. 31: pp. 220-228.
Y. Yuan and W. Sun, Optimization Theory and Methods, Science Press, Beijing, 1997.
Q. Zhou, W. Sun and H. Zhang, A new simple model trust-region method with generalized Barzilai-Borwein parameter for large-scale optimization, Science China-mathematics, 2016, vol. 59: pp. 2265-2280.
H. Zhu, Q. Ni, L. Zhang and W. Yang, A fractional trust region method for linear equality constrained optimization, Discrete Dynamics in Nature and Society, 2016, pp. 1-10.