×

zbMATH — the first resource for mathematics

A nonmonotone adaptive trust region method for unconstrained optimization based on conic model. (English) Zbl 1210.65123
Recently, nonmonotone line search techniques have been studied by many authors. Many authors generalized the nonmonotone technique to trust region methods and proposed nonmonotone trust region methods. Theoretical analysis and numerical results show that the algorithms with nonmonotone properties are more efficient than the usual monotone algorithms.
In this paper a nonmonotone adaptive trust region method for unconstrained optimization based on conic model is presented. A simple way is given to determine the trust region radius at each iterate point. The local and global convergence properties of algorithm are proved under some reasonable assumptions. The numerical results show the efficiency of the new algorithm.

MSC:
65K05 Numerical mathematical programming methods
90C30 Nonlinear programming
90C51 Interior-point methods
Software:
minpack
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Fletcher, R., Practical method of optimization, Unconstrained optimization, vol. 1, (1980), John Wiley New York · Zbl 0439.93001
[2] Powell, M.J.D., On the global convergence of trust region algorithms for unconstrained optimization, Math. prog., 29, 297-303, (1984) · Zbl 0569.90069
[3] Schultz, G.A.; Schnabel, R.B.; Byrd, R.H., A family of trust region based algorithms for unconstrained minimization with strong global convergence, SIAM J. numer. anal., 22, 47-67, (1985) · Zbl 0574.65061
[4] Yuan, Y., On the convergence of trust region algorithms, Math. numer. sinica, 16, 333-346, (1996)
[5] Yuan, Y.; Sun, W., Optimization theory and methods, (1997), Science Press Beijing, China
[6] J. Nocedal, Y. Yuan, Combining trust region and line search techniques. Report NAM 07, Department of EECS, Northwestern University, 1991. · Zbl 0909.90243
[7] Davidon, W.C., Conic approximation and collinear scaling for optimizers, SIAM J. numer. anal., 17, 268-281, (1980) · Zbl 0424.65026
[8] Di, S.; Sun, W., Trust region method for conic model to solve unconstrained optimization problems, Optim. meth. software., 6, 237-263, (1996)
[9] Sun, W.; Yuan, J.; Yuan, Y., Trust region method of conic model for linearly constrained optimization, J. comput. math., 21, 295-304, (2003) · Zbl 1049.65052
[10] Sun, W.; Yuan, Y., A conic trust region method for nonlinearly constrained optimization, Ann. oper. res., 103, 175-191, (2001) · Zbl 1008.90062
[11] Q. Ni, S. Hu, A new derivative free algorithm based on conic interpolation model. Technical report, Faculty of Science, Nanjing University of Aeronautics and Astronautics, 2001.
[12] Grippo, L.; Lampariello, F.; Lucidi, S., A nonmonotone line search technique for newton’s method, SIAM J. numer. anal., 23, 707-716, (1986) · Zbl 0616.65067
[13] Zhang, J.; Zhang, X., A nonmonotone adaptive trust region method and its convergence, Comput. math. appl., 45, 1469-1477, (2003) · Zbl 1065.90071
[14] Mo, J.; Zhang, K.; Wei, Z., A nonmonotone trust region method for unconstrained optimization, Appl. math. comput., 171, 371-384, (2005) · Zbl 1094.65059
[15] Chen, Z.; Han, J.; Xu, D., A nonmonotone trust region method for nonlinear programming with simple bound constraints, Appl. math. optim., 43, 63-85, (2001) · Zbl 0973.65049
[16] Sun, W., Nonmonotone trust region method for optimization, Appl. math. comput., 156, 159-174, (2004) · Zbl 1059.65055
[17] Qu, S.; Zhang, K.; Zhang, J., A nonmonotone trust region method of conic model for unconstrained optimization, J. comput. appl. math., 220, 119-128, (2008) · Zbl 1151.65055
[18] Sartenaer, A., Automatic determination of an initial trust region in nonlinear programming, SIAM J. sci. comp., 18, 1788-1803, (1997) · Zbl 0891.90151
[19] Zhang, X.; Zhang, J.; Liao, L., An adaptive trust region method and its convergence, Sci. China, 45, 620-631, (2002) · Zbl 1105.90361
[20] Zhang, X.; Chen, Z.; Zhang, J., A self-adaptive trust region method for unconstrained optimization, OR trans., 5, 53-62, (2001)
[21] Fu, J.; Sun, W., Nonmonotone adaptive trust region method for unconstrained optimization problems, Appl. math. comput., 163, 489-504, (2005) · Zbl 1069.65063
[22] Fu, J.; Sun, W.; Sampaio, R.J.B., An adaptive approach of conic trust region method for unconstrained optimization problems, J. appl. math. comput., 19, 165-177, (2005) · Zbl 1084.65060
[23] Han, Q.; Sun, W.; Han, J.; Sampaio, R.J.B., An adaptive trust region method for unconstrained optimization, Optim. meth. software, 20, 665-677, (2005) · Zbl 1127.90415
[24] Sorensen, D.C., The q-superlinear convergence of a collinear scaling algorithm for unconstrained optimization, SIAM J. numer. anal., 17, 84-114, (1980) · Zbl 0428.65040
[25] Ariyawansa, K.A., Deriving collinear scaling algorithms as extensions of quasi-Newton methods and the local convergence of DFP and BFGS related collinear scaling algorithm, Math. prog., 49, 23-48, (1990) · Zbl 0724.90059
[26] Zhang, J.; Wang, Y.; Zhang, X., Superlinearly convergent trust region method without the assumption of positive-definite Hessian, J. optim. theory appl., 129, 201-218, (2006) · Zbl 1139.90032
[27] Xu, C.; Yang, X., Convergence of conic quasi-Newton trust region methods for unconstrained minimization, Math. appl., 11, 71-76, (1998) · Zbl 0954.90029
[28] Moré, J.J.; Garbow, B.S.; Hilstron, K.E., Testing unconstrained optimization software, ACM trans. math. software., 7, 17-41, (1981) · Zbl 0454.65049
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.