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Combining nonmonotone conic trust region and line search techniques for unconstrained optimization. (English) Zbl 1215.65107
The authors propose a trust region method for solving a general unconstrained optimization problem. After outlining the necessary background and an overview of the literature in the first section, they proceed to describe a new trust region algorithm which can be regarded as a combination of the conic model, non-monotone and line-search techniques. The third and fourth sections study the convergence properties of the proposed algorithm, whereas the last section presents the results of numerical experimentation using the proposed algorithm.

##### MSC:
 65K05 Numerical mathematical programming methods 90C30 Nonlinear programming 90C51 Interior-point methods
L-BFGS; minpack
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##### References:
 [1] Fletcher, R., Practical methods of optimization, (1987), Wiley New York · Zbl 0905.65002 [2] Nocedal, J.; Wright, S.J., Numerical optimization, (1999), Springer New York · Zbl 0930.65067 [3] Powell, M.J.D., On the global convergence of trust region algorithms for unconstrained optimization, Math. program., 29, 297-303, (1984) · Zbl 0569.90069 [4] Yuan, Y.X.; Sun, W.Y., Optimization theory and methods, (1997), Science Press Beijing [5] Nocedal, J.; Yuan, Y., Combining trust region and line search techniques, (), 153-175 · Zbl 0909.90243 [6] E.M. Gertz, Combination Trust-Region Line Search Methods for Unconstrained Optimization, University of California, San Diego, 1999. [7] Mo, J.T.; Zhang, K.C.; Wei, Z.X., A nonmonotone trust region methods for unconstrained optimization, Appl. math. comput., 171, 371-384, (2005) · Zbl 1094.65059 [8] Grippo, L.; Lampariello, F.; Lucidi, S., A nonmonotone line search technique for newton’s method, SIAM J. numer. anal., 23, 4, 707-716, (1986) · Zbl 0616.65067 [9] Deng, N.Y.; Xiao, Y.; Zhou, F.J., Nonmonotonic trust region algorithm, J. optim. theory appl., 76, 259-285, (1993) · Zbl 0797.90088 [10] Gu, N.Z.; Mo, J.T., Incorporating nonmonotone strategies into the trust region method for unconstrained optimization, Comput. math. appl., 55, 2158-2172, (2008) · Zbl 1183.90387 [11] Ke, X.; Han, J., A class of nonmonotone trust region algorithms for unconstrained optimization, Sci. China ser. A, 41, 9, 927-932, (1998) · Zbl 0917.90271 [12] Mo, J.; Liu, C.; Yan, S., A nonmonotone trust region method based on non-increasing technique of weighted average of the successive function values, J. comput. appl. math., 97-108, (2006) [13] Sun, W.Y., Nonmonotone trust region method for solving optimization problems, Appl. math. comput., 156, 159-174, (2004) · Zbl 1059.65055 [14] Conn, A.R.; Gould, N.I.M.; Toint, Ph.L., Trust-region methods, society for industrial and applied mathematics, (2000), SIAM Philadelphia, PA · Zbl 0643.65031 [15] Davidon, W.C., Conic approximation and collinear scaling for optimizers, SIAM J. numer. anal., 17, 268-281, (1980) · Zbl 0424.65026 [16] Ji, Y.; Qu, S.J.; Wang, Y.J.; Li, H.M., A conic trust-region method for optimization with nonlinear equality and inequality 4 constrains via active-set strategy, Appl. math. comput., 183, 217-231, (2006) · Zbl 1112.65052 [17] Qu, S.J.; Jiang, S.D., 2008A trust-region with a conic model for unconstrained optimization, Math. methods appl. sci., 31, 1780-1808, (2008) · Zbl 1146.49026 [18] 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 [19] Qu, S.J.; Zhang, K.C.; Zhang, J., A nonmonotone trust region method of conic model for unconstrained optimization, J. comput. appl. math., 220, 119-128, (2008) · Zbl 1151.65055 [20] Zhang, H.; Hager, W.W., A nonmonotone line search technique and its application to unconstrained optimization, SIAM J. optim., 14, 4, 1043-1056, (2004) · Zbl 1073.90024 [21] Moré, J.J.; Grabow, B.S.; Hillstrom, K.E., Testing unconstrained optimization software, ACM trans. math. software, 7, 17-41, (1981) · Zbl 0454.65049
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