×

zbMATH — the first resource for mathematics

Incorporating nonmonotone strategies into the trust region method for unconstrained optimization. (English) Zbl 1183.90387
Summary: This paper concerns a nonmonotone line search technique and its application to the trust region method for unconstrained optimization problems. In our line search technique, the current nonmonotone term is a convex combination of the previous nonmonotone term and the current objective function value, instead of an average of the successive objective function values that was introduced by H. Zhang and W. W. Hager [SIAM J. Optim. 14, No. 4, 1043–1056 (2004; Zbl 1073.90024)]. We incorporate this nonmonotone scheme into the traditional trust region method such that the new algorithm possesses nonmonotonicity. Unlike the traditional trust region method, our algorithm performs a nonmonotone line search to find a new iteration point if a trial step is not accepted, instead of resolving the subproblem. Under mild conditions, we prove that the algorithm is global and superlinear convergence holds. Primary numerical results are reported.

MSC:
90C30 Nonlinear programming
Software:
minpack
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Levenberg, K., A method for the solution of certain nonlinear problems in least squares, Quart. appl. math., 2, 164-168, (1994) · Zbl 0063.03501
[2] Marquardt, D.W., An algorithm for least squares estimation with nonlinear parameters, SIAM J. appl. math., 11, 431-441, (1963) · Zbl 0112.10505
[3] Goldfeld, S.; Quandt, R.; Trotter, H., Maximization by quadratic Hill climbing, Econometrica, 34, 541-551, (1966) · Zbl 0145.40901
[4] Powell, M.J.D., Convergence properties of a class of minimization algorithm, (), 1-27 · Zbl 0908.65007
[5] Fletcher, R., An algorithm for solving linearly constrained optimization problems, Math. program., 2, 133-165, (1972) · Zbl 0249.90067
[6] Yuan, Y., On a subproblem of trust region algorithms for constrained optimization, Math. program., 47, 53-63, (1990) · Zbl 0711.90062
[7] Powell, M.J.D.; Yuan, Y., A trust region algorithm for equality constrained optimization, Math. program., 49, 189-213, (1990) · Zbl 0816.90121
[8] Conn, A.R.; Gould, N.I.M.; Toint, Ph.L., Trust-region methods, (2000), SIAM Publications Philadelphia, PA · Zbl 0643.65031
[9] Nocedal, J.; Yuan, Y., Combining trust region and line search techniques, (), 153-175 · Zbl 0909.90243
[10] E.M. Gertz, Combination trust-region line search methods for unconstrained optimization, University of California, San Diego, 1999
[11] 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
[12] Deng, N.Y.; Xiao, Y.; Zhou, F.J., Nonmonotone trust region algorithm, J. optim. theory appl., 76, 2, 259-285, (1993) · Zbl 0797.90088
[13] 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
[14] Sun, W., Nonmonotone trust region method for solving optimization problems, Appl. math. comput., 156, 1, 159-174, (2004) · Zbl 1059.65055
[15] Fu, J.; Sun, W., Nonmonotone adaptive trust-region method for unconstrained optimization problems, Appl. math. comput., 163, 1, 489-504, (2005) · Zbl 1069.65063
[16] 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
[17] Toint, P.L., An assessment of non-monotone line search techniques for unconstrained optimization, SIAM J. sci. comput., 17, 725-739, (1996) · Zbl 0849.90113
[18] Dai, Y.H., On the nonmonotone line search, J. optim. theory appl., 112, 2, 315-330, (2002) · Zbl 1049.90087
[19] Mo, J.; Liu, C.; Yan, S., A nonmonotone trust region method based on nonincreasing technique of weighted average of the successive function values, J. comput. appl. math., (2006)
[20] Bertsekas, D.P., Nonlinear programming, (1999), Athena Scientific Belmont, MA · Zbl 0935.90037
[21] Moré, J.J.; Garbow, B.S.; Hillstrome, K.E., Testing unconstrained optimization software, ACM trans. math. software, 7, 1, 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.