In this paper, an adaptive nonmonotone line search method for unconstrained minimization problems is proposed. At every iteration, the new algorithm selects only one of the two directions: a Newton-type direction and a negative curvature direction, to perform the line search. The nonmonotone technique is included in the backtracking line search when the Newton-type direction is the search direction. Furthermore, if the negative curvature direction is the search direction, we increase the steplength under certain conditions. The global convergence to a stationary point with second-order optimality conditions is established. Some numerical results which show the efficiency of the new algorithm are reported.
ZHOU Qunyan, SUN Wenyu
. Adaptive nonmonotone line search method for unconstrained
optimization[J]. Frontiers of Mathematics in China, 2008
, 3(1)
: 133
-148
.
DOI: 10.1007/s11464-008-0001-5
1. Bunch J R Parlett B N Direct methods for solvingsymmetric indefinite systems of linear equationsSIAM J Nume Anal 1971 8639655. doi:10.1137/0708060
2. Deng N Y Xiao Y Zhou F Y A nonmonotonic trust region algorithmJournal of Optimization Theory and Applications 1993 76259285. doi:10.1007/BF00939608
3. Ferris M C Lucidi S Roma M Nonmonotone curvilinear line search methods for unconstrainedoptimizationComputational Optimizationand Applications 1996 6117136
4. Fu J Sun W Nonmonotone adaptive trust-regionmethod for unconstrained optimization problemsApplied Mathematics and Computation 2005 163489504. doi:10.1016/j.amc.2004.02.011
5. Grippo L Lamparillo F Lucidi S A nonmonotone line search technique for New-ton's methodSIAM J Numer Anal 1986 23707716. doi:10.1137/0723046
6. Gould N I M Lucidi S Roma M et al.Exploiting negative curvature directions in linesearchmethods for unconstrained optimizationOptimizationMethods and Software 2000 147598. doi:10.1080/10556780008805794
7. Lucidi S Rochetich F Roma M Curvilinear stabilization techniques for truncate Newtonmethods in large scale unconstrained optimizationSIAM J Optimization 1998 3916939. doi:10.1137/S1052623495295250
8. Lampariello F Sciandrone M Use of the minimum-norm searchdirection in a non-monotone version of the Gauss-Newton methodJournal of Optimization Theory and Applications 2003 1196582. doi:10.1023/B:JOTA.0000005041.99777.af
9. McCormick G P Amodification of Armijo's step-size rule for negative curvatureMathematical Programming 1977 13111115. doi:10.1007/BF01584328
10. Moré J J Garbow B S Hillstrom K E Testing unconstrained optimization softwareACM Trans Math Software 1981 71741. doi:10.1145/355934.355936
11. Moré J J Sorensen D C On the use of directions ofnegative curvature in a modified Newton methodMathematical Programming 1979 16120. doi:10.1007/BF01582091
12. Nocedal J Wright S J Numerical OptimizationNew YorkSpringer 1999
13. Sun W Nonmonotoneoptimization methods: motivation and developmentIn: International Conference on Numerical Linear Algebra and Optimization,Guilin, China October 7–10, 2003 (Invited talk)
14. Sun W Nonmonotonetrust-region method for solving optimization problemsApplied Mathematics and Computation 2004 156159174. doi:10.1016/j.amc.2003.07.008
15. Sun W Han J Sun J On global convergence of nonmonotone decent methodJ Comput Appl Math 2002 1468998. doi:10.1016/S0377‐0427(02)00420‐X
16. Yuan Y Sun W Optimization Theory and MethodsBeijingSciencePress 1997
17. Zhou H Sun W Nonmonotone descent algorithmfor nonsmooth unconstrained optimization problemsInternational Journal of Pure and Applied Mathematics 2003 9153163
18. Zhou Q Sun W A nonmonotone second ordersteplength method for unconstrained optimizationJournal of Computational Mathematics 2007 23104112
19. Zhang J Xu C A class of indefinite doglegpath methods for unconstrained minimizationSIAM J Optimization 1999 3646667. doi:10.1137/S105262349627523X
20. Zhang J L Zhang X S A nonmonotone adaptive trustregion method and its convergenceComputersand Mathematics with Applications 2003 4514691477. doi:10.1016/S0898‐1221(03)00130‐5