Nonmonotone Linear Searches And Their Applications In Conjugate Gradient Methods And Quasi-Newton Methods

Optimization problems have a wide range of practical background in economics, man-agement, engineering and so on. Conjugate gradient methods and quasi-Newton methodsare two kinds of most commonly used methods for solving optimization problems. Con-jugate gradient methods needs lower storage and are particularly welcome in the solutionof large scale problems. Broyden class of quasi-Newton methods are welcome methods forsolving middle and lower size problems due to their good numerical performance and fastconvergence property.In general, an iterative method for solving optimization problems uses monotone lin-ear search technique. An advantage of this kind of linear search lies in that the generatedsequence of function values is monotonically decreasing. On the other hand, however, amonotone linear search may generate a small step-length in some cases. As a remedy,Grippo et al proposed a non-monotone linear search technique. Numerical results haveshown that this kind of linear search works very well. In this thesis, we introduce thenon-monotone linear search technique proposed by Grippo et al to the MFR, MPRP,MBFGS and CBFGS methods.In Chapter 2, we propose a non-monotone MFR method and establish its globalconvergence. We also do numerical experiments to test the method and compare theperformance of the method with existing monotone MFR method. The numerical resultsshow that the non-monotone MFR method outperforms the monotone MFR method.In Chapter 3, we study the non-monotone MPRP method and establish its globalconvergence. We also do numerical experiments to test the method and compare theperformance of the method with existing monotone MPRP method. The numerical resultsshow that the non-monotone MFR method outperforms the monotone MPRP method.In Chapter 4, we propose a non-monotone CBFGS and a non-monotone MBFGSmethods. We show that these non-monotone methods are globally convergent under mildconditions. We also do some numerical experiments to test the proposed methods. Theresults show that the non-monotone methods performs very well.