Modified Quasi-Newton Methods For Large-Scale Unconstrained Optimization

Abstract

The focus of this thesis is on finding the unconstrained minimizer of a function, when the dimension n is large. Specifically, we will focus on the wellknown class of optimization methods called the quasi-Newton methods. First we briefly give some mathematical background. Then we discuss the quasi-Newton's methods, the fundamental method in underlying most approaches to the problems of large-scale unconstrained optimization, as well as the related so-called line search methods. A review of the optimization methods currently available that can be used to solve large-scale problems is also given. The mam practical deficiency of quasi-Newton methods is the high computational cost for search directions, which is the key issue in large-scale unconstrained optimization. Due to the presence of this deficiency, we introduce a variety of techniques for improving the quasi-Newton methods for large-scale problems, including scaling the SR1 update, matrix-storage free methods and the extension of modified BFGS updates to limited-memory scheme. Comprehensive theoretical and experimental results are also given. Finally we comment on some achievements in our researches. Possible extensions are also given to conclude this thesis.