Newton’s and quasi-Newton methods for minimising multi-variable functions

Abstract

Newton’s method (also known as the Newton-Raphson method) is an optimisation method that can be used for obtaining the minimiser of objective functions. This method has superior convergence when compared to other optimisation techniques such as the method of steepest descent and the conjugate gradient method (provided the initial point is taken close to the minimiser). In this thesis, the Newton method is discussed in detail by looking at the implementation of this method to obtaining the minimiser of single variable as well as multi-variable functions. Some modifications of the method are also analysed. The applicability of this method to solving problems is exhibited through a novel application example. Some quasi-Newton methods namely, the rank-one correction, DFP and BFGS methods, are also considered in an attempt to remedy the drawbacks of Newton’s method. Algorithms implemented in MATLAB for all the methods discussed in this thesis can be found in the Appendix section of this dissertation.