An improvement of BFGS by applying n-th section method for solving unconstrained optimization / Nurul Atikah Mohamed Ramli - Mohamed Ramli, Nurul Atikah (2019)
Optimization is one of mathematics field that greatly developed when Quasi-newton method was presented to solve the unconstrained optimization problem. An iterative method is used to solve the problem by finding the value of search direction, dk and step size, ak. There are different methods that can be an option to solve the dk and ak in optimization problem to get the best results in term of number of iterations and CPU time. Thus, in this research, an improvement of one of the Quasi-newton method which is Broyden-Fletcher-Goldfarb-Shanno (BFGS) method has been done by employing new inexact line search method. From the previous researches, there are a few of inexact line search method that have been introduced such as Goldstein, Armijo and Wolfe line search. However, these line searches are complicated and its complexity will burden the CPU time even though the number of iteration is reduced in BFGS method. The improvement is finding the step size ak to solve the problem by employing a new inexact line search method. This new inexact line search is known as M-th section method. The n-th section method is the modification of the original bisection method. As in bisection method, this simple n-th section method divides each interval section with an even number of interval which is greater than two. Thus, n-th section method in this project is fourth and sixth section method. This new proposed algorithm is compared with the original bisection and newton method in terms of number of iteration and CPU time. Numerical results are obtained based on eight test functions with two different tolerances. This research shows that the proposed algorithm is efficient when compared to bisection and newton method as inexact line search method. Besides, this proposed algorithm has achieved the global solution and possessed sufficient descent condition. The result is analysed based on number of iterations and CPU times. It is concluded that the n-th section method which is the sixth section method is the best method that can be used as the inexact line search in BFGS to lower the number of iterations and solve most of the problems.