A Modified Spectral Gradient Method for Solving Non Linear System

Koay, Yeong Lin (2020) A Modified Spectral Gradient Method for Solving Non Linear System. Final Year Project, UTAR.

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Abstract

The objective of this project is to modify the spectral gradient method in solving the nonlinear systems. The multiple damping spectral gradient method with line search has been proposed for making improvements to the slow convergence issues. It operates separately on the gradient vector norm and the objective function at the same time and can be considered as an alternative for solving large-scale optimization problems. The results show that the spectral gradient method provides the best performance in solving the optimization problems, compared to the steepest descent method and conjugate gradient method, under the backtracking line search with Armijo condition (BTA). The main difference between these methods is the calculation of direction vector, dk. Besides, there is a relationship between solving a series of nonlinear equations and finding the optimal solutions to the problems. Most of the methods used for solving nonlinear systems are optimization-based methods. Therefore, the spectral gradient method with the BTA line search technique has been modified in order to solve solving the nonlinear systems. The efficiency of the modified spectral gradient method is tested by comparing the number of iterations, the number of function call and the computational time, with the BFGS method, steepest descent method and conjugate gradient method. The step length of these methods is selected by using the modified BTA line search technique. Finally, the modified spectral gradient method shows a better performance compared to the steepest descent method and the conjugate gradient method. The modified method gives more stable results compared to the BFGS method because numerous papers from different researchers have suggested that the BFGS method is not an appropriate method in solving the large-scale problems. Furthermore, the SG method is popular due to the fact that less storage is needed for the calculation. The modified SG method can be used in solving some nonlinear application problems. Thus, the modified spectral gradient method can be considered as an alternative method for solving nonlinear systems. The improvements in the amount of tested problem, line search strategy and search direction are recommended, in order to increase the efficiency of the modified method.

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