A globally convergent hybrid conjugate gradient method for solving unconstrained optimization problems / Ain Aqiela Azamuddin

The Conjugate Gradient (CG) method is utilized among researchers in solving optimization problems. It possesses characteristics that distinguish the steepest descent and Newton’s method. For example, it has a faster convergence rate than steepest descent method and a lesser computational cost than Newton’s method. In 2016, the Aini-Rivaie-Mustafa (ARM) method, a modified CG method, was introduced and presented a good performance compared to the previous CG method that it was tested with. However, sometimes the ARM method generates a negative CG coefficient value. Therefore, this paper intends to propose a new hybrid CG method for solving an unconstrained optimization problem, where the main focus of this study is to improve the ARM method. It is combined with another CG method to solve this problem that always generates a positive CG coefficient value. The proposed method demonstrates that it could globally converge towards the minimizer and possessed sufficient descent conditions under a strong Wolfe line search. Besides that, the numerical observation was made by testing the method with 20 standard test functions that vary in shape. The testing was also made on four different variables on three different initial points, ranging from close to far from the solution point. Moreover, the purpose is to test the method's efficiency and reliability in solving different types of functions with various ranges and numbers of variables. The testing was made using MATLAB R2013A, and the results were recorded and compared employing the performance profile introduced by Dolan and More (2002). The result indicates that it could outperform both original methods regarding the Central Processing Unit (CPU) time and the number of problems solved where the proposed method (A-ARM method) could solve 100% of the test functions surpassing both of its original methods.