Performance analysis of selected spectral conjugate gradient methods for large-scale unconstrained optimization problems with strong-Wolfe line search

The conjugate gradient (CG) method is crucial for solving unconstrained optimization especially large-scale unconstrained optimization problems. Although many studies have proposed new spectral CG method and demonstrated their superior performance, identifying the most efficient method among them remains challenging. The spectral CG method improves the convergence behaviour of the classical CG method through the incorporation of the spectral parameters, which ensures better descent properties while remaining computationally efficient for large-scale optimization problems. Therefore, this study is done by investigating the performance of four selected spectral CG methods, which are spectral Hestenes-Steifel (SHS), spectral Conjugate Descent-Fletcher (SCD), spectral Rivaie, Mustafa, Ismail, and Leong (SRMIL), and spectral Sri-Mustafa-2 (SSM-2), under inexact strong-Wolfe line search. The main objective of this study is to analyse and compare the numerical performance and evaluate efficiency of these methods. Numerical experiments are conducted on 20 standard unconstrained optimization test functions with five different problem dimensions and various initial points, implemented using MATLAB programming. The performance is analysed based on the number of iterations and Central Processing Unit (CPU) time. In addition, a mathematical proof is provided to verify that all four spectral CG methods satisfy the sufficient descent condition. The results indicate that the SHS method gives the best performance compared to the other methods, which shows that it is the fastest method and able to solve all test problems.