A novel four-term conjugate gradient method for large-scale optimization problems involving formulation and application

The conjugate gradient (CG) method is widely employed for solving unconstrained optimization problems due to its independence from second derivatives or their approximations. It has found extensive applications in fields such as image restoration, neural networks, deep learning, and machine learning. In this study, we propose a novel four-term CG method designed to improve both convergence and descent properties. The proposed method builds upon the WYL and Dai-Liao CG frameworks, with the latter preserving the conjugacy condition under inexact line searches. Numerical experiments demonstrate that the proposed method outperforms several well-established CG methods, including CG-Descent, Dai-Liao, and the WYL family, in terms of iteration count, function and gradient evaluations, and CPU time. Furthermore, the method is extended to address real-world application problems, highlighting its practical effectiveness.