A three-term conjugate gradient method with a new descent structure for unconstrained optimization and low-carbon supply chain optimization.

Optimization methods play a vital role in solving large-scale problems across engineering and environmental fields. Among them, the conjugate gradient (CG) method is popular for its computational efficiency, and recent developments-such as the three-term CG variant-have shown improved convergence and numerical stability. This paper proposes an improved three-term CG method for unconstrained optimization. Inspired by the Three-Term Zheng-Huang-Shi (ZHS) CG parameterization and Three-Term Rivaie-Mustafa-Ismail-Leong (TTRMIL), this method is specifically designed to improve the computational performance of CG methods. In the proposed method, sufficient descent conditions and global convergence properties for general functions are mathematically proven under assumption and criteria from the strong Wolfe line search. Numerical experiments conducted on several unconstrained optimization problems highlight the superiority of the new method over certain CG methods with similar characteristics. In real-world applications, the proposed method is extended to address the challenge of reducing carbon emissions for sustainability, particularly within the Low Carbon Supply Chain (LCSC) optimization problem.