Global convergence property with inexact line search for a new conjugate gradient method

To develop new conjugate gradient (CG) methods that are both theoretically robust and practically effective for solving unconstrained optimization problems, we propose novel hybrid conjugate gradient algorithms. In these algorithms, the scale parameter /3k is defined as a convex combination of /3HZ k (from Hager and Zhang's method) and /3kBA (from Al-Bayati and Al-Assady's method). In one hybrid algorithm, the parameter in the convex combination is determined to satisfy the conjugacy condition, independent of the line search.In the other algorithm, the parameter is computed to ensure that the conjugate gradient direction aligns with the Newton direction. Under certain conditions, the proposed methods guarantee a sufficient descent at each iteration and exhibit global convergence properties. Furthermore, numerical results demonstrate that the hybrid computational scheme based on the conjugacy condition is efficient and performs favorably compared to some well-known algorithms.