Some three-term RMIL conjugate gradient methods with descent property for solving optimization problems with application

Abstract

To propose a conjugate gradient (CG) scheme with a promising structure, it has been observed that Newton's direction is optimal when the current iteration is near the solution, and the objective function behaves like a quadratic. However, for large-scale problems, a method that does not require second-derivative information is often necessary. Therefore, to develop a more effective scheme for handling complex problems, we apply the standard secant equation to construct a combination of three-term CG search directions using the β k RMIL method. This combination approximates the quasi-Newton direction and ensures sufficient descent. Furthermore, we establish the global convergence of the scheme under mild assumptions, demonstrating that the algorithm is robust and reliable compared to earlier CG methods. Finally, we showcase the efficiency of the proposed scheme by applying it to solve a three degrees of freedom (3DOF) motion control model.