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

¹ Center of Excellence in Theoretical and Computational Science (TaCS-CoE), Fixed Point Research Laboratory, Fixed Point Theory and Applications Research Group, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), Bangkok 10140, Thailand
² Department of Mathematics, Faculty of Sciences, Modibbo Adama University, Yola 652105, Nigeria
³ Department of Computer Science, Faculty of Science, Gombe State University, Gombe, Nigeria
⁴ Faculty of Science Energy and Environment, King Mongkut's University of Technology North Bangkok, Rayong Campus (KMUTNB), Rayong 21120, Thailand

^* Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it. , This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 18 October 2023
Accepted: 26 March 2026

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.