Some three-term RMIL conjugate gradient methodswith descent property for solving optimizationproblems with application

To propose a conjugate gradient (CG) scheme with a promising structure, ithas been observed that Newton’s direction is the optimal search direction to follow whenthe current iteration is close to the solution, and the objective function behaves like aquadratic function. However, when dealing with large-scale problems, it is often necessaryto choose a method that does not require second-derivative information. Therefore, in oureffort to develop a more effective scheme for handling complex problems, we apply thestandard secant equation to construct a combination of three-term CG search directionsusing the βRMIL k method. This combination approximates the quasi-Newton directionand ensures sufficient descent. Furthermore, we establish the global convergence of thescheme with mild assumptions, and the algorithm is robust and reliable when comparedto certain earlier CG techniques. Finally, we showcase the efficiency of the proposedscheme by applying it to solve a three degrees of freedom (3DOF) motion control models.