Improved conjugate gradient methods and application to nonparametric estimation

The conjugate gradient (CG) method is one of the most important ideas in scientific computing, applied to solving linear systems of equations and nonlinear optimization problems. In this paper, based on a variant of Dai–Yuan (DY) method and Fletcher–Reeves (FR) method, two modified CG methods (named IDY and IFR) are presented and analyzed. The search direction of the presented methods fulfills the sufficient descent condition at each iteration. We establish the global convergence of the proposed algorithms under normal assumptions and strong Wolfe line search. Preliminary elementary numerical experiment results are presented, demonstrating the effectiveness of the methods. Finally, the methods are extended to solve the problem of conditional model regression function.