Applying New Preconditioned Conjugated Gradient Algorithms to Unconstrained Optimization Problems

In this paper, we study a new and improved preconditioned conjugate gradient (PCG) algorithm based on Dai and Liao's procedure to enhance the CG algorithm of (Maulana). The new PCG algorithm satisfies the coupling condition and the sufficient descent condition. This work proposes improved conjugate gradient methods to enhance the efficiency and robustness of classical conjugate gradient methods. The study changes the diagonal of the inverse Hessian approximation to quasi-Newton Broyden-Fletcher-Goldfarb-Shano (BFGS) updating to make a preconditioner for nonlinear conjugate gradient (NCG) methods used to solve large-scale optimization problems with no constraints. We will calculate the step size of this two-term algorithm by accelerating the Wolfe-Powell line searching technique. The proposed new PCG algorithms have proven their global convergence in certain specific conditions reported in this paper.