A three-terms Polak–Ribière–Polyak conjugate gradient algorithm for large-scale nonlinear equations

In this paper, a conjugate gradient algorithm for systems of large-scale nonlinear equations is designed by the following steps: (i) A three-terms conjugate gradient direction dkdk is presented where the direction possesses the sufficient descent property and the trust region property independent of line search technique; (ii) A backtracking line search technique along the direction is proposed to get the step length αkαk and construct a point; (iii) If the point satisfies the given condition then it is the next point, otherwise the projection-proximal technique is used and get the next point. Both the direction and the line search technique are the derivative-free approaches, then the large-scale nonlinear equations are successfully solved (100,000 variables). The global convergence of the given algorithm is established under suitable conditions. Numerical results show that the proposed method is efficient for large-scale problems.