Sufficient Descent Condition of the Polak-Ribière-Polyak (PRP) Conjugate Gradient Method without Line Search

Nonlinear conjugate gradient methods for unconstrained optimization problems are used in many aspects of theoretical and applied sciences. They are iterative methods, so at any iteration a step length is computed using a method called line search. In most cases, the sufficient descent condition plays an important role to prove the global convergence of a conjugate gradient method. Due to its outperformance in practical computation, the Polak-Ribière-Polyak (PRP) conjugate gradient method is widely used for solving nonlinear unconstrained optimization problems. However, the sufficient descent condition of PRP has not established without line search yet. In this study, we established the sufficient descent condition without line search based on the conditions 0 < b_k^PRP £ x b_k^FR and 1 < b_k^PRP £mb_k^FR, where 0 < x < 1 and m > 1. As a result, we found that under certain conditions, the sufficient descent condition is satisfied when the PRP implemented without line search