A short note on the global convergence of the unmodified PRP method

It is well-known that the search direction generated by the standard (unmodified) PRP nonlinear conjugate gradient method is not necessarily a descent direction of the objective function, which brings difficulty for its global convergence for general functions. However, to our surprise, it is easily proved in this short note that the unmodified PRP method still converges globally even for nonconvex optimization by the use of an approximate descent line search.