A modified Hestenes-Stiefel conjugate gradient method with sufficient descent condition and conjugacy condition

In this paper, by taking a little modification to the Hestenes-Stiefel method, we propose a new way to construct descent directions satisfying the sufficient descent condition. Also, an adaptive conjugacy condition and a intrinsic self-restarting mechanism are revealed, a dynamical adjustment can be regarded as the inheritance and development of properties of standard Hestenes-Stiefel method. Furthermore, we establish global convergence for general nonconvex objective function under mild condition. Numerical results show that our presented methods can be efficient for solving large-scale test problems and therefore is promising.