Sufficient descent nonlinear conjugate gradient methods with conjugacy condition

In this paper, by the use of Gram-Schmidt orthogonalization, we propose a class of modified conjugate gradient methods. The methods are modifications of the well-known conjugate gradient methods including the PRP, the HS, the FR and the DY methods. A common property of the modified methods is that the direction generated by any member of the class satisfies g_k^Td_k=-||g_k||²g_{k}^{T}d_k=-|g_k|^2. Moreover, if line search is exact, the modified method reduces to the standard conjugate gradient method accordingly. In particular, we study the modified YT and YT+ methods. Under suitable conditions, we prove the global convergence of these two methods. Extensive numerical experiments show that the proposed methods are efficient for the test problems from the CUTE library.