A Hybird Conjugate Gradient Method On Stiefel Manifold

In this paper,drawing from spectral conjugate gradient method and modified Fletcher-Reeves(FR)conjugate gradient method,we construct a new hybird conjuagte gradient method based on Fletcher-Reeves(FR)conjugate gradient method and Dai-Yuan(DY)conjugate gradient method.This new method combines modified FR conjugate gradient method with modified DY conjugate gradient method.The gradient of the objective function and directions generated by hybird conjugate gradient methods always satisfy the sufficient descent conditions and the properties are independent of the line research condition.The hybird conjugate gradient method is generated to Stiefel manifold and discuss its applications.Firstly,we combine with the difinition of generalized inner product in tangent space and derive the projection transformation of the objective function on Stiefel manifold,and we generalize our new method from Euclidean space to Stiefel manifold under two kinds of vector transports which based on generalized inner product and Euclidean inner product respectively.Meanwhile,the directions still satisfy the sufficient descent condition on Stiefel manifold.Under the non-monotonous line search condition,the convergence of our conjugate gradient method is proved combined with Zoutendijk condition.Finaly,we use our hybird conjugate gradient method to solve the linear eigenvalue problem on Stiefel manifold.The numerical experiments shows the efficiency and feasibility of the hybird conjugate gradient method.