The Steepest Descent Method And The Newton Method With Step Size On Riemannian Manifold

This paper mainly considers the convergence of the steepest descent method with inexact line search and Newton method with inexact line search on Rieman-nian manifolds. First, we propose two algorithms:Line search method with Gold-stein condition on Riemannian manifolds and Line Search method with Wolfe condition on Riemannian manifolds. Under suitable conditions, we prove that both algorithms have global convergence and when the descent direction is the opposite gradient direction, both methods have linear convergence rate. Second, we propose the Newton method with Goldstein (Wolfe) condition on Riemannian manifolds. Under suitable conditions, we prove that the Newton method with Goldstein(Wolfe) condition has global convergence and superlinear convergence rate. Finally, some applications of the steepest descent method with inexact line search and Newton method with inexact line search on Riemannian manifolds in maximal correlation problem are discussed.