Abstract: This paper gave the identification of convex function on Riemannian manifold by use of Penot generalized directional derivative and the Clarke generalized gradient,and gave a sufficient condition for the minimum point of convex programming on Riemannian manifolds,and Lagrange theorem,Lagrange sufficient condition,the Kuhn-Tucker theorem and sufficient condition of the minimum point of the equality constrained optimization problems,the inequality constrained optimization problems,and equality and inequality constrained optimization problem was given.

Key words: Riemannian manifold,Convex function,Optimality condition,Generalized gradient