Optimality Conditions On Riemannian Manifold Of Nonsmooth Optimization

Riemannian manifold of non-smooth problems and related issues has been one of the problems is to the hot spots of optimation, several important non-smooth analysis tools from the Euclidean space is extended to the Riemannian manifold by a number of scholars. This article innovative in the Riemannian manifold is given the concept of Penot generalized directional derivative, given the necessary conditions , a first-order and second-order sufficient condition of non-smooth unconstrained optimization on Riemannian manifold by use of Penot generalized directional derivative and the Clarke generalized gradient,promote the Lagrange theorem from the Euclidean space to the Riemannian manifold. Use of Manifold properties, given the nessary condition for equality constrained optimization problems, through the approach of non-smooth exact penalty function from inequality constrained optimization problems into unconstrained problems, given the necessary optimality conditions.Convex analysis theory is an important theoretical foundation for mathematical programming,this article based on the Riemannian manifold, additional the basic conditions for geodesic convex function, parallel extension the content and conclusions from convex analysis to Riemannian manifolds. A necessary and sufficient condition for unconstrained optimization problem is proved, and gives the necessary conditions for optimality of the equality constrained optimization problems, the inequality constrained optimization problems, and with equality and inequality constrained optimization problem, final,writer prove out the classical Lagrange multiplier theorem for the unity of conclusions.