Invex sets and preinvex functions on Riemannian manifolds

The concept of a geodesic invex subset of a Riemannian manifold is introduced. Geodesic invex and preinvex functions on a geodesic invex set with respect to particular maps are defined. The relation between geodesic invexity and preinvexity of functions on manifolds is studied. Using proximal subdifferential, certain results concerning extremum points of a non smooth geodesic preinvex function on a geodesic invex set are obtained. The main value inequality and the mean value theorem in invexity analysis are extended to Cartan-Hadamard manifolds.