Generalized Invex Sets and Preinvex Functions on Riemannian Manifolds

In this paper, a geodesic α-invex subset of a Riemannian manifold is introduced. Geodesic α-invex and α-preinvex functions on a geodesic α-invex set with respect to particular maps are also defined. Further, we study the relationships between geodesic α-invex and α-preinvex functions on Riemannian manifolds. Some results of a non smooth geodesic α-preinvex function are also discussed using proximal subdifferentiation. At the end, mean value inequality and the mean value theorem in α-invexity analysis are extended to Cartan-Hadamard manifolds. Our results extend and generalize the known results in the literature.