GENERALIZED INVEX SETS AND PREINVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS

Abstract

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