A Study of Geodesic (E, F)-Preinvex Functions on Riemannian Manifolds

In this manuscript, we define the (E,F)-invex set, (E,F)-invex functions, and (E,F)-preinvex functions on Euclidean space, i.e., simply vector space. We extend these concepts on the Riemannian manifold. We also detail the fundamental properties of (E,F)-preinvex functions and provide some examples that illustrate the concepts well. We have established a relation between (E,F)-invex and (E,F)-preinvex functions on Riemannian manifolds. We introduce the conditions A and define the (E,F)-proximal sub-gradient. (E,F)-preinvex functions are also used to demonstrate their applicability in optimization problems. In the last, we establish the points of extrema of a non-smooth (E,F)-preinvex functions on (E,F)-invex subset of the Riemannian manifolds by using the (E,F)-proximal sub-gradient.