A Study of Geodesic (E, F)-Preinvex Functions on Riemannian Manifolds - Ehtesham Akhter; Mohd Bilal; Musavvir Ali (Mathematics)
In this manuscript, we define the <semantics>(E,F)</semantics>-invex set, <semantics>(E,F)</semantics>-invex functions, and <semantics>(E,F)</semantics>-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 <semantics>(E,F)</semantics>-preinvex functions and provide some examples that illustrate the concepts well. We have established a relation between <semantics>(E,F)</semantics>-invex and <semantics>(E,F)</semantics>-preinvex functions on Riemannian manifolds. We introduce the conditions <semantics>A</semantics> and define the <semantics>(E,F)</semantics>-proximal sub-gradient. <semantics>(E,F)</semantics>-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 <semantics>(E,F)</semantics>-preinvex functions on <semantics>(E,F)</semantics>-invex subset of the Riemannian manifolds by using the <semantics>(E,F)</semantics>-proximal sub-gradient.