Generalized Geodesic Convexity on Riemannian Manifolds

Abstract

We introduce log-preinvex and log-invex functions on a Riemannian manifold. Some properties and relationships of these functions are discussed. A characterization for the existence of a global minimum point of a mathematical programming problem is presented. Moreover, a mean value inequality under geodesic log-preinvexity is extended to Cartan-Hadamard manifolds.

Keywords

• geodesic log-invex function
• geodesic log-preinvex function
• global minimum
• mean value inequality
• Riemannian manifolds