Geodesic Convex Functions

Abstract

In this part, the local-global property (every local optimum is global) of a smooth nonlinear optimization problem is studied in certain nonconvex cases, more precisely, if the set of feasible points is not convex (e.g., equality constraints) or the problem functions are not convex and generalized convex in the classical sense, or a Riemannian metric may be introduced related to a nonlinear coordinate representation (a nonlinear coordinate transformation) on the constraint manifold or to the improvement of the problem structure. The local-global property is in connection with the concept of generalized convexity (invexity) which plays an important role in mathematical optimization theory.