Geodesically Convex Optimization

This lecture introduces the concept of geodesically convex optimization on a Riemannian manifold, defining geodesic convexity and providing examples. It covers the properties of geodesically convex sets, functions, and linear functions, as well as the conditions for a function to be geodesically convex or concave. The lecture also explores the relationship between convexity and minimization, presenting proofs and claims related to global and local minimizers. Additionally, it discusses the convexity of functions on geodesic segments and the implications of strong convexity. The study of Riemannian manifolds and continuous functions in the context of convex optimization is highlighted.