Geodesic Convex Optimization: Differentiation on Manifolds, Geodesics, and Convexity

Convex optimization is a vibrant and successful area due to the existence of a variety of efficient algorithms that leverage the rich structure provided by convexity. Convexity of a smooth set or a function in a Euclidean space is defined by how it interacts with the standard differential structure in this space -- the Hessian of a convex function has to be positive semi-definite everywhere. However, in recent years, there is a growing demand to understand non-convexity and develop computational methods to optimize non-convex functions. Intriguingly, there is a type of non-convexity that disappears once one introduces a suitable differentiable structure and redefines convexity with respect to the straight lines, or {\em geodesics}, with respect to this structure. Such convexity is referred to as {\em geodesic convexity}. Interest in studying it arises due to recent reformulations of some non-convex problems as geodesically convex optimization problems over geodesically convex sets. Geodesics on manifolds have been extensively studied in various branches of Mathematics and Physics. However, unlike convex optimization, understanding geodesics and geodesic convexity from a computational point of view largely remains a mystery. The goal of this exposition is to introduce the first part of geodesic convex optimization -- geodesic convexity -- in a self-contained manner. We first present a variety of notions from differential and Riemannian geometry such as differentiation on manifolds, geodesics, and then introduce geodesic convexity. We conclude by showing that certain non-convex optimization problems such as computing the Brascamp-Lieb constant and the operator scaling problem have geodesically convex formulations.