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

Understand

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.