Defining Geodesic Convexity
In my previous blog post, I discussed the notion of geodesic convexity, which is a generalization of ordinary convexity in Euclidean space to Riemannian manifolds. This is essential for disciplined optimization over manifolds; hence, it's very relevant to my research in optimization-based motion planning over manifolds.
Unfortunately, in the current literature, there are various definitions of geodesic convexity, with key differing properties. This is chiefly because if a geodesic connecting two points exists (it may not exist), it may not be unique, and it may not be minimizing. In this brief blog post, I'll discuss the various definitions that are used, and then explain why we use the one we did in our GGCS paper.
Nicholas Boumal's textbook An introduction to optimization on smooth manifolds is an excellent reference on the subject, and he discusses the various definitions of geodesic convexity in Section 11.3.
The first definition of geodesic convexity is the one used throughout Boumal's textbook. I'll be referring to it as "weakly" g-convex, to distinguish from the definition of g-convex we use in our paper. A set S is weakly g-convex if for every pair of points x and y in S, there is a geodesic segment connecting them that is contained in S. (Definition 11.2 of Boumal.) This is a very permissive definition for convexity, but is still suitable for optimization. However, it leads to a nasty side-effect: the intersection of two weakly g-convex sets may not be weakly g-convex!
The next two definitions handle this issue. A set S is totally convex if for every pair of points x and y in S, there is at least one geodesic segment connecting them, and all such segments are contained in S. (Definition 11.16 of Boumal.) In other sources, this definition may also include the requirement that S is nonempty, or that every pair of points in the manifold is connected by at least one geodesic ("geodesic completeness"). Finally, we reach the definition used for GGCS. A set S is strongly convex (which we call g-convex) if for every pair of points x and y in S, there is a unique minimizing geodesic connecting them, and that geodesic is contained in S.
Both strong convexity and total convexity directly imply weak convexity. But no other implications hold. For a certain special class of manifolds, called Cartan-Hadamard manifolds, all three definitions are equivalent. This includes Euclidean space with the canonical metric, and in that case, it also coincides with the ordinary definition of convexity.
Why We Use G-Convexity
To use geodesic convexity with the graph of convex sets framework, we desired two properties. For one, we needed the intersection of two convex sets to be convex, as we often need to constrain points to lie in the intersection of convex sets. For another, we needed two points in a convex set to uniquely describe a path between them (since we're optimization for trajectories). Thus, strong convexity (which we call g-convexity) was the appropriate definition for our purposes.