Optimisation Geometry: Is There a Central Role for Differential Geometry in Optimisation?
by Professor Jonathan Manton
Abstract: The theme of this talk is the role, in optimisation, differential geometry has
played, currently plays, and may in the future play.
The past decade has seen a surge of interest in optimisation on manifolds, driven by the fact that some real-world problems are best formulated on a manifold such as a Grassmann or Stiefel manifold, or even on a more structured space such as a Lie group. A basic question to ask, and one with a long history, is how the Newton method can be extended to a manifold. A partial answer, in the form of a general framework, has been given recently
( http://arxiv.org/abs/1207.5087 ). Interestingly, studying the more general context of optimisation on manifolds leads to new insight into the Euclidean case. Having reviewed this, the talk then focuses on what we call real-time optimisation. Here, geometry plays a prominent role because the class of cost functions as a whole can be treated as a fibre bundle. This research is in its infancy but it is speculated there will be close connections between geometry and the complexity of (real-time) optimisation; a link with Smale's
notion of topological complexity can already be made. Finally, the
question is asked whether the current dichotomy in optimisation, that convex problems are easy and all other problems are hard, is an artifact of focusing on the Euclidean case and neglecting other rich geometric structures.