Optimization on manifolds is about minimizing a cost function over a smooth manifold, such as spheres, low-rank matrices, orthonormal frames, rotations, etc. I will present the basic framework as well as some of the more general convergence results, including recent complexity results. (Toolbox: http://www.manopt.org)

This is non-convex optimization; there are no global optimality guarantees in general. Nevertheless, I will discuss a class of problems related to semidefinite relaxations for Max-Cut and community detection in the stochastic block model for which standard optimization algorithms on manifolds converge to global optimizers. This is part of a growing set of applications on manifolds with stronger theoretical backup.

Joint work with P.-A. Absil, A. Bandeira, C. Cartis and V. Voroninski.