Seminar • Algorithms and Complexity • Interior-point Methods on Manifolds: Theory and Applications

Please note: This seminar will take place in M3 4206 and online.

Harold Nieuwboer, PhD student
Korteweg-de Vries Institute and QuSoft
University of Amsterdam

Interior-point methods offer a highly versatile framework for convex optimization that is effective in theory and practice. A key notion in their theory is that of a self-concordant barrier. We give a suitable generalization of self-concordance to Riemannian manifolds and show that it gives the same structural results and guarantees as in the Euclidean setting, in particular local quadratic convergence of Newton's method. We analyze a path-following method for optimizing compatible objectives over a convex domain for which one has a self-concordant barrier, and obtain the standard complexity guarantees as in the Euclidean setting.

We provide general constructions of barriers, and show that on the space of positive-definite matrices and other symmetric spaces, the squared distance to a point is self-concordant. To demonstrate the versatility of our framework, we give algorithms with state-of-the-art complexity guarantees for the general class of scaling and non-commutative optimization problems, which have been of much recent interest, and we provide the first algorithms for efficiently finding high-precision solutions for computing minimal enclosing balls and geometric medians in nonpositive curvature.

This is joint work with Hiroshi Hirai and Michael Walter, based on https://arxiv.org/abs/2303.04771.

To attend this seminar in person, please go to M3 4206. You can also attend virtually using Zoom at https://uwaterloo.zoom.us/j/93639893659.