### V5E4 - Selected Topics in Scientific Computing (Summer Term 2017)

## Riemannian Optimization

Due to its wide range of applications, optimization on manifolds, or Riemannian optimization, is a
fast growing research topic in the field of nonlinear optimization.
In considering an optimization problem on a Riemannian manifold, fundamental challenges occur due to the nonlinear nature of the manifold.
Indeed, any method which is based on the linear structure of the spaces-that is, virtually any method known in optimization- breaks down.
A manifold is locally isomorphic to a linear space via chart maps.
For this reason one might wonder whether it is enough to simply work in a chart
domain and use classical linear algorithms. Unfortunately, such an approach does
in general not lead to useful algorithms: First of all symmetries of the underlying
Riemannian manifold will in general not be respected by such algorithms. Moreover,
there often does not exist a canonical useful representation for charts. But more
fundamentally, localizing to a chart inevitably leads to distortions in the metric
which leads to much slower convergence. Finally, we are interested in establishing
global convergence of the algorithms. Such a property is clearly out
of reach by working in a chart domain, which is a local procedure. It is also
clear that the mathematical analysis of global convergence requires global
arguments from Riemannian geometry and cannot be deduced from corresponding
linear results. For all those reasons, together with the evident need for efficient
and reliable Riemannian optimization algorithms, the construction of
intrinsic algorithms has become a thriving area of research in the past few years. Intended topics include:

- Smooth optimization methods such as the steepest descent method, Newton's method, trust-region
methods and conjugate gradient methods
- Derivative-free schemes such as the Powell's method, direct local search methods and particle swarm optimization algorithms

Recommended literature:

- P.-A. Absil, R. Mahony, and R. Sepulchre, Optimization Algorithms on Matrix Manifolds, Princeton University Press , 2008.
- C. Udriste, Convex Functions and Optimization Methods on Riemannian Manifolds, Mathematics and its Applications, 297. Kluwer Academic Publishers Group, Dordrecht, 1994.

Prerequisites: analysis and linear algebra (Bachelor level), smooth optimization methods (Steepest descent methods, Newton methods etc.).

Date & time: | Wednesday | 14:15–15:45 Uhr, | Wegelerstr. 6, Room 5.002 |

Begin: | Wednesday, | 19.04.2017 |