Fixed-point filter design and Riemannian geometry

Integer programming is a notoriously hard problem even in case of linear 0-1 programming. On the other hand, there are numerous problems in engineering and applied mathematics that require fast solvers, typically in an approximate sense. For example, fixed-point implementations of existing floating point algorithms can be reformulated using integer programming. Here, a certain notion of nearness is used to find the closest grid point to a given non-grid point. The underlying metric is typically highly warped and rounding to the nearest neighbor is very often doomed to fail. We present a novel approach that is a combination of three ideas. (1) Riemannian geometry is used to describe the underlying metric. (2) A well-known theorem by Minkowski suggests the existence of a good approximation in a certain neighborhood of the optimal floating point. Other neighborhoods are generated by using semi-definite Boolean optimization techniques. (3) Such neighborhoods are sampled in a highly efficient way to find the aforementioned approximation. Examples chosen from the field of digital signal processing explain some implementation details.