Fast Interpolation-based Globality Certificates for Computing Kreiss Constants and the Distance to Uncontrollability
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The Kreiss constant of a matrix and the distance to uncontrollability can both be defined by global minimization problems of certain singular value functions in two real variables, which often have multiple local minima. The state-of-the-art for computing both of these quantities uses optimization to first find minimizers and then computes globality certificates to either assert that a given minimizer is a global one, or when not, provide new starting points for another round of optimization. These existing globality certificates are expensive to compute, which limits them to rather small problems, and for Kreiss constants, they also have high memory requirements. In this paper, we propose alternative globality certificates for both Kreiss constants and the distance to uncontrollability, based on the idea of building interpolant approximations to certain one-variable distance functions. Our new certificates can be orders of magnitude faster to compute, have relatively low memory requirements, and seem to be more reliable in practice.
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