Hybrid Expansion-contraction: a Robust Scaleable Method for Approximating the H∞ Norm

We present a new scaleable algorithm for approximating the H∞ norm, an important robust stability measure for linear dynamical systems with input and output. Our spectral value set based method uses a novel hybrid expansion-contraction scheme that, under reasonable assumptions, is guaranteed to converge to a stationary point of the optimization problem defining the H∞ norm, and, in practice, typically returns local or global maximizers. We prove that the hybrid expansion-contraction method has a quadratic rate of convergence that is also confirmed in practice. In comprehensive numerical experiments, we show that our new method is not only robust but exceptionally fast, successfully completing a large-scale test set 25 times faster than an earlier method [N. Guglielmi, M. Gürbüzbalaban, and M.L. Overton, Fast approximation of the H∞ norm via optimization over spectral value sets, SIAM Journal on Matrix Analysis and Applications 34:709–737, 2013], which occasionally breaks down far from a stationary point of the underlying optimization problem.