Computing Robustness to Sparse Diagonal Uncertainty

Abstract

A new robustness metric $\nu$ was recently proposed as a substitute for the structured singular value $\mu$ to better capture robustness to sparse diagonal uncertainty, but its computation has remained an open problem. In this paper, we show that computing $\nu$ is equivalent to maximizing the spectral radius of a nonnegative matrix product. This equivalence allows us to transfer existing results on spectral-radius maximization to $\nu$, including a refined upper bound and conditions under which the bounds coincide. We then provide reformulations and structural results that enable an algorithm to solve the nonconvex optimization problem for nontrivial problems using global solvers. This also enables identification of the most fragile parts of the system. Our results do not yet provide a scalable solution for computing $\nu$, but they are an important step toward computability and interpretability.