On the Closest Stable/Unstable Nonnegative Matrix and Related Stability Radii

We consider the problem of computing the closest stable/unstable nonnegative matrix to a given real matrix. The distance between matrices is measured in the Frobenius norm. The problem is addressed for two types of stability: the Schur stability (the matrix is stable if its spectral radius is smaller than one) and the Hurwitz stability (the matrix is stable if its spectral abscissa is negative). We show that the closest unstable matrix can always be explicitly found. The problem of computing the closest stable matrix to a nonnegative matrix is a hard problem even if the stable matrix is not constrained to be nonnegative. Adding the nonnegativity constraint makes the problem even more difficult. For the closest stable matrix, we present an iterative algorithm which converges to a local minimum with a linear rate. It is shown that the total number of local minima can be exponential in the dimension. Numerical results and the complexity estimates are presented. Read More: https://epubs.siam.org/doi/10.1137/18M1172454