ИСТИНА

: The problem of minimizing and of maximizing the spectral radius on a set of matrices is notoriously hard even for relatively ``good’’ sets (for example, on convex sets of positive matrices). Nevertheless, for some sets of a special structure that problem admits an efficient solution. We consider sets of nonnegative matrices with independent row uncertainties. Remarkable spectral properties of such sets were discovered by V.Blondel and Y.Nesterov in 2009, although their applications in the game theory, in the theory of asynchronous systems, etc. were studied before. In 2013, in a joint paper with Y.Nesterov we suggested two methods for minimizing/maximizing the spectral radius over sets of matrices with independent row uncertainties. One of them, the so-called ``spectral simplex method’’ demonstrated its surprising efficiency. We show possible ways to adopt this method to sparse matrices and to non-poyhedral uncertainty sets, present some theoretical estimates and discuss applications to the graph theory, dynamical systems, mathematical economics, etc. One surprising link to a well-known problem of indicators in the population dynamics will also be shown.