Hourglass alternative and the finiteness conjecture for the spectral characteristics of sets of non-negative matrices

Abstract

Recently Blondel, Nesterov and Protasov proved [1,2] that the finiteness conjecture holds for the generalized and the lower spectral radii of the sets of non-negative matrices with independent row/column uncertainty. We show that this result can be obtained as a simple consequence of the so-called hourglass alternative used in [3], by the author and his companions, to analyze the minimax relations between the spectral radii of matrix products. Axiomatization of the statements that constitute the hourglass alternative makes it possible to define a new class of sets of positive matrices having the finiteness property, which includes the sets of non-negative matrices with independent row uncertainty. This class of matrices, supplemented by the zero and identity matrices, forms a semiring with the Minkowski operations of addition and multiplication of matrix sets, which gives means to construct new sets of non-negative matrices possessing the finiteness property for the generalized and the lower spectral radii.

The Operator Approach to Entropy Games

“…Chen and Han [CH14] developed a related convex programming approach to solve the entropy maximization problem for Markov chains with uncertain parameters. We also note that the present Collatz-Wielandt approach, building on [AGN11], yields an alternative to the approach of [ACD + 16] using the "hourglass alternative" of [Koz15] to produce concise certificates allowing one to bound the value of entropy games. By comparison with [ACD + 16], a essential difference is the use of o-minimality arguments: these are needed because we study the more precise version of the game, in which the initial state is fixed.…”