Rank-one Characterization of Joint Spectral Radius of Finite Matrix Family

In this paper we study the joint/generalized spectral radius of a finite set of matrices in terms of its rank-one approximation by singular value decomposition. In the first part of the paper, we show that any finite set of matrices with at most one element's rank being greater than one satisfies the finiteness property under the framework of (invariant) extremal norm. Formula for the computation of joint/generalized spectral radius for this class of matrix family is derived. Based on that, in the second part, we further study the joint/generalized spectral radius of finite sets of general matrices through constructing rank-one approximations in terms of singular value decomposition, and some new characterizations of joint/generalized spectral radius are obtained. Several benchmark examples from applications as well as corresponding numerical computations are provided to illustrate the approach.Comment: Submitted for review on April 23, 2011; Final version is updated on Jan. 4, 2013 and Dec. 29, 201

Tropical polyhedra are equivalent to mean payoff games

We show that several decision problems originating from max-plus or tropical convexity are equivalent to zero-sum two player game problems. In particular, we set up an equivalence between the external representation of tropical convex sets and zero-sum stochastic games, in which tropical polyhedra correspond to deterministic games with finite action spaces. Then, we show that the winning initial positions can be determined from the associated tropical polyhedron. We obtain as a corollary a game theoretical proof of the fact that the tropical rank of a matrix, defined as the maximal size of a submatrix for which the optimal assignment problem has a unique solution, coincides with the maximal number of rows (or columns) of the matrix which are linearly independent in the tropical sense. Our proofs rely on techniques from non-linear Perron-Frobenius theory.Comment: 28 pages, 5 figures; v2: updated references, added background materials and illustrations; v3: minor improvements, references update

Multiorder, Kleene stars and cyclic projectors in the geometry of max cones

This paper summarizes results on some topics in the max-plus convex geometry, mainly concerning the role of multiorder, Kleene stars and cyclic projectors, and relates them to some topics in max algebra. The multiorder principle leads to max-plus analogues of some statements in the finite-dimensional convex geometry and is related to the set covering conditions in max algebra. Kleene stars are fundamental for max algebra, as they accumulate the weights of optimal paths and describe the eigenspace of a matrix. On the other hand, the approach of tropical convexity decomposes a finitely generated semimodule into a number of convex regions, and these regions are column spans of uniquely defined Kleene stars. Another recent geometric result, that several semimodules with zero intersection can be separated from each other by max-plus halfspaces, leads to investigation of specific nonlinear operators called cyclic projectors. These nonlinear operators can be used to find a solution to homogeneous multi-sided systems of max-linear equations. The results are presented in the setting of max cones, i.e., semimodules over the max-times semiring.Comment: 26 pages, a minor revisio

A new graph perspective on max-min fairness in Gaussian parallel channels

In this work we are concerned with the problem of achieving max-min fairness in Gaussian parallel channels with respect to a general performance function, including channel capacity or decoding reliability as special cases. As our central results, we characterize the laws which determine the value of the achievable max-min fair performance as a function of channel sharing policy and power allocation (to channels and users). In particular, we show that the max-min fair performance behaves as a specialized version of the Lovasz function, or Delsarte bound, of a certain graph induced by channel sharing combinatorics. We also prove that, in addition to such graph, merely a certain 2-norm distance dependent on the allowable power allocations and used performance functions, is sufficient for the characterization of max-min fair performance up to some candidate interval. Our results show also a specific role played by odd cycles in the graph induced by the channel sharing policy and we present an interesting relation between max-min fairness in parallel channels and optimal throughput in an associated interference channel.Comment: 41 pages, 8 figures. submitted to IEEE Transactions on Information Theory on August the 6th, 200