Approximating the joint spectral radius using a genetic algorithm framework ⋆ Chia-Tche Chang ∗ , Vincent D. Blondel ∗ ∗ ICTEAM Institute, Universit´ e catholique de Louvain Avenue Georges Lemaˆ ıtre, 4, B-1348 Louvain-la-Neuve, Belgium (e-mail: {chia-tche.chang, vincent.blondel}@uclouvain.be) Abstract: The joint spectral radius of a set of matrices is a measure of the maximal asymptotic growth rate of products of matrices in the set. This quantity appears in many applications but is known to be difficult to approximate. Several approaches to approximate the joint spectral radius involve the construction of long products of matrices, or the construction of an appropriate extremal matrix norm. In this article we present a brief overview of several recent approximation algorithms and introduce a genetic algorithm approximation method. This new method does not give any accuracy guarantees but is quite fast in comparison to other techniques. The performances of the different methods are compared and are illustrated on some benchmark examples. Our results show that, for large sets of matrices or matrices of large dimension, our genetic algorithm may provide better estimates or estimates for situations where these are simply too expensive to compute with other methods. As an illustration of this we compute in less than a minute a bound on the capacity of a code avoiding a given forbidden pattern that improves the bound currently reported in the literature. Keywords: Joint spectral radius, generalized spectral radius, genetic algorithms, product of matrices, dynamic systems, discrete-time systems. 1. INTRODUCTION The joint spectral radius (jsr) ρ(Σ) of a set of matrices Σ ⊂ R n×n is a quantity characterizing the maximal asymptotic growth rate of products of matrices in the set. More precisely, it is defined by: ρ(Σ) = lim t→∞ ρ t (Σ), (1) with: ρ t (Σ) = max  ‖M ‖ 1/t | M ∈ Σ t  , independently of the chosen submultiplicative matrix norm. Here, Σ t denotes the set of products of length t of matrices in Σ. In the particular case where Σ = {M }, the jsr is equal to the usual spectral radius, i.e., the largest magnitude of the eigenvalues. The jsr was introduced by Rota and Strang in Rota and Strang (1960) and has since then appeared in many applications such as stability of switched systems (Gurvits (1995)), continuity of wavelets (Daubechies and Lagarias (1992)), combinatorics and language theory (Jungers et al. (2009)), capacity of codes (Moision et al. (2001)), etc. The issue of approximating the jsr has been widely studied. The first algorithms proposed consisted in constructing products of increasing length and using (1) as upper ⋆ This paper presents research results of the Belgian Network DYSCO (Dynamical Systems, Control, and Optimization), funded by the Interuniversity Attraction Poles Programme, initiated by the Belgian State, Science Policy Office. The scientific responsibility rests with the authors. Chia-Tche Chang is a F.R.S.-FNRS Research Fellow (Belgian Fund for Scientific Research). bounds. Lower bounds can be obtained using the general- ized spectral radius ¯ ρ(Σ), defined by: ¯ ρ(Σ) = lim sup t→∞ ¯ ρ t (Σ), with: ¯ ρ t (Σ) = max  ρ(M ) 1/t | M ∈ Σ t  . Indeed, we have the following inequalities for all t (see Jungers (2009)): ¯ ρ t (Σ) ≤ ¯ ρ(Σ) ≤ ρ(Σ) ≤ ρ t (Σ). (2) The generalized spectral radius is equal to the jsr if Σ is a bounded (in particular, a finite) set (see Berger and Wang (1992)). This generalizes thus the well-known Gelfand formula for the spectral radius of a single matrix. Unfortunately, the sequence of bounds in (2) converges slowly to ρ(Σ) except in some particular cases, and so any approximation algorithm directly based on these in- equalities is bound to be inefficient. The problem of ap- proximating the jsr is indeed NP-Hard (see Tsitsiklis and Blondel (1997)). Several branch-and-bound methods have been designed, some of which even allowing arbitrarily accurate approximations (see Gripenberg (1996)), but this is thus at the expense of a high computation time. In order to speed up this procedure, one could try to find an appropriate norm that gives a fast convergence rate. In some cases one can even find a norm that is extremal for some set of matrices, that is, a norm such that the jsr is reached with t = 1. More precisely, we have the following definition: Definition 1. (Extremal norm). A norm ‖·‖ is said to be extremal for a set of matrices Σ if Preprints of the 18th IFAC World Congress Milano (Italy) August 28 - September 2, 2011 Copyright by the International Federation of Automatic Control (IFAC) 8681