Finding Extremal Complex Polytope Norms for Families of Real Matrices

Guglielmi, Nicola; Zennaro, Marino

doi:10.1137/080715718

Cited by 29 publications

(25 citation statements)

References 27 publications

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“…We shall also remark that there are powerful techniques for approximation of the JSR that do not use semidefinite programming, such as approaches based on computation of a polytopic norm [24], [25], [26]. Research in the computation of the JSR continues to be an active area and each novel technique has the potential to enhance not only our ability to solve certain instances more efficiently, but also our understanding of the relations between the different approaches.…”

mentioning

confidence: 99%

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Joint Spectral Radius and Path-Complete Graph Lyapunov Functions

Ahmadi¹,

Jungers²,

Parrilo³

et al. 2014

SIAM J. Control Optim.

105

133

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Abstract. We introduce the framework of path-complete graph Lyapunov functions for approximation of the joint spectral radius. The approach is based on the analysis of the underlying switched system via inequalities imposed among multiple Lyapunov functions associated to a labeled directed graph. Inspired by concepts in automata theory and symbolic dynamics, we define a class of graphs called path-complete graphs, and show that any such graph gives rise to a method for proving stability of the switched system. This enables us to derive several asymptotically tight hierarchies of semidefinite programming relaxations that unify and generalize many existing techniques such as common quadratic, common sum of squares, path-dependent quadratic, and maximum/minimumof-quadratics Lyapunov functions. We compare the quality of approximation obtained by certain classes of path-complete graphs including a family of dual graphs and all path-complete graphs with two nodes on an alphabet of two matrices. We derive approximation guarantees for several families of path-complete graphs, such as the De Bruijn graphs. This provides worst-case perfomance bounds for path-dependent quadratic Lyapunov functions and a constructive converse Lyapunov theorem for maximum/minimum-of-quadratics Lyapunov functions.

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mentioning

confidence: 99%

“…Motivated by the abundance of applications, there has been much work on efficient computation of the joint spectral radius; see e.g. [21], [11], [10], [36], [43], [38], [25], [26], [24] and references therein. Unfortunately, the negative results in the literature certainly restrict the horizon of possibilities.…”

mentioning

confidence: 99%

Joint Spectral Radius and Path-Complete Graph Lyapunov Functions

Ahmadi¹,

Jungers²,

Parrilo³

et al. 2014

SIAM J. Control Optim.

105

133

View full text Add to dashboard Cite

show abstract

“…Another geometric approach involving polytopes has been studied in [11] and in [7]. The method is based on balanced polytopes, which are polytopes P ⊂ K n such that there exists a finite set of vectors V = {v 1 , .…”

Section: Extremal Norm Constructionmentioning

confidence: 99%

“…More details can be found in [11]. This method requires a good initial guess, but if the candidate product is indeed optimal, then the algorithm stops after a finite number of steps and provides a certificate for the product optimality.…”

Section: Extremal Norm Constructionmentioning

confidence: 99%

An experimental study of approximation algorithms for the joint spectral radius

Chang

Blondel

2012

Numer Algor

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We describe several approximation algorithms for the joint spectral radius and compare their performance on a large number of test cases. The joint spectral radius of a set Σ of n × n matrices is the maximal asymptotic growth rate that can be obtained by forming products of matrices from Σ. This quantity is NP-hard to compute and appears in many areas, including in system theory, combinatorics and information theory. A dozen algorithms have been proposed this last decade for approximating the joint spectral radius but little is known about their practical efficiency. We overview these approximation algorithms and classify them in three categories: approximation obtained by examining long products, by building a specific matrix norm, and by using optimization-based techniques. All these algorithms are now implemented in a (freely available) MATLAB toolbox that was released in 2011. This tool-

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“…This idea was put to good use in [22,30] (polyhedral norms), [8,31,41] (Kronecker lifting), and [29] (sums of squares). The computational complexity of these methods grows exponentially with the dimension of the matrices.…”

Section: K}mentioning

confidence: 99%

Joint Spectral Characteristics of Matrices: A Conic Programming Approach

Protasov¹,

Jungers²,

Blondel³

2010

SIAM J. Matrix Anal. & Appl.

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Abstract. We propose a new method to compute the joint spectral radius and the joint spectral subradius of a set of matrices. We first restrict our attention to matrices that leave a cone invariant. The accuracy of our algorithm, depending on geometric properties of the invariant cone, is estimated. We then extend our method to arbitrary sets of matrices by a lifting procedure, and we demonstrate the efficiency of the new algorithm by applying it to several problems in combinatorics, number theory, and discrete mathematics.

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Finding Extremal Complex Polytope Norms for Families of Real Matrices

Cited by 29 publications

References 27 publications

Joint Spectral Radius and Path-Complete Graph Lyapunov Functions

Joint Spectral Radius and Path-Complete Graph Lyapunov Functions

An experimental study of approximation algorithms for the joint spectral radius

Joint Spectral Characteristics of Matrices: A Conic Programming Approach

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