Eigenvalues of dynamic max-min systems

Olsder, Geert Jan

doi:10.1007/bf01805562

Cited by 76 publications

(40 citation statements)

References 9 publications

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“…. , α), which subsumes earlier results in the literature, such as [17]. The results of the present paper provide an independent proof of this fixed point theorem.…”

Section: Min-max Functionssupporting

confidence: 88%

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The duality theorem for min-max functions

Gaubert

Gunawardena

1998

Comptes Rendus de l'Académie des Sciences - Series I - Mathemat

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Abstract. The set of min-max functions F : R n → R n is the least set containing coordinate substitutions and translations and closed under pointwise max, min, and function composition. The Duality Conjecture asserts that the trajectories of a min-max function, considered as a dynamical system, have a linear growth rate (cycle time) and shows how this can be calculated through a representation of F as an infimum of max-plus linear functions. We prove the conjecture using an analogue of Howard's policy improvement scheme, carried out in a lattice ordered group of germs of affine functions at infinity. The methods yield an efficient algorithm for computing cycle times. LE THÉORÈME DE DUALITÉ POUR LES FONCTIONS MIN-MAXRésumé. L'ensemble des fonctions min-max F : R n → R n est le plus petit ensemble de fonctions qui contient les substitutions de coordonnées et les translations, et qui est stable par les opérations min et max (point par point), ainsi que par composition. La Conjecture de Dualité affirme que les trajectoires d'un système récurrent gouverné par une dynamique min-max ont un taux de croissance linéaire (temps de cycle), qui se calculeà partir d'une représentation de F comme infimum de fonctions max-plus linéaires. Nous montrons cette conjecture en utilisant une itération sur les politiquesà la Howard,à valeurs dans un groupe réticulé de germes de fonctions affinesà l'infini. On a ainsi un algorithme efficace pour calculer le temps de cycle.

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“…. , α), which subsumes earlier results in the literature, such as [17]. The results of the present paper provide an independent proof of this fixed point theorem.…”

Section: Min-max Functionssupporting

confidence: 88%

“…Min-max functions were introduced in [9] following earlier work on special cases in [17]. Functions which are homogeneous and nonexpansive in the supremum norm -so called topical functions-have appeared in the the work of several authors, in particular [3,16,11,14].…”

Section: Min-max Functionsmentioning

confidence: 99%

The duality theorem for min-max functions

Gaubert

Gunawardena

1998

Comptes Rendus de l'Académie des Sciences - Series I - Mathemat

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“…More generally, we shall say that a map from R d to R d is a homogeneous min-max self-map if its coordinates are of the form of Definition 4.1. This definition is inspired by the min-max functions considered by Gunawardena [Gun94] and Olsder [Ols91]. The terms of this form comprise the semidifferentials of the min-max functions considered there.…”

Section: A Policy Iteration Algorithm To Compute the Smallest Fixed Pmentioning

confidence: 99%

Computing the smallest fixed point of order-preserving nonexpansive mappings arising in positive stochastic games and static analysis of programs

Adjé¹,

Gaubert²,

Goubault³

2014

Journal of Mathematical Analysis and Applications

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The problem of computing the smallest fixed point of an orderpreserving map arises in the study of zero-sum positive stochastic games. It also arises in static analysis of programs by abstract interpretation. In this context, the discount rate may be negative. We characterize the minimality of a fixed point in terms of the nonlinear spectral radius of a certain semidifferential. We apply this characterization to design a policy iteration algorithm, which applies to the case of finite state and action spaces. The algorithm returns a locally minimal fixed point, which turns out to be globally minimal when the discount rate is nonnegative.

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“…Considérez la i-ième équation scalaire de (4) Alors on a À max < A < Àmin-Hormis le théorème 8, dont la preuve a été donnée en [9], peu de propriétés de (1), (2) sont réellement connues. Dans ce travail nous considérons les points suivants.…”

Section: La Théorie : Point De Départ Et Les Problèmes éTudiésunclassified

“…Comment définir de tels circuits? Ici on parle d'un comportement périodique si il existe des scalaires d, c, K, tel que [9] et [3]). Est-ce qu'une propriété similaire existe pour des systèmes min-max?…”

Section: La Théorie : Point De Départ Et Les Problèmes éTudiésunclassified

Analyse de systèmes min-max

Olsder¹

1996

RAIRO-Oper. Res.

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Eigenvalues of dynamic max-min systems

Cited by 76 publications

References 9 publications

The duality theorem for min-max functions

The duality theorem for min-max functions

Computing the smallest fixed point of order-preserving nonexpansive mappings arising in positive stochastic games and static analysis of programs

Analyse de systèmes min-max

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