Results 1  10
of
33
Methods and Applications of (max,+) Linear Algebra
 STACS'97, NUMBER 1200 IN LNCS, LUBECK
, 1997
"... Exotic semirings such as the "(max, +) semiring" (R # {#},max,+), or the "tropical semiring" (N #{+#},min,+), have been invented and reinvented many times since the late fifties, in relation with various fields: performance evaluation of manufacturing systems and discrete ev ..."
Abstract

Cited by 93 (30 self)
 Add to MetaCart
(Show Context)
Exotic semirings such as the "(max, +) semiring" (R # {#},max,+), or the "tropical semiring" (N #{+#},min,+), have been invented and reinvented many times since the late fifties, in relation with various fields: performance evaluation of manufacturing systems and discrete event system theory; graph theory (path algebra) and Markov decision processes, HamiltonJacobi theory; asymptotic analysis (low temperature asymptotics in statistical physics, large deviations, WKB method); language theory (automata with multiplicities) . Despite this apparent profusion, there is a small set of common, nonnaive, basic results and problems, in general not known outside the (max, +) community, which seem to be useful in most applications. The aim of this short survey paper is to present what we believe to be the minimal core of (max, +) results, and to illustrate these results by typical applications, at the frontier of language theory, control, and operations research (performance evaluation of...
Numerical computation of spectral elements in maxplus algebra
, 1998
"... We describe the specialization to maxplus algebra of Howard’s policy improvement scheme, which yields an algorithm to compute the solutions of spectral problems in the maxplus semiring. Experimentally, the algorithm shows a remarkable (almost linear) average execution time. ..."
Abstract

Cited by 55 (7 self)
 Add to MetaCart
We describe the specialization to maxplus algebra of Howard’s policy improvement scheme, which yields an algorithm to compute the solutions of spectral problems in the maxplus semiring. Experimentally, the algorithm shows a remarkable (almost linear) average execution time.
The duality theorem for minmax functions
 C. R. Acad.Sci.Paris.326, Série I
, 1998
"... Abstract. The set of minmax 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 minmax function, considered as a dynamical system, have ..."
Abstract

Cited by 45 (16 self)
 Add to MetaCart
(Show Context)
Abstract. The set of minmax 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 minmax 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 maxplus 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 MINMAX Résumé. L’ensemble des fonctions minmax 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 minmax ont un taux de croissance linéaire (temps de cycle), qui se calcule à partir d’une représentation de F comme infimum de fonctions maxplus linéaires. Nous montrons cette conjecture en utilisant une itération sur les politiques à la Howard, à valeurs dans un groupe réticulé de germes de fonctions affines à l’infini. On a ainsi un
Maxplus convex sets and functions
, 2003
"... We consider convex sets and functions over idempotent semifields, like the maxplus semifield. We show that if K is a conditionally complete idempotent semifield, with completion ¯ K, a convex function Kn → ¯ K which is lower semicontinuous in the order topology is the upper hull of supporting f ..."
Abstract

Cited by 44 (15 self)
 Add to MetaCart
We consider convex sets and functions over idempotent semifields, like the maxplus semifield. We show that if K is a conditionally complete idempotent semifield, with completion ¯ K, a convex function Kn → ¯ K which is lower semicontinuous in the order topology is the upper hull of supporting functions defined as residuated differences of affine functions. This result is proved using a separation theorem for closed convex subsets of Kn, which extends earlier results of Zimmermann, Samborski, and Shpiz.
A spectral theorem for convex monotone homogeneous maps
 In Proceedings of the Satellite Workshop on MaxPlus Algebras, IFAC SSSC’01
, 2001
"... Abstract. We consider convex maps f: R n → R n that are monotone (i.e., that preserve the product ordering of R n), and nonexpansive for the supnorm. This includes convex monotone maps that are additively homogeneous (i.e., that commute with the addition of constants). We show that the fixed point ..."
Abstract

Cited by 27 (15 self)
 Add to MetaCart
(Show Context)
Abstract. We consider convex maps f: R n → R n that are monotone (i.e., that preserve the product ordering of R n), and nonexpansive for the supnorm. This includes convex monotone maps that are additively homogeneous (i.e., that commute with the addition of constants). We show that the fixed point set of f, when it is nonempty, is isomorphic to a convex infsubsemilattice of R n, whose dimension is at most equal to the number of strongly connected components of a critical graph defined from the tangent affine maps of f. This yields in particular an uniqueness result for the bias vector of ergodic control problems. This generalizes results obtained previously by Lanery, Romanovsky, and Schweitzer and Federgruen, for ergodic control problems with finite state and action spaces, which correspond to the special case of piecewise affine maps f. We also show that the length of periodic orbits of f is bounded by the cyclicity of its critical graph, which implies that the possible orbit lengths of f are exactly the orders of elements of the symmetric group
On Rational Series in One Variable over certain Dioids
, 1994
"... We give a characterization of rational series in one variable over certain idempotent semirings (commutative dioids) such as for instance the "(max; +)" semiring. We show that a series is rational iff it is merge of ultimately geometric series. As a byproduct, we obtain a new proof of th ..."
Abstract

Cited by 26 (5 self)
 Add to MetaCart
We give a characterization of rational series in one variable over certain idempotent semirings (commutative dioids) such as for instance the "(max; +)" semiring. We show that a series is rational iff it is merge of ultimately geometric series. As a byproduct, we obtain a new proof of the periodicity theorem for powers of irreducible matrices and also some more general auxiliary results. We apply this characterization of rational series to the minimal realization problem for which we obtain an upper bound. We also obtain a lower bound in terms of minors in a symmetrized semiring.
Minplus methods in eigenvalue perturbation theory and generalised LidskiiVishikLjusternik theorem
, 2005
"... Abstract. We extend the perturbation theory of Viˇsik, Ljusternik and Lidskiĭ for eigenvalues of matrices, using methods of minplus algebra. We show that the asymptotics of the eigenvalues of a perturbed matrix is governed by certain discrete optimisation problems, from which we derive new perturba ..."
Abstract

Cited by 21 (2 self)
 Add to MetaCart
Abstract. We extend the perturbation theory of Viˇsik, Ljusternik and Lidskiĭ for eigenvalues of matrices, using methods of minplus algebra. We show that the asymptotics of the eigenvalues of a perturbed matrix is governed by certain discrete optimisation problems, from which we derive new perturbation formulæ, extending the classical ones and solving cases which where singular in previous approaches. Our results include general weak majorisation inequalities, relating leading exponents of eigenvalues of perturbed matrices and minplus analogues of eigenvalues. 1.
A nonlinear hierarchy for discrete event dynamical systems
, 1998
"... Dynamical systems of monotone homogeneous functions appear in Markov decision theory, in discrete event systems and in PerronFrobenius theory. We consider the case when these functions are given by finite algebraic expressions involving the operations max, min, convex hull, translations, and an inf ..."
Abstract

Cited by 21 (8 self)
 Add to MetaCart
Dynamical systems of monotone homogeneous functions appear in Markov decision theory, in discrete event systems and in PerronFrobenius theory. We consider the case when these functions are given by finite algebraic expressions involving the operations max, min, convex hull, translations, and an infinite family of binary operations, of which max and min are limit cases. We set up a hierarchy of monotone homogeneous functions that reflects the complexity of their defining algebraic expressions. For two classes of this hierarchy, we show that the trajectories of the corresponding dynamical systems admit a linear growth rate (cycle time). The first class generalizes the minmax functions considered previously in the literature. The second class generalizes both maxplus linear maps and ordinary nonnegative linear maps.
Linear Projector in the maxplus Algebra
 IEEE MEDITERRANEEN CONFERENCE ON CONTROL, CHYPRE
, 1997
"... In general semimodules, we say that the image of a linear operator B and the kernel of a linear operator C are direct factors if every equivalence class modulo C crosses the image of B at a unique point. For linear maps represented by matrices over certain idempotent semifields such as the (max, +) ..."
Abstract

Cited by 19 (10 self)
 Add to MetaCart
In general semimodules, we say that the image of a linear operator B and the kernel of a linear operator C are direct factors if every equivalence class modulo C crosses the image of B at a unique point. For linear maps represented by matrices over certain idempotent semifields such as the (max, +)semiring, we give necessary and sufficient conditions for an image and a kernel to be direct factors. We characterize the semimodules that admit a direct factor (or equivalently, the semimodules that are the image of a linear projector): their matrices have a ginverse. We give simple effective tests for all these properties, in terms of matrix residuation.
Approximating the spectral radius of sets of matrices in the maxalgebra is NPhard
 THE IEEE TRANS. ON AUTOMATIC CONTROL
, 1999
"... The lower and average spectral radii measure the minimal and average growth rates, respectively, of long products of matrices taken from a finite set. The logarithm of the average spectral radius is traditionally called the Lyapunov exponent. When one forms these products in the maxalgebra, we obtai ..."
Abstract

Cited by 17 (5 self)
 Add to MetaCart
The lower and average spectral radii measure the minimal and average growth rates, respectively, of long products of matrices taken from a finite set. The logarithm of the average spectral radius is traditionally called the Lyapunov exponent. When one forms these products in the maxalgebra, we obtain quantities that measure the performance of Discrete Event Systems. We show that approximating the lower and average maxalgebraic spectral radii is NPhard.