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32
Maxplus algebra
, 2006
"... Maxplus algebra has been discovered more or less independently by several schools, in relation with various mathematical fields. This chapter is limited to finite dimensional linear algebra. For more information, the reader may consult the books [CG79, Zim81, CKR84, BCOQ92, KM97, GM02]. The collect ..."
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Cited by 37 (5 self)
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Maxplus algebra has been discovered more or less independently by several schools, in relation with various mathematical fields. This chapter is limited to finite dimensional linear algebra. For more information, the reader may consult the books [CG79, Zim81, CKR84, BCOQ92, KM97, GM02]. The collections of articles [MS92, Gun98, LM05] give a good idea of current developments.
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 ..."
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Cited by 27 (15 self)
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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
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 ..."
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Cited by 21 (2 self)
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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.
The MaxPlus Martin Boundary
 DOCUMENTA MATH.
, 2009
"... We develop an idempotent version of probabilistic potential theory. The goal is to describe the set of maxplus harmonic functions, which give the stationary solutions of deterministic optimal control problems with additive reward. The analogue of the Martin compactification is seen to be a general ..."
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Cited by 16 (9 self)
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We develop an idempotent version of probabilistic potential theory. The goal is to describe the set of maxplus harmonic functions, which give the stationary solutions of deterministic optimal control problems with additive reward. The analogue of the Martin compactification is seen to be a generalisation of the compactification of metric spaces using (generalised) Busemann functions. We define an analogue of the minimal Martin boundary and show that it can be identified with the set of limits of “almostgeodesics”, and also the set of (normalised) harmonic functions that are extremal in the maxplus sense. Our main result is a maxplus analogue of the Martin representation theorem, which represents harmonic functions by measures supported on the minimal Martin boundary. We illustrate it by computing the eigenvectors of a class of LaxOleinik semigroups with nondifferentiable Lagrangian: we relate extremal eigenvector to Busemann points of normed spaces.
On visualization scaling, subeigenvectors and Kleene stars in max algebra
 Linear Algebra Appl
"... The purpose of this paper is to investigate the interplay arising between max algebra, convexity and scaling problems. The latter, which have been studied in nonnegative matrix theory, are strongly related to max algebra. One problem is that of strict visualization scaling, defined as, for a given n ..."
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Cited by 15 (7 self)
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The purpose of this paper is to investigate the interplay arising between max algebra, convexity and scaling problems. The latter, which have been studied in nonnegative matrix theory, are strongly related to max algebra. One problem is that of strict visualization scaling, defined as, for a given nonnegative matrix A, a diagonal matrix X such that all elements of X −1 AX are less than or equal to the maximum cycle geometric mean of A, with strict inequality for the entries which do not lie on critical cycles. In this paper such scalings are described by means of the max algebraic subeigenvectors and Kleene stars of nonnegative matrices as well as by some concepts of convex geometry.
Reducible spectral theory with applications to the robustness of matrices in maxalgebra
, 2009
"... Let a ⊕ b = max(a, b) and a ⊗ b = a + b for a, b ∈ R: = R ∪ {−∞}. By maxalgebra we understand the analogue of linear algebra developed for the pair of operations (⊕,⊗), extended to matrices and vectors. The symbol Ak stands for the kth maxalgebraic power of a square matrix A. Let us denote by ε t ..."
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Cited by 11 (3 self)
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Let a ⊕ b = max(a, b) and a ⊗ b = a + b for a, b ∈ R: = R ∪ {−∞}. By maxalgebra we understand the analogue of linear algebra developed for the pair of operations (⊕,⊗), extended to matrices and vectors. The symbol Ak stands for the kth maxalgebraic power of a square matrix A. Let us denote by ε the maxalgebraic “zero ” vector, all the components of which are −∞. The maxalgebraic eigenvalueeigenvector problem is the following: Given A ∈ Rn×n, find all λ ∈ R and x ∈ Rn, x = ε, such that A⊗x = λ⊗x. Certain problems of scheduling lead to the following question: Given A ∈ Rn×n, is there a k such that Ak ⊗ x is a maxalgebraic eigenvector of A? If the answer is affirmative for every x = ε, then A is called robust. First, we give a complete account of the reducible maxalgebraic spectral theory, and then we apply it to characterize robust matrices.
Pairwise ranking: choice of method can produce arbitrarily different rank order
, 2011
"... We examine three methods for ranking by pairwise comparison: Principal Eigenvector, HodgeRank and Tropical Eigenvector. It is shown that the choice of method can produce arbitrarily different rank order.To be precise, for any two of the three methods, and for any pair of rankings of at least four ..."
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Cited by 6 (4 self)
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We examine three methods for ranking by pairwise comparison: Principal Eigenvector, HodgeRank and Tropical Eigenvector. It is shown that the choice of method can produce arbitrarily different rank order.To be precise, for any two of the three methods, and for any pair of rankings of at least four items, there exists a comparison matrix for the items such that the rankings found by the two methods are the prescribed ones. We discuss the implications of this result in practice, study the geometry of the methods, and state some open problems.
Series Expansions of Lyapunov Exponents and Forgetful Monoids
, 2000
"... We consider Lyapunov exponents of random iterates of monotone homogeneous maps. We assume that the images of some iterates are lines, with positive probability. Using this memoryloss property which holds generically for random products of matrices over the maxplus semiring, and in particular, for ..."
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Cited by 5 (0 self)
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We consider Lyapunov exponents of random iterates of monotone homogeneous maps. We assume that the images of some iterates are lines, with positive probability. Using this memoryloss property which holds generically for random products of matrices over the maxplus semiring, and in particular, for Tetrislike heaps of pieces models, we give a series expansion formula for the Lyapunov exponent, as a function of the probability law. In the case of rational probability laws, we show that the Lyapunov exponent is an analytic function of the parameters of the law, in a domain that contains the absolute convergence domain of a partition function associated to a special "forgetful" monoid, defined by generators and relations.
The Analytic Hierarchy Process, Max Algebra and Multiobjective Optimisation. arXiv:1207.6572v1
, 2012
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Generic Asymptotics Of Eigenvalues Using MinPlus Algebra
 in &quot;Proceedings of the Satellite Workshop on MaxPlus Algebras, IFAC SSSC’01
, 2001
"... We consider a square matrix A # whose entries have first order asymptotics of the form (A # ) i j when # goes to 0, for some a i j C and A i j R. We show that under a nondegeneracy condition, the order of magnitudes of the different eigenvalues of A # are given by minplus eigenvalues of m ..."
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Cited by 3 (1 self)
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We consider a square matrix A # whose entries have first order asymptotics of the form (A # ) i j when # goes to 0, for some a i j C and A i j R. We show that under a nondegeneracy condition, the order of magnitudes of the different eigenvalues of A # are given by minplus eigenvalues of minplus Schur complements built from A (A i j ), or equivalently by generalized minimal mean weights of circuits. This construction gives, in non singular cases, a graph interpretation to the slopes of the Newton polygon of the characteristic polynomial of A # . It explains the order of magnitudes of eigenvalues in the perturbation formula of Lidski, Visik and Ljusternik, and it allows us to solve some cases which are singular in this theory. Copyright 2001 IFAC Keywords: Perturbation theory, Maxplus algebra, Spectral theory, Graphs, Schur complements. 1.