4,376 research outputs found
Max-plus (A,B)-invariant spaces and control of timed discrete event systems
The concept of (A,B)-invariant subspace (or controlled invariant) of a linear
dynamical system is extended to linear systems over the max-plus semiring.
Although this extension presents several difficulties, which are similar to
those encountered in the same kind of extension to linear dynamical systems
over rings, it appears capable of providing solutions to many control problems
like in the cases of linear systems over fields or rings. Sufficient conditions
are given for computing the maximal (A,B)-invariant subspace contained in a
given space and the existence of linear state feedbacks is discussed. An
application to the study of transportation networks which evolve according to a
timetable is considered.Comment: 24 pages, 1 Postscript figure, proof of Lemma 1 and some references
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Cyclic classes and attraction cones in max algebra
In max algebra it is well-known that the sequence A^k, with A an irreducible
square matrix, becomes periodic at sufficiently large k. This raises a number
of questions on the periodic regime of A^k and A^k x, for a given vector x.
Also, this leads to the concept of attraction cones in max algebra, by which we
mean sets of vectors whose ultimate orbit period does not exceed a given
number. This paper shows that some of these questions can be solved by matrix
squaring (A,A^2,A^4, ...), analogously to recent findings concerning the orbit
period in max-min algebra. Hence the computational complexity of such problems
is of the order O(n^3 log n). The main idea is to apply an appropriate diagonal
similarity scaling A -> X^{-1}AX, called visualization scaling, and to study
the role of cyclic classes of the critical graph. For powers of a visualized
matrix in the periodic regime, we observe remarkable symmetry described by
circulants and their rectangular generalizations. We exploit this symmetry to
derive a concise system of equations for attraction cpne, and we present an
algorithm which computes the coefficients of the system.Comment: 38 page
The level set method for the two-sided eigenproblem
We consider the max-plus analogue of the eigenproblem for matrix pencils
Ax=lambda Bx. We show that the spectrum of (A,B) (i.e., the set of possible
values of lambda), which is a finite union of intervals, can be computed in
pseudo-polynomial number of operations, by a (pseudo-polynomial) number of
calls to an oracle that computes the value of a mean payoff game. The proof
relies on the introduction of a spectral function, which we interpret in terms
of the least Chebyshev distance between Ax and lambda Bx. The spectrum is
obtained as the zero level set of this function.Comment: 34 pages, 4 figures. Changes with respect to the previous version: we
explain relation to mean-payoff games and discrete event systems, and show
that the reconstruction of spectrum is pseudopolynomia
The Analytic Hierarchy Process, Max Algebra and Multi-objective Optimisation
The Analytic Hierarchy Process (AHP) is widely used for decision making
involving multiple criteria. Elsner and van den Driessche introduced a
max-algebraic approach to the single criterion AHP. We extend this to the
multi-criteria AHP, by considering multi-objective generalisations of the
single objective optimisation problem solved in these earlier papers. We relate
the existence of globally optimal solutions to the commutativity properties of
the associated matrices; we relate min-max optimal solutions to the generalised
spectral radius; and we prove that Pareto optimal solutions are guaranteed to
exist.Comment: 1 figur
Multiorder, Kleene stars and cyclic projectors in the geometry of max cones
This paper summarizes results on some topics in the max-plus convex geometry,
mainly concerning the role of multiorder, Kleene stars and cyclic projectors,
and relates them to some topics in max algebra. The multiorder principle leads
to max-plus analogues of some statements in the finite-dimensional convex
geometry and is related to the set covering conditions in max algebra. Kleene
stars are fundamental for max algebra, as they accumulate the weights of
optimal paths and describe the eigenspace of a matrix. On the other hand, the
approach of tropical convexity decomposes a finitely generated semimodule into
a number of convex regions, and these regions are column spans of uniquely
defined Kleene stars. Another recent geometric result, that several semimodules
with zero intersection can be separated from each other by max-plus halfspaces,
leads to investigation of specific nonlinear operators called cyclic
projectors. These nonlinear operators can be used to find a solution to
homogeneous multi-sided systems of max-linear equations. The results are
presented in the setting of max cones, i.e., semimodules over the max-times
semiring.Comment: 26 pages, a minor revisio
Tropical polar cones, hypergraph transversals, and mean payoff games
We discuss the tropical analogues of several basic questions of convex
duality. In particular, the polar of a tropical polyhedral cone represents the
set of linear inequalities that its elements satisfy. We characterize the
extreme rays of the polar in terms of certain minimal set covers which may be
thought of as weighted generalizations of minimal transversals in hypergraphs.
We also give a tropical analogue of Farkas lemma, which allows one to check
whether a linear inequality is implied by a finite family of linear
inequalities. Here, the certificate is a strategy of a mean payoff game. We
discuss examples, showing that the number of extreme rays of the polar of the
tropical cyclic polyhedral cone is polynomially bounded, and that there is no
unique minimal system of inequalities defining a given tropical polyhedral
cone.Comment: 27 pages, 6 figures, revised versio
Quasi-cluster algebras from non-orientable surfaces
With any non necessarily orientable unpunctured marked surface (S,M) we
associate a commutative algebra, called quasi-cluster algebra, equipped with a
distinguished set of generators, called quasi-cluster variables, in bijection
with the set of arcs and one-sided simple closed curves in (S,M). Quasi-cluster
variables are naturally gathered into possibly overlapping sets of fixed
cardinality, called quasi-clusters, corresponding to maximal non-intersecting
families of arcs and one-sided simple closed curves in (S,M). If the surface S
is orientable, then the quasi-cluster algebra is the cluster algebra associated
with the marked surface (S,M) in the sense of Fomin, Shapiro and Thurston. We
classify quasi-cluster algebras with finitely many quasi-cluster variables and
prove that for these quasi-cluster algebras, quasi-cluster monomials form a
linear basis. Finally, we attach to (S,M) a family of discrete integrable
systems satisfied by quasi-cluster variables associated to arcs in the
quasi-cluster algebra and we prove that solutions of these systems can be
expressed in terms of cluster variables of type A.Comment: 38 pages, 14 figure
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