113 research outputs found
Characteristic polynomials of supertropical matrices
Supertropical matrix theory was investigated in [6], whose terminology we
follow. In this work we investigate eigenvalues, characteristic polynomials and
coefficients of characteristic polynomials of supertropical matrices and their
powers, and obtain the analog to the basic property of matrices that any power
of an eigenvalue of a matrix is an eigenvalue of the corresponding power of the
matrix.Comment: Department of Mathematics, Bar-Ilan University, Ramat Gan 52900,
Israel. Email: [email protected]. This paper is part of the author's
Ph.D thesis, which was written at Bar-Ilan University under the supervision
of Prof. L. H. Rowe
Tropical polyhedra are equivalent to mean payoff games
We show that several decision problems originating from max-plus or tropical
convexity are equivalent to zero-sum two player game problems. In particular,
we set up an equivalence between the external representation of tropical convex
sets and zero-sum stochastic games, in which tropical polyhedra correspond to
deterministic games with finite action spaces. Then, we show that the winning
initial positions can be determined from the associated tropical polyhedron. We
obtain as a corollary a game theoretical proof of the fact that the tropical
rank of a matrix, defined as the maximal size of a submatrix for which the
optimal assignment problem has a unique solution, coincides with the maximal
number of rows (or columns) of the matrix which are linearly independent in the
tropical sense. Our proofs rely on techniques from non-linear Perron-Frobenius
theory.Comment: 28 pages, 5 figures; v2: updated references, added background
materials and illustrations; v3: minor improvements, references update
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 max-plus finite element method for solving deterministic optimal control problems: basic properties and convergence analysis
We introduce a max-plus analogue of the Petrov-Galerkin finite element method
to solve finite horizon deterministic optimal control problems. The method
relies on a max-plus variational formulation. We show that the error in the sup
norm can be bounded from the difference between the value function and its
projections on max-plus and min-plus semimodules, when the max-plus analogue of
the stiffness matrix is exactly known. In general, the stiffness matrix must be
approximated: this requires approximating the operation of the Lax-Oleinik
semigroup on finite elements. We consider two approximations relying on the
Hamiltonian. We derive a convergence result, in arbitrary dimension, showing
that for a class of problems, the error estimate is of order or , depending on the
choice of the approximation, where and are respectively the
time and space discretization steps. We compare our method with another
max-plus based discretization method previously introduced by Fleming and
McEneaney. We give numerical examples in dimension 1 and 2.Comment: 31 pages, 11 figure
Stability and convergence in discrete convex monotone dynamical systems
We study the stable behaviour of discrete dynamical systems where the map is
convex and monotone with respect to the standard positive cone. The notion of
tangential stability for fixed points and periodic points is introduced, which
is weaker than Lyapunov stability. Among others we show that the set of
tangentially stable fixed points is isomorphic to a convex inf-semilattice, and
a criterion is given for the existence of a unique tangentially stable fixed
point. We also show that periods of tangentially stable periodic points are
orders of permutations on letters, where is the dimension of the
underlying space, and a sufficient condition for global convergence to periodic
orbits is presented.Comment: 36 pages, 1 fugur
Factorization of Tropical Matrices
In contrast to the situation in classical linear algebra, not every
tropically non-singular matrix can be factored into a product of tropical
elementary matrices. We do prove the factorizability of any tropically
non-singular 2x2 matrix and, relating to the existing Bruhat decomposition,
determine which 3x3 matrices are factorizable. Nevertheless, there is a closure
operation, obtained by means of the tropical adjoint, which is always
factorizable, generalizing the decomposition of the closure operation * of a
matrix.Comment: This paper is part of the author's PhD thesis, which was written at
Bar-Ilan University under the supervision of Prof. L. H. Rowe
Combinatorial simplex algorithms can solve mean payoff games
A combinatorial simplex algorithm is an instance of the simplex method in
which the pivoting depends on combinatorial data only. We show that any
algorithm of this kind admits a tropical analogue which can be used to solve
mean payoff games. Moreover, any combinatorial simplex algorithm with a
strongly polynomial complexity (the existence of such an algorithm is open)
would provide in this way a strongly polynomial algorithm solving mean payoff
games. Mean payoff games are known to be in NP and co-NP; whether they can be
solved in polynomial time is an open problem. Our algorithm relies on a
tropical implementation of the simplex method over a real closed field of Hahn
series. One of the key ingredients is a new scheme for symbolic perturbation
which allows us to lift an arbitrary mean payoff game instance into a
non-degenerate linear program over Hahn series.Comment: v1: 15 pages, 3 figures; v2: improved presentation, introduction
expanded, 18 pages, 3 figure
The tropical shadow-vertex algorithm solves mean payoff games in polynomial time on average
We introduce an algorithm which solves mean payoff games in polynomial time
on average, assuming the distribution of the games satisfies a flip invariance
property on the set of actions associated with every state. The algorithm is a
tropical analogue of the shadow-vertex simplex algorithm, which solves mean
payoff games via linear feasibility problems over the tropical semiring
. The key ingredient in our approach is
that the shadow-vertex pivoting rule can be transferred to tropical polyhedra,
and that its computation reduces to optimal assignment problems through
Pl\"ucker relations.Comment: 17 pages, 7 figures, appears in 41st International Colloquium, ICALP
2014, Copenhagen, Denmark, July 8-11, 2014, Proceedings, Part
Minimal half-spaces and external representation of tropical polyhedra
We give a characterization of the minimal tropical half-spaces containing a
given tropical polyhedron, from which we derive a counter example showing that
the number of such minimal half-spaces can be infinite, contradicting some
statements which appeared in the tropical literature, and disproving a
conjecture of F. Block and J. Yu. We also establish an analogue of the
Minkowski-Weyl theorem, showing that a tropical polyhedron can be equivalently
represented internally (in terms of extreme points and rays) or externally (in
terms of half-spaces containing it). A canonical external representation of a
polyhedron turns out to be provided by the extreme elements of its tropical
polar. We characterize these extreme elements, showing in particular that they
are determined by support vectors.Comment: 19 pages, 4 figures, example added with a new figure, figures
improved, references update
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