308 research outputs found
Policy iteration for perfect information stochastic mean payoff games with bounded first return times is strongly polynomial
Recent results of Ye and Hansen, Miltersen and Zwick show that policy
iteration for one or two player (perfect information) zero-sum stochastic
games, restricted to instances with a fixed discount rate, is strongly
polynomial. We show that policy iteration for mean-payoff zero-sum stochastic
games is also strongly polynomial when restricted to instances with bounded
first mean return time to a given state. The proof is based on methods of
nonlinear Perron-Frobenius theory, allowing us to reduce the mean-payoff
problem to a discounted problem with state dependent discount rate. Our
analysis also shows that policy iteration remains strongly polynomial for
discounted problems in which the discount rate can be state dependent (and even
negative) at certain states, provided that the spectral radii of the
nonnegative matrices associated to all strategies are bounded from above by a
fixed constant strictly less than 1.Comment: 17 page
Tropical Kraus maps for optimal control of switched systems
Kraus maps (completely positive trace preserving maps) arise classically in
quantum information, as they describe the evolution of noncommutative
probability measures. We introduce tropical analogues of Kraus maps, obtained
by replacing the addition of positive semidefinite matrices by a multivalued
supremum with respect to the L\"owner order. We show that non-linear
eigenvectors of tropical Kraus maps determine piecewise quadratic
approximations of the value functions of switched optimal control problems.
This leads to a new approximation method, which we illustrate by two
applications: 1) approximating the joint spectral radius, 2) computing
approximate solutions of Hamilton-Jacobi PDE arising from a class of switched
linear quadratic problems studied previously by McEneaney. We report numerical
experiments, indicating a major improvement in terms of scalability by
comparison with earlier numerical schemes, owing to the "LMI-free" nature of
our method.Comment: 15 page
Tropical totally positive matrices
We investigate the tropical analogues of totally positive and totally
nonnegative matrices. These arise when considering the images by the
nonarchimedean valuation of the corresponding classes of matrices over a real
nonarchimedean valued field, like the field of real Puiseux series. We show
that the nonarchimedean valuation sends the totally positive matrices precisely
to the Monge matrices. This leads to explicit polyhedral representations of the
tropical analogues of totally positive and totally nonnegative matrices. We
also show that tropical totally nonnegative matrices with a finite permanent
can be factorized in terms of elementary matrices. We finally determine the
eigenvalues of tropical totally nonnegative matrices, and relate them with the
eigenvalues of totally nonnegative matrices over nonarchimedean fields.Comment: The first author has been partially supported by the PGMO Program of
FMJH and EDF, and by the MALTHY Project of the ANR Program. The second author
is sported by the French Chateaubriand grant and INRIA postdoctoral
fellowshi
Dobrushin ergodicity coefficient for Markov operators on cones, and beyond
The analysis of classical consensus algorithms relies on contraction
properties of adjoints of Markov operators, with respect to Hilbert's
projective metric or to a related family of seminorms (Hopf's oscillation or
Hilbert's seminorm). We generalize these properties to abstract consensus
operators over normal cones, which include the unital completely positive maps
(Kraus operators) arising in quantum information theory. In particular, we show
that the contraction rate of such operators, with respect to the Hopf
oscillation seminorm, is given by an analogue of Dobrushin's ergodicity
coefficient. We derive from this result a characterization of the contraction
rate of a non-linear flow, with respect to Hopf's oscillation seminorm and to
Hilbert's projective metric
Non-linear eigenvalue problems arising from growth maximization of positive linear dynamical systems
We study a growth maximization problem for a continuous time positive linear
system with switches. This is motivated by a problem of mathematical biology
(modeling growth-fragmentation processes and the PMCA protocol). We show that
the growth rate is determined by the non-linear eigenvalue of a max-plus
analogue of the Ruelle-Perron-Frobenius operator, or equivalently, by the
ergodic constant of a Hamilton-Jacobi (HJ) partial differential equation, the
solutions or subsolutions of which yield Barabanov and extremal norms,
respectively. We exploit contraction properties of order preserving flows, with
respect to Hilbert's projective metric, to show that the non-linear eigenvector
of the operator, or the "weak KAM" solution of the HJ equation, does exist. Low
dimensional examples are presented, showing that the optimal control can lead
to a limit cycle.Comment: 8 page
Minimax representation of nonexpansive functions and application to zero-sum recursive games
We show that a real-valued function on a topological vector space is
positively homogeneous of degree one and nonexpansive with respect to a weak
Minkowski norm if and only if it can be written as a minimax of linear forms
that are nonexpansive with respect to the same norm. We derive a representation
of monotone, additively and positively homogeneous functions on
spaces and on , which extend results of Kolokoltsov, Rubinov,
Singer, and others. We apply this representation to nonconvex risk measures and
to zero-sum games. We derive in particular results of representation and
polyhedral approximation for the class of Shapley operators arising from games
without instantaneous payments (Everett's recursive games)
Hypergraph conditions for the solvability of the ergodic equation for zero-sum games
The ergodic equation is a basic tool in the study of mean-payoff stochastic
games. Its solvability entails that the mean payoff is independent of the
initial state. Moreover, optimal stationary strategies are readily obtained
from its solution. In this paper, we give a general sufficient condition for
the solvability of the ergodic equation, for a game with finite state space but
arbitrary action spaces. This condition involves a pair of directed hypergraphs
depending only on the ``growth at infinity'' of the Shapley operator of the
game. This refines a recent result of the authors which only applied to games
with bounded payments, as well as earlier nonlinear fixed point results for
order preserving maps, involving graph conditions.Comment: 6 pages, 1 figure, to appear in Proc. 54th IEEE Conference on
Decision and Control (CDC 2015
Tropical Cramer Determinants Revisited
We prove general Cramer type theorems for linear systems over various
extensions of the tropical semiring, in which tropical numbers are enriched
with an information of multiplicity, sign, or argument. We obtain existence or
uniqueness results, which extend or refine earlier results of Gondran and
Minoux (1978), Plus (1990), Gaubert (1992), Richter-Gebert, Sturmfels and
Theobald (2005) and Izhakian and Rowen (2009). Computational issues are also
discussed; in particular, some of our proofs lead to Jacobi and Gauss-Seidel
type algorithms to solve linear systems in suitably extended tropical
semirings.Comment: 41 pages, 5 Figure
Tropical bounds for eigenvalues of matrices
We show that for all k = 1,...,n the absolute value of the product of the k
largest eigenvalues of an n-by-n matrix A is bounded from above by the product
of the k largest tropical eigenvalues of the matrix |A| (entrywise absolute
value), up to a combinatorial constant depending only on k and on the pattern
of the matrix. This generalizes an inequality by Friedland (1986),
corresponding to the special case k = 1.Comment: 17 pages, 1 figur
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