45 research outputs found
Aggregation and residuation
In this paper, we give a characterization of meet-projections in simple atomistic lattices that generalizes results on the aggregation of partitions in cluster analysis.Aggregation theory ; dependence relation ; meet projection ; partition ; residual map ; simple lattice
Model reference control for timed event graphs in dioids
This paper deals with feedback controller synthesis for timed event graphs in dioids. We discuss here the existence and the computation of a controller which leads to a closed-loop system whose behavior is as close as possible to the one of a given reference model and which delays as much as possible the input of tokens inside the (controlled) system. The synthesis presented here is mainly based on residuation theory results and some Kleene star properties
The max-plus finite element method for optimal control problems: further approximation results
We develop the max-plus finite element method to solve finite horizon
deterministic optimal control problems. This method, that we introduced in a
previous work, relies on a max-plus variational formulation, and exploits the
properties of projectors on max-plus semimodules. We prove here a convergence
result, in arbitrary dimension, showing that for a subclass of problems, the
error estimate is of order , where and
are the time and space steps respectively. We also show how the
max-plus analogues of the mass and stiffness matrices can be computed by convex
optimization, even when the global problem is non convex. We illustrate the
method by numerical examples in dimension 2.Comment: 13 pages, 2 figure
A max-plus finite element method for solving finite horizon deterministic optimal control problems
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, and exploits the
properties of projectors on max-plus semimodules. We obtain a nonlinear
discretized semigroup, corresponding to a zero-sum two players game. We give an
error estimate of order , for a
subclass of problems in dimension 1. We compare our method with a max-plus
based discretization method previously introduced by Fleming and McEneaney.Comment: 13 pages, 5 figure
The tropical analogue of polar cones
We study the max-plus or tropical analogue of the notion of polar: the polar
of a cone represents the set of linear inequalities satisfied by its elements.
We establish an analogue of the bipolar theorem, which characterizes all the
inequalities satisfied by the elements of a tropical convex cone. We derive
this characterization from a new separation theorem. We also establish variants
of these results concerning systems of linear equalities.Comment: 21 pages, 3 figures, example added, figures improved, notation
change
Aggregation and residuation
URL des Documents de travail : http://centredeconomiesorbonne.univ-paris1.fr/bandeau-haut/documents-de-travail/ To appear in Order 2012.Documents de travail du Centre d'Economie de la Sorbonne 2011.85 - ISSN : 1955-611XIn this paper, we give a characterization of meet-projections in simple atomistic lattices that generalizes results on the aggregation of partitions in cluster analysis.Dans ce papier, nous donnons une caractérisation des inf-projections dans un treillis atomistique simple, résultat qui généralise des résultats sur l'agrégation des partitions en théorie de la classification
The Fr\'echet Contingency Array Problem is Max-Plus Linear
In this paper we show that the so-called array Fr\'echet problem in
Probability/Statistics is (max, +)-linear. The upper bound of Fr\'echet is
obtained using simple arguments from residuation theory and lattice
distributivity. The lower bound is obtained as a loop invariant of a greedy
algorithm. The algorithm is based on the max-plus linearity of the Fr\'echet
problem and the Monge property of bivariate distribution
Residuation algebras with functional duals
We employ the theory of canonical extensions to study residuation algebras
whose associated relational structures are functional, i.e., for which the
ternary relations associated to the expanded operations admit an interpretation
as (possibly partial) functions. Providing a partial answer to a question of
Gehrke, we demonstrate that no universal first-order sentence in the language
of residuation algebras is equivalent to the functionality of the associated
relational structures