Abstract not found

this paper. The link between the weak convergence and the epigraph convergence used in convex analysis is done. The Cramer transform used in the large deviation literature is defined as the composition of the Laplace transform by the logarithm by the Fenchel transform. It transforms convolution into inf-convolution. Probabilistic results about processes with independent increments are then transformed into similar results on dynamic programming equations. Cramer transform gives new insight on the Hopf method used to compute explicit solutions of some HJB equations. It also explains the limit theorems obtained directly as the image of the classic limit theorems of probability. Bibliographic notes are given at the end of the paper. 2 Cost Measures and Decision Variables

We consider equations of the form Bf = g, where B is a Galois connection between lattices of functions. This includes the case where B is the Fenchel transform, or more generally a Moreau conjugacy. We characterize the existence and uniqueness of a solution f in terms of generalized subdifferentials, which extends K. Zimmermann’s covering theorem for max-plus linear equations, and give various illustrations.

Considering probability theory in which the semifield of positive real numbers is replaced by the idempotent semifield of real numbers (union infinity) endowed with the min and plus laws leads to a new formalism for optimization. Probability measures correspond to minimums of functions that we call cost measures, whereas random variables correspond to constraints on these optimization problems that we call decision variables. We review in this context basic notions of probability theory -- random variables, convergence of random variables, characteristic functions, L p norms. Whenever it is possible, results and definitions are stated in a general idempotent semiring.

The concept of the lower limit for vector-valued mappings is the main focus of this work. We first introduce a new definition of adequate lower and upper level sets for vector-valued mappings and establish some of their topological and geometrical properties. Characterization of semicontinuity for vector-valued mappings is thereafter presented. Then, we define the concept of vector lower limit, proving its lower semicontinuity, and furnishing in this way a concept of lower semicontinuous regularization for mappings taking their values in a complete lattice. The results obtained in the present work subsume the standard ones when the target space is finite dimensional. In particular, we recapture the scalar case with a new flexible proof. In addition, extensions of usual operations of lower and upper limits for vector-valued mappings are explored. The main result is finally applied to obtain a continuous D.C. decomposition of continuous D.C. mappings.

We develop an idempotent version of probabilistic potential theory. The goal is to describe the set of max-plus 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 “almost-geodesics”, and also the set of (normalised) harmonic functions that are extremal in the max-plus sense. Our main result is a max-plus 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 Lax-Oleinik semigroups with nondifferentiable Lagrangian: we relate extremal eigenvector to Busemann points of normed spaces.

Abstract — We generalise the Gärtner-Ellis theorem of large deviations theory. Our results allow us to derive large deviation type results in stochastic optimal control from the convergence of generalised logarithmic moment generating functions. They rely on the characterisation of the uniqueness of the solutions of max-plus linear equations. We give an illustration for a simple investment model, in which logarithmic moment generating functions represent risk-sensitive values. I.

We consider equations of the form Bf = g, where B is a Galois connection between lattices of functions. This includes the case where B is the Fenchel transform, or more generally a Moreau conjugacy. We characterize the existence and uniqueness of a solution f in terms of generalized subdifferentials, which extends K. Zimmermann's covering theorem for maxplus linear equations. c 2002 Academie des sciences/ Editions scientifiques et medicales Elsevier SAS Inversibilit e des correspondances de Galois fonctionnelles R esum e. On considere des equations de la forme Bf = g, ou B est une correspondance de Galois entre des treillis de fonctions, ce qui inclut le cas ou B est la transformation de Fenchel, ou plus generalement une conjugaison de Moreau. Nous caracterisons l'existence et l'unicite d'une solution f , en termes de sous-differentiels generalises, et etendons ainsi le theoreme de couverture de K. Zimmermann pour les equations lineaires max-plus. c 2002 Academie des sciences/ Editions scientifiques et medicales Elsevier SAS Version francaise abr eg ee Soient (F ; F ) et (G ; G ) deux ensembles partiellement ordonnes, et B : F ! G , C : G ! F . On dit que (B; C) est une correspondance de Galois si la propriete (7b) ci-dessous est satisfaite. L'application C, qui est unique, est notee B . On dit aussi que B et C sont des correspondances de Galois. On s'interesse au cas ou F = sci(Y; R) est l'ensemble des fonctions semi-continues inferieurement d'un espace topologique separe Y dans R, et ou G = R , pour un espace topologique separe X . On montre en particulier que B et B s'ecrivent sous la forme (8), ou b et b sont des applications X Y R ! R, et ou pour tout x 2 X; y 2 Y , (b(x; y; ); b (x; y; )) est une correspondance de Galois....

We revisit some results obtained recently in min-plus algebra following theideasofstatisticalmechanics. Computationofgeodesicsinagraphcanbedoneby min-plus matrix products. A min-plus matrixisseenasakindoffinitestatesmechanicalsystem. Theenergyofthis systemistheeigenvalueofitsmin-plusmatrix. Thegraphinterpretation oftheeigenvaluemaybeseenasakindofMariottelaw. The Cramer transformisintroducedbystatisticsonpopulationsof independentminpluslinearsystemsseenasakindofperfectgas. It transformsprobabilitycalculusinwhatwecalldecisioncalculus. Then, dynamicprogrammingequations, which are min-plus linearrecurrences, may be seen as min-plus Kolmogorov equations for Markov chains. An ergodic theorem for Bellman chains, analogue of Markov chains, is given. The min-plus counterparts of aggregation coherency and reversibility of Markov chains are then studied. They provide new decomposition results to compute solutions of dynamic programming equations.

.Werevisitsomeresultsobtainedrecentlyinmin-plusalge- brafollowingtheideasofstatisticalmechanics.Computationofgeodesicsinagraphcanbedonebymin -plusmatrixproducts.Amin-plusmatrixisseenasakindoffinitestatesmechanicalsystem. Theenergyofthis systemistheeigenvalueofitsmin-plusmatrix.Thegraphinterpretation oftheeigenvaluemaybeseenasakindofMariottelaw.TheCramer transformisintroducedbystatisticsonpopulationsofindependentminpluslinearsystemsseenasakindofperfectgas. Ittransformsprobabilitycalculusinwhatwecalldecisioncalculus. Then,dynamicprogrammingequations, whicharemin-pluslinearrecurrences,maybeseenas min-plusKolmogorovequationsforMarkovchains.Anergodictheorem forBellmanchains,analogueofMarkovchains,isgiven.Themin-plus counterpartsofaggregationcoherencyandreversibilityofMarkovchains arethenstudied.Theyprovidenewdecompositionresultstocomputesolutionsofdynamicprogrammingequations. 1.INTRODUCTION Min-plusalgebra,whichisthesetofrealnumbersendowedwiththemin andtheplusoperations,hasbeenstudiedfor...