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28
The Maslov Dequantization, Idempotent and Tropical Mathematics: a Very Brief Introduction
, 2005
"... ..."
Duality between Probability and Optimization
 In Proceedings of the workshop "Idempotency
, 1997
"... 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 ..."
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Cited by 17 (8 self)
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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 infconvolution. 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
Set Coverings and Invertibility of Functional Galois Connections
, 2004
"... 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 ..."
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Cited by 13 (3 self)
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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, and give various illustrations.
Theory of Cost Measures: Convergence of Decision Variables
 INRIA REPORT N
, 1995
"... 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 ..."
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Cited by 11 (6 self)
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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 MaxPlus Martin Boundary
 DOCUMENTA MATH.
, 2009
"... We develop an idempotent version of probabilistic potential theory. The goal is to describe the set of maxplus 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 general ..."
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Cited by 10 (5 self)
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We develop an idempotent version of probabilistic potential theory. The goal is to describe the set of maxplus 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 “almostgeodesics”, and also the set of (normalised) harmonic functions that are extremal in the maxplus sense. Our main result is a maxplus 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 LaxOleinik semigroups with nondifferentiable Lagrangian: we relate extremal eigenvector to Busemann points of normed spaces.
Lower Semicontinuous Regularization For VECTORVALUED MAPPINGS
, 2004
"... The concept of the lower limit for vectorvalued mappings is the main focus of this work. We first introduce a new definition of adequate lower and upper level sets for vectorvalued mappings and establish some of their topological and geometrical properties. Characterization of semicontinuity for v ..."
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Cited by 9 (1 self)
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The concept of the lower limit for vectorvalued mappings is the main focus of this work. We first introduce a new definition of adequate lower and upper level sets for vectorvalued mappings and establish some of their topological and geometrical properties. Characterization of semicontinuity for vectorvalued 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 vectorvalued mappings are explored. The main result is finally applied to obtain a continuous D.C. decomposition of continuous D.C. mappings.
MinPlus linearity and statistical mechanics
, 1996
"... We revisit some results obtained recently in minplus algebra following theideasofstatisticalmechanics. Computationofgeodesicsinagraphcanbedoneby minplus matrix products. A minplus matrixisseenasakindoffinitestatesmechanicalsystem. Theenergyofthis systemistheeigenvalueofitsminplusmatrix. Thegraph ..."
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Cited by 7 (1 self)
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We revisit some results obtained recently in minplus algebra following theideasofstatisticalmechanics. Computationofgeodesicsinagraphcanbedoneby minplus matrix products. A minplus matrixisseenasakindoffinitestatesmechanicalsystem. Theenergyofthis systemistheeigenvalueofitsminplusmatrix. Thegraphinterpretation oftheeigenvaluemaybeseenasakindofMariottelaw. The Cramer transformisintroducedbystatisticsonpopulationsof independentminpluslinearsystemsseenasakindofperfectgas. It transformsprobabilitycalculusinwhatwecalldecisioncalculus. Then, dynamicprogrammingequations, which are minplus linearrecurrences, may be seen as minplus Kolmogorov equations for Markov chains. An ergodic theorem for Bellman chains, analogue of Markov chains, is given. The minplus counterparts of aggregation coherency and reversibility of Markov chains are then studied. They provide new decomposition results to compute solutions of dynamic programming equations.
Solutions of maxplus linear equations and large deviations
 in "Proceedings of the joint 44th IEEE Conference on Decision and Control and European Control Conference ECC 2005 (CDCECC’05), Seville, Espagne", Also arXiv:math.PR/0509279, 2005, http://hal.inria.fr/inria00000218/en/. Maxplus 37
"... Abstract — We generalise the GärtnerEllis 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 ..."
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Cited by 7 (1 self)
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Abstract — We generalise the GärtnerEllis 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 maxplus linear equations. We give an illustration for a simple investment model, in which logarithmic moment generating functions represent risksensitive values. I.
Invertibility of Functional Galois Connections
, 2002
"... 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 subdifferenti ..."
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Cited by 5 (3 self)
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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 sousdifferentiels generalises, et etendons ainsi le theoreme de couverture de K. Zimmermann pour les equations lineaires maxplus. 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) cidessous 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 semicontinues 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....
The tropical analogue of polar cones
 Linear Algebra and Appl
, 2009
"... Abstract. We study the maxplus 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 conv ..."
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Cited by 5 (5 self)
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Abstract. We study the maxplus 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. 1.