Theory of cost measures : convergence of decision variables
نویسنده
چکیده
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,Lp norms. Whenever it is possible, results and definitions are stated in a general idempotent semiring. Key-words: Max-plus algebra, Dioid, Idempotent semiring, Idempotent measure, Decision variable, Fenchel transform, Optimization, Probability. (Résumé : tsvp) e-mail : [email protected] Unité de recherche INRIA Rocquencourt Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex (France) Téléphone : (33 1) 39 63 55 11 – Télécopie : (33 1) 39 63 53 30 Théorie des mesures de coût : sur la convergence des variables de décision Résumé : Si l’on considére la théorie des probabilités dans laquelle le demi-corps des réels positifs est remplacé par le demi-corps idempotent des réels (union l’infini) munis des lois min et plus, on obtient un nouveau formalisme pour l’optimisation. Les mesures de probabilité deviennent des minimums de fonctions que nous appellerons mesures de coût tandis que les variables aléatoires correspondent à des contraintes sur ces problèmes d’optimisation que nous appellerons variables de décision. Nous considérons dans ce contexte les notions de base de la théorie des probabilités – variables aléatoires, convergence de variables aléatoires, fonctions caracteristiques, normes Lp – et dès que cela est possible, nous établissons les théorèmes et les définitions dans un demi-anneau général. Mots-clé : Algebra max-plus, Dioı̈de, Demi-anneau idempotent, Mesure idempotente, Variable de decision, Transformée de Fenchel, Optimisation, Probabilité. Theory of cost measures : convergence of decision variables 3
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