Recuit simulé partiel

Laurent Miclo

Stochastic Processes and their Applications, 1996, vol. 65, issue 2, 281-298

Abstract: Let (L[theta])[theta][epsilon]N be a family of elliptic diffusion operators on a compact and connected smooth manifold M, whose terms of first order are indexed by a parameter [theta] living in N, the n-dimensional torus. For each fixed [theta], we associate to L[theta] its invariant probability [mu][theta]. Let f be a smooth function on M x N and define for [theta] [epsilon] N, F([theta]) = [integral operator] f(x, [theta])[mu][theta](dx). We study partial simulated annealing algorithms (using only quite directly L[theta] and f) to find the global minima of F. This paper presents a new proof of the convergence of these algorithms, using n + 2 partial entropies associated naturally to the problem. This approach is simpler than the one exposed previously in (Miclo, 1994), which furthermore was restricted to the case n = 1, but we need to speed up much more the diffusion interacting with the simulated annealing algorithm (and in practice, this is embarrassing).Résumé On s'intéresse aux interactions entre des diffusions et des algorithmes de recuit simulé qui permettent de trouver les minima globaux (sur N, le tore de dimension n) de potentiels de la forme F([theta]) = [integral operator] f(x, [theta])[mu][theta](dx) où [mu][theta] est la probabilité invariante associée à une diffusion non dégénérée sur une variété riemannienne compacte et connexe, dont la dérive est paramétrée par [theta] [epsilon] N. On présente une nouvelle démonstration de la convergence de ces algorithmes, plus simple que celle restreinte au cas n = 1 que nous avions déjà exposée dans (Miclo, 1994), car basée sur l'étude des évolutions conjointes de n + 2 entropies partielles associées naturellement au problème. Cependant, cette méthode exige que l'on accélère fortement la diffusion adjointe, ce qui en pratique est génant.

Keywords: Diffusions; interacting; with; simulated; annealing; processes; Evolution; of; partial; entropies (search for similar items in EconPapers)
Date: 1996
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