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Recuit simulé partiel. (Partial simulated annealing). (French) Zbl 0889.60073

Summary: Let \((L_{\theta })_{\theta \in N}\) 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 \(M \times N\) and define for \(\theta \in N\), \(F(\theta ) = \int f(x,\theta ) \mu _{\theta } (dx)\). 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 by L. Miclo [Stochastics Stochastics Rep. 46, No. 3/4, 193-268 (1994; Zbl 0826.60068)], 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).

MSC:

60H30 Applications of stochastic analysis (to PDEs, etc.)
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)

Citations:

Zbl 0826.60068
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References:

[1] Holley, R.; Kusuoka, S.; Stroock, D., Asymptotics of the spectral gap with applications to the theory of simulated annealing, J. Funct. Anal., 83, 333-347 (1989) · Zbl 0706.58075
[2] Holley, R.; Stroock, D., Annealing via Sobolev inequalities, Comm. Math. Phys., 115, 553-569 (1988) · Zbl 0643.60092
[3] Ikeda, N.; Watanabe, S., Stochastic Differential Equations and Diffusion Processes (1981), North-Holland: North-Holland Amsterdam · Zbl 0495.60005
[4] Miclo, L., Recuit simulé sans potentiel sur une variété riemannienne compacte, Stochastics Stochastics Rep., 41, 23-56 (1992) · Zbl 0758.60033
[5] Miclo, L., Un algorithme de recuit simulé couplé avec une diffusion, Stochastics Stochastics Rep., 46, 193-268 (1994) · Zbl 0826.60068
[6] Miclo, L., Recuit simulé partiel: Le cas dégénéré, (Prépublication 10-96 (1996), Laboratoire de Statistique et Probabilités, Université Paul Sabatier: Laboratoire de Statistique et Probabilités, Université Paul Sabatier Toulouse, France) · Zbl 0889.60073
[7] Stroock, D., The Malliavin calculus, a functional approach, J. Funct. Anal., 44, 212-257 (1981) · Zbl 0475.60060
[8] Taniguchi, S., Malliavin’s stochastic calculus of variations for manifold-valued Wiener functionals and its applications, Z. Wahrs. Verwandte Gebiete, 65, 269-290 (1983) · Zbl 0507.60068
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