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Annealing diffusions in a potential function with a slow growth

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  • Zitt, Pierre-André

Abstract

Consider a continuous analogue of the simulated annealing algorithm in , namely the solution of the SDE , where V is a function called the potential. We prove a convergence result, similar to the one in [L. Miclo, Thèse de doctorat, Ph.D. Thesis, Université Paris VI, 1991], under weaker hypotheses on the potential function. In particular, we cover cases where the gradient of the potential goes to zero at infinity. The main idea is to replace the Poincaré and log-Sobolev inequalities used in [L. Miclo, Thèse de doctorat, Ph.D. Thesis, Université Paris VI, 1991; C.-R. Hwang, T.-S. Chiang, S.-J. Sheu, Diffusion for global optimization in Rn, SIAM J. Control Optim. 25 (1987) 737-753.] by the weak Poincaré inequalities (introduced in [M. Röckner, F.-Y. Wang, Weak Poincaré inequalities and L2 convergence rates of Markov semigroups, J. Funct. Anal. 185 (2001) 564-603]), and to estimate constants with measure-capacity criteria. We show that the convergence still holds for the 'classical' schedule [sigma](t)=c/ln(t), where c is bigger than a constant related to V (namely the height of the largest potential barrier).

Suggested Citation

  • Zitt, Pierre-André, 2008. "Annealing diffusions in a potential function with a slow growth," Stochastic Processes and their Applications, Elsevier, vol. 118(1), pages 76-119, January.
    • Handle: RePEc:eee:spapps:v:118:y:2008:i:1:p:76-119
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    References listed on IDEAS

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    1. Jacquot, Sophie, 1996. "Simulated annealing for stochastic semilinear equations on Hilbert spaces," Stochastic Processes and their Applications, Elsevier, vol. 64(1), pages 73-91, November.
    2. M. Locatelli, 2000. "Simulated Annealing Algorithms for Continuous Global Optimization: Convergence Conditions," Journal of Optimization Theory and Applications, Springer, vol. 104(1), pages 121-133, January.
    3. Jacquot, S., 1992. "Comportement asymptotique de la seconde valeur propre des processus de Kolmogorov," Journal of Multivariate Analysis, Elsevier, vol. 40(2), pages 335-347, February.
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    Cited by:

    1. Monmarché, Pierre & Fournier, Nicolas & Tardif, Camille, 2021. "Simulated annealing in Rd with slowly growing potentials," Stochastic Processes and their Applications, Elsevier, vol. 131(C), pages 276-291.

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