Abstract
In these notes, written for a D.E.A. course at University Paris XI during the first term of 1995, we prove the essentials about stochastic optimisation algorithms based on Markov chains with rare transitions, under the weak assumption that the transition matrix obeys a large deviation principle. We present a new simplified line of proofs based on the Freidlin and Wentzell graphical approach. The case of Markov chains with a periodic behaviour at null temperature is considered. We have also included some pages about the spectral gap approach where we follow Diaconis and Stroock [13] and Ingrassia [23] in a more conventional way, except for the application to non reversible Metropolis algorithms (subsection 6.2.2) where we present an original result.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Date: May 1995, English translation January 1997, in revised form November 1998.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Azencott Robert (1988) Simulated Annealing, Séminaire Bourbaki 40ième année, 1987–1988 697
Azencott Robert (1992) Sequential Simulated Annealing: Speed of Convergence and Acceleration Techniques, in Simulated Annealing: Parallelization Techniques, R. Azencott Ed., Wiley Interscience.
Azencott Robert (1992) A Common Large Deviations Mathematical Framework for Sequential Annealing and Parallel Annealing, in Simulated Annealing: Parallelization Techniques, R. Azencott Ed., Wiley Interscience.
Azencott Robert and Graffigne Christine (1992) Parallel Annealing by Periodically Interacting Multiple Searches: Acceleration Rates, in Simulated Annealing: Parallelization Techniques, R. Azencott Ed., Wiley Interscience.
Catoni Olivier (1991) Exponential Triangular Cooling Schedules for Simulated Annealing Algorithms: a case study, Applied Stochastic Analysis, Proceedings of a US-French Workshop, Rutgers University, April 29–May 2, 1991, Karatzas I. and Ocone D. eds., Lecture Notes in Control and Information Sciences No 177, Springer Verlag, 1992.
Catoni Olivier (1992) Rough Large Deviation Estimates for Simulated Annealing: Application to Exponential Schedules, The Annals of Probability, Vol. 20, nb. 3, pp. 1109–1146.
Catoni Olivier, (1998) The Energy Transformation Method for the Metropolis Algorithm Compared with Simulated Annealing. Probab. Theory Related Fields 110 (1998), no. 1, pages 69–89.
Catoni Olivier and Cerf Raphael (1997) The Exit Path of a Markov Chain with Rare Transitions, ESAIM: P&S, vol. 1, pp. 95–144, http://www emath.fr/Maths/Ps/ps.html
Catoni Olivier (1998) Solving Scheduling Problems by Simulated Annealing. SIAM J. Control Optim. 36, no. 5, (electronic), pages 1539–1575.
Catoni Olivier (1996) Metropolis, Simulated Annealing and I.E.T. Algorithms: Theory and Experiments. Journal of Complexity 12, special issue on the conference Foundation of Computational Mathematics, January 5–12 1997, Rio de Janeiro, pages 595–623, December 1996.
Cot Cécile and Catoni Olivier (1998) Piecewise constant triangular cooling schedules for generalized simulated annealing algorithms. Ann. Appl. Probab. 8, no. 2, pages 375–396.
Deuschel J.D. and Mazza C. (1994) L 2 convergence of time nonhomogeneous Markov processes: I. Spectral Estimates, The annals of Applied Probability, vol. 4, no. 4, 1012–1056.
Diaconis Persi and Stroock Daniel (1991) Geometric Bounds for Eigenvalues of Markov Chains, The Annals of Applied Probability, Vol. 1, No. 1, 36–61.
Duflo M. (1996) Algorithmes Stochastiques, Mathématiques & Applications (Paris), Springer Verlag.
Fill J. A. (1991) Eigenvalue bounds on the convergence to stationarity for nonreversible Markov chains, with an application to the exclusion process, Ann. Applied Probab., 1.
Freidlin, M. I. and Wentzell, A. D. (1984). Random Perturbations of Dynamical Systems. Springer, New York.
Geman S., Geman D., Stochastic relaxation, Gibbs distribution, and the Bayesian restoration of images, I.E.E.E. Transaction on Pattern Analysis and Machine Intelligence, 6, 721–741, 1984.
Götze F. (1991) Rate of Convergence of Simulated Annealing Processes, preprint.
Graffigne Christine (1992) Parallel Annealing by Periodically Interacting Multiple Searches: An Experimental Study, in Simulated Annealing: Parallelization Techniques, R. Azencott Ed., Wiley Interscience
Holley R. and Stroock D. (1988) Annealing via Sobolev inequalities, Comm. Math. Phys., 115:553–559.
Holley, R. A., Kusuoka, S. and Stroock, D. W. (1989), Asymptotics of the spectral gap with applications to the theory of simulated annealing, Journal of functional analysis, 83, 333–347.
Hwang, C. R. and Sheu, S. J. (1992) Singular perturbed Markov chains and exact behaviour of simulated annealing processes. J. Theoret. Prob., 5, 2, 223–249.
Ingrassia S. (1994) On the rate of convergence of the Metropolis algorithm and Gibbs sampler by geometric bounds, Ann. Appl. Probab. 4, no. 2, 347–389.
Kirchhoff G. (1847) Über die Auflösung der Gleichungen, auf welche man beider Untersuchung der linearen Verteilung galvanischer Ströme gefuhrt wird, Ann. Phys. Chem., 72, pp. 497–508. (English transl. IRE Trans. Circuit Theory CT-5 (1958) 4–7).
Kirkpatrick, S., Gelatt C. D. and Vecchi M. P., (1983) Optimization by simulated annealing, Science, 220, 621–680, 1983.
Miclo Laurent (1991) Evolution de l'énergie libre. Application à l'étude de la convergence des algorithmes du recuit simulé. Doctoral Dissertation, Université d'Orsay, February 1991.
Miclo Laurent (1996) Sur les problémes de sortie discrets inhomogènes Ann. Appl. Probab. 6, no 4, 1112–1156.
Miclo Laurent (1995) Sur les temps d'occupations des processus de Markov finis inhomogènes à basse température, submitted to Stochastics and Stochastics Reports.
Miclo Laurent (1997) Remarques sur l'hypercontractivité et l'évolution de l'entropie pour des chaînes de Markov finies, Séminaire de Probabilités XXXI, Lecture Notes in Mathematics 1655, Springer.
Saloff-Coste, Laurent (1997) Lectures on finite Markov chains Lectures on probability theory and statistics (Saint-Flour, 1996), 301–413, Lecture Notes in Math., 1665, Springer, Berlin.
Trouvé Alain (1993) Parallélisation massive du recuit simulé, Doctoral Dissertation, Université Paris 11, January 5 1993.
Trouvé Alain (1994) Cycle Decomposition and Simulated Annealing, S.I.A.M. J. Control Optim., 34(3), 1996.
Trouvé, Alain (1995) Rough Large Deviation Estimates for the Optimal Convergence Speed Exponent of Generalized Simulated Annealing Algorithms, Ann. Inst. H. Poincaré, Probab. Statist., 32(2), 1996.
Editor information
Rights and permissions
Copyright information
© 1999 Springer-Verlag
About this paper
Cite this paper
Catoni, O. (1999). Simulated annealing algorithms and Markov chains with rare transitions. In: Azéma, J., Émery, M., Ledoux, M., Yor, M. (eds) Séminaire de Probabilités XXXIII. Lecture Notes in Mathematics, vol 1709. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0096510
Download citation
DOI: https://doi.org/10.1007/BFb0096510
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-66342-3
Online ISBN: 978-3-540-48407-3
eBook Packages: Springer Book Archive