Relative Frequencies of Generalized Simulated Annealing

We consider a class of nonhomogeneous Markov chains arising in simulated annealing and related stochastic search algorithms. Using only elementary first principles, we analyze the convergence and rate of convergence of the relative frequencies of visits to states in the Markov chain. We describe in detail three examples, including the standard simulated annealing algorithm, to show how our framework applies to specific stochastic search algorithmsthese examples have not previously been recognized to be sufficiently similar to share common analytical grounds. Our analysis, though elementary, provides the strongest sample path convergence results to date for simulated annealing-type Markov chains. Our results serve to illustrate that by taking a purely sample path view, surprisingly strong statements can be made using only relatively elementary tools.

[1]  Donald Geman,et al.  Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images , 1984 .

[2]  John N. Tsitsiklis,et al.  Markov Chains with Rare Transitions and Simulated Annealing , 1989, Math. Oper. Res..

[3]  E. Chong,et al.  EQUIVALENT NECESSARY AND SUFFICIENT CONDITIONS ON NOISE SEQUENCES FOR STOCHASTIC APPROXIMATION ALGORITHMS , 1996 .

[4]  S. Kulkarni,et al.  An alternative proof for convergence of stochastic approximation algorithms , 1996, IEEE Trans. Autom. Control..

[5]  Donald Geman,et al.  Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images , 1984, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[6]  Sanjeev R. Kulkarni,et al.  Noise Conditions for Prespecified Convergence Rates of Stochastic Approximation Algorithms , 1999, IEEE Trans. Inf. Theory.

[7]  E. Chong,et al.  A deterministic analysis of stochastic approximation with randomized directions , 1998, IEEE Transactions on Automatic Control.

[8]  M. Freidlin,et al.  Random Perturbations of Dynamical Systems , 1984 .

[9]  O. Catoni.  Rough Large Deviation Estimates for Simulated Annealing: Application to Exponential Schedules , 1992 .

[10]  Yu-Chi Ho,et al.  Stochastic Comparison Algorithm for Discrete Optimization with Estimation , 1999, SIAM J. Optim..

[11]  N. Metropolis,et al.  Equation of State Calculations by Fast Computing Machines , 1953, Resonance.

[12]  Bruce E. Hajek,et al.  Cooling Schedules for Optimal Annealing , 1988, Math. Oper. Res..

[13]  C. Tsallis,et al.  Generalized simulated annealing , 1995, cond-mat/9501047.

[14]  A. Trouvé.  Convergence optimale pour les algorithmes de recuits généralisés , 1992 .

[15]  R. Durrett.  Probability: Theory and Examples , 1993 .

[16]  C. D. Gelatt,et al.  Optimization by Simulated Annealing , 1983, Science.

[17]  O. Catoni,et al.  Piecewise constant triangular cooling schedules for generalized simulated annealing algorithms , 1998 .

[18]  P. Kumar,et al.  Simulated-annealing type Markov chains and their order balance equations , 1988, Proceedings of the 27th IEEE Conference on Decision and Control.

[19]  D. Yan,et al.  Stochastic discrete optimization , 1992 .

[20]  P. Moral,et al.  On the Convergence and the Applications of the Generalized Simulated Annealing , 1999 .

[21]  Saul B. Gelfand,et al.  Simulated annealing type algorithms for multivariate optimization , 1991, Algorithmica.