Abstract
We study asymptotic properties of discrete and continuous time generalized simulated annealing processesX(·) by considering a class of singular perturbed Markov chains which are closely related to the large deviation of perturbed diffusion processes. Convergence ofX(t) in probability to a setS 0 of desired states, e.g., the set of global minima, and in distribution to a probability concentrated onS 0 are studied. The corresponding two critical constants denoted byd and Λ withd≤Λ are given explicitly. When the cooling schedule is of the formc/logt, X(t) converges weakly forc>0. Whether the weak limit depends onX(0) or concentrates onS 0 is determined by the relation betweenc, d, and Λ. Whenc>Λ, the expression for the rate of convergence for each state is also derived.
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Hwang, CR., Sheu, SJ. Singular perturbed Markov chains and exact behaviors of simulated annealing processes. J Theor Probab 5, 223–249 (1992). https://doi.org/10.1007/BF01046734
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DOI: https://doi.org/10.1007/BF01046734