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The stochastic process corresponding to the simulated annealing optimization algorithm is generalized to the case of an arbitrary state space. Conditions for the strong and weak convergence of the process are established. In addition the relation between the size of the generating distributions and the possible rate of cooling is studied.

The optimality of the one step look-ahead stopping rule is shown to hold under conditions different from those discussed by Chow, Robbins and Seigmund [5]. These results are corollaries of the following theorem: Let {Xn, n = 0, 1, …}; X0 = x be a discrete-time homogeneous Markov process with state space (E, ℬ). For any ℬ-measurable function g and α in (0, 1], define Aαg(x) = αExg(X1) – g(x) to be the infinitesimal generator of g. If τ is any stopping time satisfying the conditions: Ex[αNg(XN)I(τ > N)]→0 as as N → ∞, then Applications of the results are considered.

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Simulated annealing process in general state space

Optimality of the one step look-ahead stopping times

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2 results

Simulated annealing process in general state space

Optimality of the one step look-ahead stopping times