Balance of Recurrence Order in Time-Inhomogeneous Markov Chains with Application to Simulated Annealing.
ILLINOIS UNIV AT URBANA DECISION AND CONTROL LAB
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Considered in this thesis is the class of time inhomogeneous Markov chains whose transition probabilities are proportional to nonnegative powers of a small, time varying parameter. A new notion of order of recurrence is introduced for the states and transitions of such Markov processes. These orders of recurrence provide considerable information about the asymptotic behavior of the process. The orders of recurrence are shown to satisfy a balance equation across every edge cut the graph associated with the Markov chain. This allows for computation of the orders of recurrence. An algorithm for generating all solutions to the order balance equations is given. The method of optimization by simulated annealing falls within the framework of this class of Markov chains. It is shown that if the simulated annealing Markov chain is weakly reversible, then the sum of the order of recurrence of a state and its cost is constant on connected components of recurrent states. In the special case of symmetric neighborhoods, there holds a detailed balance across every edge in the graph of the Markov process. From the constancy of the sum of the order of recurrence of a state and its cost, a necessary and sufficient condition on the cooling schedule in order to guarantee that the minimum of the optimization problem is hit with probability one is deduced. The issue of defining a notion of recurrence orders which allows for negative values of these orders is explored, and it is shown that the balance equations need not hold when negative recurrence orders are allowed. JHD.
- Statistics and Probability