Balance of Recurrece Order in Time-Inhomogenous Markov Chains with Application to Simulated Annealing

Abstract

We define a notion of order of recurrence for the states and transitions of a general class of time-inhomogeneous Markov chains with transition probabilities proportional to powers of a small vanishing parameter. These orders are shown to satisfy a balance equation across every edge cut in the associated graph. The resulting order balance equations allow computation of the orders of recurrence of the states, and thereby the determination of the asymptotic behavior of the Markov chain.The method of optimization by simulated annealing is a special case of such Markov processes, and can therefore be treated by means of these balance equations. In particular, in this special situation we show that there holds a detailed balance of order of recurrence across every edge in the graph. Moreover, the sum of the order of recurrence of a state and its cost is shown to be a constant in each connected set of recurrent states. By this approach, we determine the necessary and sufficient condition on the “rate of cooling” to guarantee that a minimum of the optimization problem is hit with probability one. Moreover, the rates of convergence of the probabilities can be deduced from the orders of recurrence.

Quasi-Statically Cooled Markov Chains

“…By Remark 2.5 of Connors and Kumar [5], our theory also applies to the chain [x(t)\. Note that V jr > 0, since Y is recurrently connected.…”

“…Results of this nature have been shown for simulated annealing elsewhere [5,6,8,9,15,17]. One hopes that by running the timeinhomogeneous chain with the small parameter sequence e(t) reduced to zero slowly, it will also converge to g. One can then say that the chain has been quasi-statically cooled.…”

“…Similar to the order of the cooling schedule, we define the recurrence order of a state. The following property of balance of recurrence orders across cut-sets, from Connors and Kumar [5], is fundamental. The higher the value of ft, the more often the state / is visited.…”

Simulated Annealing for Optimal Resource Allocation in Wireless Networks with Imperfect Communications

“…2) We propose a new algorithmic approach that can deal more efficiently with an impact from the imperfect communications, and rigorously prove the optimality of the proposed algorithm under the standard SA framework. 3) We demonstrate that the proposed algorithm offers significant improvement in terms of its computational speed in networks with high message drop rates. We organize this paper as follows.…”

“…if p = max {c ≥ 0 : ∞ t=1 ǫ(t) c = ∞} Having defined the above terms, we summarize the main results established in [4] as well as in [3], [5], which are valid under the following mild assumptions.…”