On the problem of exit from cycles for simulated annealing processes--a backward equation approach

Abstract

For a simulated annealing process $X_t$ on S with transition rates $q_{ij}(t) = p_{ij} \exp (-(U(i, j))/T(t))$ where $i, j \epsilon S$ and $T(t) \downarrow 0$ in a suitable way, we study the exit distribution $P_{t,i}(X_{\tau} = a)$ and mean exit time $E_{t,i}(\tau)$ of $X_t$ from a cycle c as $t \to \infty$. A cycle is a particular subset of S whose precise definition will be given in Section 1. Here $\tau$ is the exit time of the process from c containing i and a is an arbitrary state not in c. We consider the differential (backward) equation of $P_{t,i}(X_{\tau} = a)$ and $E_{t,i}(\tau)$ and show that $\lim_{t\to\infty}P_{t,i}(X_{\tau} = a)/\exp (-U(c, a) - T(t))$ and $\lim_{t\to\infty E_{t,i}(\tau)/\exp(V(c)/T(t))$ exist and are independent of $i \epsilon c$. The constant $(U(c, a))$ is usually referred to as the cost from c to a and $V(c), (\leq U(c, a))$ is the minimal cost coming out of c. We also obtain estimates of $|P_{t,i}(X_{\tau} = a) - P_{t,j}(X_{\tau} = a)|$ and $|E_{t,i}(\tau)|$ for $i \not= j$ as $t \to \infty$. As an application, we shall show that similar results hold for the family of Markov processes with transition rates $q_{ij}^{\varepsilon} = p_{ij} \exp(-U(i, j)/\varepsilon)$ where $\varepsilon > 0$ is small.