Open Access
August 1998 On the problem of exit from cycles for simulated annealing processes--a backward equation approach
Tzuu-Shuh Chiang, Yunshyong Chow
Ann. Appl. Probab. 8(3): 896-916 (August 1998). DOI: 10.1214/aoap/1028903456

Abstract

For a simulated annealing process Xt on S with transition rates qij(t)=pijexp((U(i,j))/T(t)) where i,jϵS and T(t)0 in a suitable way, we study the exit distribution Pt,i(Xτ=a) and mean exit time Et,i(τ) of Xt from a cycle c as t. A cycle is a particular subset of S whose precise definition will be given in Section 1. Here τ 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 Pt,i(Xτ=a) and Et,i(τ) and show that limtPt,i(Xτ=a)/exp(U(c,a)T(t)) and $\lim_{t\to\infty E_{t,i}(\tau)/\exp(V(c)/T(t))existandareindependentofi \epsilon c.Theconstant(U(c, a))$ is usually referred to as the cost from c to a and V(c),(U(c,a)) is the minimal cost coming out of c. We also obtain estimates of |Pt,i(Xτ=a)Pt,j(Xτ=a)| and |Et,i(τ)| for ij as t. As an application, we shall show that similar results hold for the family of Markov processes with transition rates qijε=pijexp(U(i,j)/ε) where ε>0 is small.

Citation

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Tzuu-Shuh Chiang. Yunshyong Chow. "On the problem of exit from cycles for simulated annealing processes--a backward equation approach." Ann. Appl. Probab. 8 (3) 896 - 916, August 1998. https://doi.org/10.1214/aoap/1028903456

Information

Published: August 1998
First available in Project Euclid: 9 August 2002

zbMATH: 0937.60067
MathSciNet: MR1627807
Digital Object Identifier: 10.1214/aoap/1028903456

Subjects:
Primary: 60J27 , 60J99
Secondary: 15A18‎ , 15A51 , 90B40

Keywords: backward equations , cycles , Simulated annealing process

Rights: Copyright © 1998 Institute of Mathematical Statistics

Vol.8 • No. 3 • August 1998
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