A piecewise Markovian model for simulated annealing with stochastic cooling schedules

1995 ◽  
Vol 32 (3) ◽  
pp. 649-658 ◽  
Author(s):  
M. Kolonko
Keyword(s):  
Simulated Annealing ◽  
Random Number ◽  
Stopping Time ◽  
Markovian Model ◽  
Countable State Space ◽  
Global Character ◽  
Cooling Schedules ◽  
State History ◽  
Countable State ◽  
Recent Part

We introduce a stochastic process with discrete time and countable state space that is governed by a sequence of Markov matrices . Each Pm is used for a random number of steps Tm and is then replaced by Pm+1. Tm is a randomized stopping time that may depend on the most recent part of the state history. Thus the global character of the process is non-Markovian.This process can be used to model the well-known simulated annealing optimization algorithm with randomized, partly state depending cooling schedules. Generalizing the concept of strong stationary times (Aldous and Diaconis [1]) we are able to show the existence of optimal schedules and to prove some desirable properties. This result is mainly of theoretical interest as the proofs do not yield an explicit algorithm to construct the optimal schedules.

A piecewise Markovian model for simulated annealing with stochastic cooling schedules

1995 ◽  
Vol 32 (03) ◽  
pp. 649-658 ◽  
Author(s):  
M. Kolonko
Keyword(s):  
Simulated Annealing ◽  
Random Number ◽  
Stopping Time ◽  
Markovian Model ◽  
Countable State Space ◽  
Global Character ◽  
Cooling Schedules ◽  
State History ◽  
Countable State ◽  
Recent Part

We introduce a stochastic process with discrete time and countable state space that is governed by a sequence of Markov matrices . Each Pm is used for a random number of steps Tm and is then replaced by Pm +1. Tm is a randomized stopping time that may depend on the most recent part of the state history. Thus the global character of the process is non-Markovian. This process can be used to model the well-known simulated annealing optimization algorithm with randomized, partly state depending cooling schedules. Generalizing the concept of strong stationary times (Aldous and Diaconis [1]) we are able to show the existence of optimal schedules and to prove some desirable properties. This result is mainly of theoretical interest as the proofs do not yield an explicit algorithm to construct the optimal schedules.

Examples for the Theory of Strong Stationary Duality with Countable State Spaces

1990 ◽  
Vol 4 (2) ◽  
pp. 157-180 ◽  
Author(s):  
Persi Diaconis ◽  
James Allen Fill
Keyword(s):  
Stopping Time ◽  
Renewal Theory ◽  
Hitting Times ◽  
State Spaces ◽  
Ergodic Markov Chain ◽  
Countable State Space ◽  
Strong Stationary Duality ◽  
Countable State ◽  
Infinite State ◽  
First Hitting Times

Let X1,X2,… be an ergodic Markov chain on the countable state space. We construct a strong stationary dual chain X* whose first hitting times give sharp bounds on the convergence to stationarity for X. Examples include birth and death chains, queueing models, and the excess life process of renewal theory. This paper gives the first extension of the stopping time arguments of Aldous and Diaconis [1,2] to infinite state spaces.

THE SHANNON–MCMILLAN THEOREM FOR MARKOV CHAINS INDEXED BY A CAYLEY TREE IN RANDOM ENVIRONMENT

2017 ◽  
Vol 32 (4) ◽  
pp. 626-639 ◽  
Author(s):  
Zhiyan Shi ◽  
Pingping Zhong ◽  
Yan Fan
Keyword(s):  
Markov Chains ◽  
State Space ◽  
Random Environment ◽  
Cayley Tree ◽  
Law Of Large Numbers ◽  
Strong Law ◽  
Countable State Space ◽  
Large Numbers ◽  
Countable State ◽  
Definition Of

In this paper, we give the definition of tree-indexed Markov chains in random environment with countable state space, and then study the realization of Markov chain indexed by a tree in random environment. Finally, we prove the strong law of large numbers and Shannon–McMillan theorem for Markov chains indexed by a Cayley tree in a Markovian environment with countable state space.

scholarly journals Markov chains with exponential return times are finitary

2020 ◽  
pp. 1-9
Author(s):  
OMER ANGEL ◽  
YINON SPINKA
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Renewal Process ◽  
Proper Subgroup ◽  
Ergodic Markov Chain ◽  
Countable State Space ◽  
Return Times ◽  
Exponential Tails ◽  
Stationary Version ◽  
Countable State

Abstract Consider an ergodic Markov chain on a countable state space for which the return times have exponential tails. We show that the stationary version of any such chain is a finitary factor of an independent and identically distributed (i.i.d.) process. A key step is to show that any stationary renewal process whose jump distribution has exponential tails and is not supported on a proper subgroup of ℤ is a finitary factor of an i.i.d. process.

Criteria for the non-ergodicity of stochastic processes: application to the exponential back-off protocol

1987 ◽  
Vol 24 (02) ◽  
pp. 347-354 ◽  
Author(s):  
Guy Fayolle ◽  
Rudolph Iasnogorodski
Keyword(s):  
Stochastic Process ◽  
Markov Chain ◽  
Stochastic Processes ◽  
State Space ◽  
Discrete Time ◽  
Random Variables ◽  
Countable State Space ◽  
Countable State ◽  
New Criteria

In this paper, we present some simple new criteria for the non-ergodicity of a stochastic process (Yn ), n ≧ 0 in discrete time, when either the upward or downward jumps are majorized by i.i.d. random variables. This situation is encountered in many practical situations, where the (Yn ) are functionals of some Markov chain with countable state space. An application to the exponential back-off protocol is described.

scholarly journals Two Approaches to the Construction of Perturbation Bounds for Continuous-Time Markov Chains

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 253 ◽  
Author(s):  
Alexander Zeifman ◽  
Victor Korolev ◽  
Yacov Satin
Keyword(s):  
Markov Chains ◽  
State Space ◽  
Continuous Time ◽  
Countable State Space ◽  
Perturbation Bounds ◽  
Queuing Models ◽  
Continuous Time Markov Chains ◽  
Quantitative Estimates ◽  
Countable State

This paper is largely a review. It considers two main methods used to study stability and to obtain appropriate quantitative estimates of perturbations of (inhomogeneous) Markov chains with continuous time and a finite or countable state space. An approach is described to the construction of perturbation estimates for the main five classes of such chains associated with queuing models. Several specific models are considered for which the limit characteristics and perturbation bounds for admissible “perturbed” processes are calculated.

scholarly journals Undiscounted Markov Chain BSDEs to Stopping Times

2014 ◽  
Vol 51 (1) ◽  
pp. 262-281
Author(s):  
Samuel N. Cohen
Keyword(s):  
Markov Chain ◽  
Differential Equations ◽  
Continuous Time ◽  
Stopping Time ◽  
Stopping Times ◽  
Hitting Times ◽  
Continuous Time Markov Chain ◽  
Countable State ◽  
Growth Bound ◽  
Novel Applications

We consider backward stochastic differential equations in a setting where noise is generated by a countable state, continuous time Markov chain, and the terminal value is prescribed at a stopping time. We show that, given sufficient integrability of the stopping time and a growth bound on the terminal value and BSDE driver, these equations admit unique solutions satisfying the same growth bound (up to multiplication by a constant). This holds without assuming that the driver is monotone in y, that is, our results do not require that the terminal value be discounted at some uniform rate. We show that the conditions are satisfied for hitting times of states of the chain, and hence present some novel applications of the theory of these BSDEs.

Kolmogorov's differential equations for non-stationary, countable state Markov processes with uniformly continuous transition probabilities

1973 ◽  
Vol 73 (1) ◽  
pp. 119-138 ◽  
Author(s):  
Gerald S. Goodman ◽  
S. Johansen
Keyword(s):  
Markov Chain ◽  
Differential Equations ◽  
State Space ◽  
Markov Processes ◽  
Transition Probabilities ◽  
The State ◽  
Continuous Transition ◽  
Countable State Space ◽  
Countable State ◽  
Uniformly Continuous

1. SummaryWe shall consider a non-stationary Markov chain on a countable state space E. The transition probabilities {P(s, t), 0 ≤ s ≤ t <t0 ≤ ∞} are assumed to be continuous in (s, t) uniformly in the state i ε E.

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