Suppose p is irreducible aperiodic and has stationary

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Unformatted text preview: a another place on the list or reverse the order of a sequence of cities. When is large the stationary distribution will concentrate on optimal and near optimal tours. As in the Ising model, is thought of as inverse temperature. The name derives from the fact that to force the chain to better solution we increase (i.e., reduce the temperature) as we run the simulation. One must do this slowly or the process will get stuck in local minima. For more of simulated annealing see Kirkpatrick et al. (1983) 1.7 Proofs of the Main Theorems* To prepare for the proof of the convergence theorem, Theorem 1.19, we need the following: Lemma 1.26. If there is a stationary distribution, then all states y that have ⇡ (y ) > 0 are recurrent. P1 Proof. Lemma 1.12 tells us that Ex N (y ) = n=1 pn (x, y ), so X ⇡ (x)Ex N (y ) = x X ⇡ (x) x 1 X pn (x, y ) n=1 Interchanging the order of summation and then using ⇡ pn = ⇡ , the above = 1 XX ⇡ (x)pn (x, y ) = n=1 x 1 X n=1 ⇡ (y ) = 1 since ⇡ (y ) > 0. Using Lemma 1.11 now gives Ex N (y )...
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