E reduce the temperature as we run the simulation one

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Unformatted text preview: (⇠ x )/⇡ (⇠ ) is easy to compute because Z ( ) cancels out, as do all the terms in the sum that do not involve x and its neighbors. Since 39 1.7. PROOFS OF THE MAIN THEOREMS* ⇠ x (x) = ⇠ (x). ⇡ (⇠ x ) = exp ⇡ (⇠ ) 2 X ⇠x ⇠y y ⇠x ! If x agrees with k of its four neighbors the ratio is exp( 2(4 2k )). In words p(x, y ) can be described by saying that we accept the proposed move with probability 1 if it lowers the energy and with probability ⇡ (y )/⇡ (x) if not. Example 1.38. Simulated annealing. The Metropolis-Hastings algorithm can also be used to minimize complicated functions. Consider for example the traveling salesman problem, which is to find the shortest (or least expensive) route that allows one to visit all of the cities on a list. In this case the state space will be lists of cities, x and ⇡ (x) = exp( `(x)) where `(x) is the length of the tour. The proposal kernel q is chosen to modify the list in some way. For example, we might move a city to...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).

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