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## SIMULATED ANNEALING

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**SIMULATED ANNEALING**Cole Ott**The next 15 minutes**• I’ll tell you what Simulated Annealing is. • We’ll walk through how to apply it to a Markov Chain (via an example). • We’ll play with a simulation.**The Problem**• State space • Scoring function • Objective: to find minimizing**Idea**• Construct a Markov Chain on with a stationary distribution that gives higher probabilities to lower-scoring states. • Run this chain for a while, until it is likely to be in a low-scoring state • Construct a new chain whose is even more concentrated on low-scoring states • Run this chain from • Repeat**Idea**• Construct a Markov Chain on with a stationary distribution that gives higher probabilities to lower-scoring states. • Run this chain for a while, until it is likely to be in a low-scoring state • Construct a new chain whose is even more concentrated on low-scoring states • Run this chain from • Repeat**Idea**• Construct a Markov Chain on with a stationary distribution that gives higher probabilities to lower-scoring states. • Run this chain for a while, until it is likely to be in a low-scoring state • Construct a new chain with an even stronger preference for states minimizing • Run this chain from • Repeat**Idea**• Construct a Markov Chain on with a stationary distribution that gives higher probabilities to lower-scoring states. • Run this chain for a while, until it is likely to be in a low-scoring state • Construct a new chain with an even stronger preference for states minimizing • Run this chain from • Repeat**Idea**• Construct a Markov Chain on with a stationary distribution that gives higher probabilities to lower-scoring states. • Run this chain for a while, until it is likely to be in a low-scoring state • Construct a new chain with an even stronger preference for states minimizing • Run this chain from • Construct another chain …**Idea (cont’d)**• Basically, we are constructing an inhomogeneous Markov Chain that gradually places more probability on -minimizing states • We hope that if we do this well, then approaches as**Boltzmann Distribution**• We will use the Boltzmann Distribution for our stationary distributions**Boltzmann Distribution (cont’d)**• is a normalization term • We won’t have to care about it.**Boltzmann Distribution (cont’d)**• is the Temperature • When is very large, approaches the uniform distribution • When is very small, concentrates virtually all probability on -minimizing states**Boltzmann Distribution (cont’d)**• When we want to maximize rather than minimize, we replace with**Def: Annealing**FUN FACT! • Annealing is a process in metallurgy in which a metal is heated for a period of time and then slowly cooled. • The heat breaks bonds and causes the atoms to diffuse, moving the metal towards its equilibrium state and thus getting rid of impurities and crystal defects. • When performed correctly (i.e. with a proper cooling schedule), the process makes a metal more homogenous and thus stronger and more ductile as a whole. • Parallels abound! Source: Wikipedia**Theorem (will not prove)**• Let denote the probability that a random element chosen according to is a global minimum • Then Source: Häggström**Designing an Algorithm**• Design a MC on our chosen state space with stationary distribution • Design a cooling schedule—a sequence of integers and a sequence of strictly decreasing temperature values such that for each in sequence we will run our MC at temperature for steps**Notes on Cooling Schedules**• Picking a cooling schedule is more of an art than a science. • Cooling too quickly can cause the chain to get caught in local minima • Tragically, cooling schedules with provably good chances of finding a global minima can require more time than it would take to actually enumerate every element in the state space**Example:**Traveling salesman**The Traveling Salesman Problem**• cities • Want to find a path (a permutation of ) that minimizes our distance function .**The Traveling Salesman Problem**• cities • Want to find a path (a permutation of ) that minimizes our distance function .**Our Markov Chain**• Pick u.a.r. vertices such that • With some probability, reverse the order of the substring on our path**Our Markov Chain (cont’d)**• Pick u.a.r. vertices such that • Let be the current state and let be the state obtained by reversing the substring • in • With probability • transition to , else do nothing.**Sources**"Annealing.” Wikipedia, The Free Encyclopedia, 22 Nov. 2010. Web. 3 Mar. 2011 Häggström, Olle. Finite Markov Chains and Algorithmic Applications. Cambridge University Press.