The Basics and Pseudo Code Simulated Annealing The Basics and Pseudo Code

SA Basics Fundamentally simple algorithm Not much more intelligent than guessing Relies on the algorithm’s ability to span the search space effectively Uses simple but effective means to avoid local minima Name derives from a technique in metallurgy where two metals are fused by melting them and slowly cooling the combination

SA Glossary Solution: An answer to a problem without respect to its apparent “value” Neighbor: A solution which is “next” to a given solution in the solution space (or neighborhood) Fitness: The “value” of a solution Temperature: The current average rate at which less fit solutions are accepted Annealing Schedule: The function which lowers (or raises) the temperature the algorithm uses during its search

SA Pseudo Code S0 = GenerateSolution() Temp = START_TEMP K = 0 // iteration count while(!stopping_condition(S0, Temp, K)) S1 = Neighbor(S0) // make a neighbor of S0 if(Fitness(S1) < Fitness(S0)) S0 = S1 // if S1 is better than S0 take it else if(rand() < tempFunc(S0, S1, Temp, K)) S0 = S1 // if S1 is worse than S0 // conditionally take it end if AnnealingSchedule(S0, Temp, K) // anneal the temp K = K + 1 // increase the iteration count end while This does not deal with mesa situations, where Fitness(S1) == Fitness(S0)

What is a Neighbor? A neighbor is just a permutation based on the current solution This is how the algorithm moves around the search space In the fuel bundle given to the right a neighbor would be a different arrangement of the fuel pins Image Copyright(C)1997 TOSHIBA CORPORATION. All Rights Reserved.

Why Temperature? Simulated Annealing uses temperature as a way to leave local minima Hill climbing can get stuck Unable to move outside the minima as it does not take worse neighbors SA uses an annealing schedule to determine how high the temperature should be, and how long it should remain at a given temperature

What Annealing Schedule? This is a complicated question! There are linear schedules, geometric schedules, oscillating schedules, schedules based on the rate of improvement Some SA algorithms use a GA to design their annealing schedule! A very basic, yet effective, annealing schedule is one that is geometric, but stepped T = T * 0.9996 However, this is only done every M iterations, which is either determined by the rate of improvement or by a hard figure (say every 1000 iterations) Quantitative analysis is the only way to determine the most effective schedule

Putting it all Together When the new solution is found to be worse than the old solution you must evaluate if the bad solution should be taken Another choice must be made! How do you interpret the temperature? The Metropolis-Hastings algorithm is commonly used (and has interesting ties to SA) The algorithm approximates a probability distribution and was created specifically for the Boltzmann Distribution The algorithm allows you to create samples for problems where it is hard to effectively sample the entire space This sounds exactly like what SA requires!

Metropolis-Hastings e(-(Fitness(S1) – Fitness(S0)) / (K * Temp)) This method was given by Kirkpatrick et. al and doesn’t exactly replicate what occurs in natural annealing Based on the relative difference in how bad the new solution is and how far along the algorithm is running, the Metropolis-Hastings approach takes bad guesses frequently while the temperature is high, and eventually settles into a descent until a minima is found

Conclusion Simple changes to a greedy or hill climbing algorithm can make effective use of computational resources to find solutions in a solution space too hard or large to sample fully Simulated Annealing can be applied to many applications due to extremely simple nature Simulated Annealing can be a good choice for a first guess solution to optimizing a problem For certain problems, SA is among the best solutions for finding optimal answers to a problem