Simulated annealing -a informative approach
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Simulated annealing is an optimization algorithm inspired by the annealing process in metals. It can be used to find approximate global minima in large search spaces. The algorithm works as follows: initially random solutions are accepted probabilistically, with a higher probability of accepting worse solutions at higher "temperatures"; the temperature is slowly decreased, making the probability of accepting worse solutions decrease as well, until eventually only improvements are accepted. This process allows the algorithm to avoid getting stuck in local minima. Simulated annealing has been successfully applied to problems like circuit placement, scheduling, and the traveling salesperson problem. It provides good quality solutions efficiently compared to other optimization techniques.
Simulated annealing -a informative approach
- 1. Simulated Annealing ●Presented By: Ranak Ghosh Septembar,2010 For more contact- 08013835148
- 3. Introduction ●Various algorithms proposed for placement in circuits. ●Constructive placement vs Iterative improvement. ●Simulated Annealing – an iterative improvement algorithm.
- 4. Motivation ●Annealing in metals ●Heat the solid state metal to a high temperature ●Cool it down very slowly according to a specific schedule. ●If the heating temperature is sufficiently high to ensure random state and the cooling process is slow enough to ensure thermal equilibrium, then the atoms will place themselves in a pattern that corresponds to
- 5. Motivation ●The connection between this algorithm and mathematical minimization was first noted by Pincus. ●He proposed that it forms the basis of an optimization technique for combinatorial (and other) problems. ●SA's major advantage over other methods is an ability to avoid becoming trapped at local minima. ●The algorithm employs a random search which not only accepts changes that decrease objective function f, but also some changes that increase it. ●The latter is accepted with a probability P=exp(- df/T), where df is the increase in f and T is a
- 7. Analogy ●Slowly cool down a heated solid, so that all particles arrange in the ground energy state ●At each temperature wait until the solid reaches its thermal equilibrium ●Probability of being in a state with energy E : Pr { E = E } = 1/Z(T) . exp (-E / kB.T) E Energy T Temperature kB Boltzmann constant Z(T) Normalization factor (temperature dependant)
- 11. Ball on terrain example – Simulated Annealing vs Greedy Algorithms ●The ball is initially placed at a random position on the terrain. From the current position, the ball should be fired such that it can only move one step left or right.What algorithm should we follow for the ball to finally settle at the lowest point on the terrain?
- 12. Ball on terrain example – SA vs Greedy Algorithms
- 13. Example: Travel Salesperson Problem (TSP) How to transverse n cities once and only once with a minimal total distance?
- 14. Traveling Salesperson Problem (TSP) ●TSP is famous discrete optimization problem ●Many successful uses of SAN with TSP ●Basic problem is to find best way for salesperson to hit every city in territory once and only once oSetting arises in many problems of optimization on networks (communications, transportation, etc.) ●If tour involves n cities, there are (n–1) ! /2 possible solutions oExtremely rapid growth in solution space as n increases oProblem is “NP hard” ●Perturbations in SAN steps based on three operations on network: inversion, translation, and switching
- 15. 9/8/2010 TSP ●Move sets for TSP 1-2-3-4-5-6-7-8-9-10-11-12 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12 1-2-3-4-5-9-8-7-6-10-11-12 1 2 3 4 5 6 7 8 9 10 11 12 1-2-3-4-8-7-5-9-6-10-11-12 1 2 3 4 5 6 7 8 9 10 11 12 1-2-11-4-8-7-5-9-6-10-3-12 Inversio n Switching Translatio n Inversion reverses order 6-7-8-9; translation removes section 6-7-8-9 and places 8-7 in place of 5-9; switching interchanges order of 3 and 11.
- 16. Small-Scale TSP ●10-city tour (very small by industrial standards) oKnow by enumeration that minimum cost of tour = 440 ●Randomly chose inversion, translation, or switching at each iteration oTuning required to choose “good” probabilities of selecting these operators ●8 of 10 SA runs find minimum cost tour oSample mean cost of initial tour is 700; sample mean of final tour is 444 ●Essential to success is adequate use of inversion operator; 0 of 10 SA runs find optimal tour if probability of inversion is 0.50 ●SA successfully used in much larger TSPs
- 17. Simulated Annealing for TSP- ANALOGY ●Initial configuration: permutation =>1, 2, 3, 4, …, N temperature T = 2 cooling rate α = 0.99 energy = d(1,2)+d(2,3)+…+d(N,1) ●Generate a new configuration from the current one at random ●Evaluate ΔE = current energy – previous energy ●If ΔE < 0 accept the current configuration (downhill) Else accept configuration with probability : P=exp (-∆E / kB.T) ●T= αT ●Stopping criteria if energy < threshold or number of
- 18. Case Study: 20-city problem
- 19. Applications ●Circuit partitioning and placement. ●Strategy scheduling for capital products with complex product structure. ●Umpire scheduling in US Open Tennis tournament! ●Event-based learning situations.
- 20. Jigsaw puzzles – Intuitive usage of Simulated Annealing ●Given a jigsaw puzzle such that one has to obtain the final shape using all pieces together. ●Starting with a random configuration, the human brain unconditionally chooses certain moves that tend to the solution. ●However, certain moves that may or may not lead to the solution are accepted or rejected with a certain small probability. ●The final shape is obtained as a result of a
- 21. Performance ●SA is a general solution method that is easily applicable to a large number of problems ●"Tuning" of the parameters (initial T, decrement of T, stop criterion) is relatively easy ●Generally the quality of the results of SA is good, although it can take a lot of time ●Results are generally not reproducible: another run can give a different result ●SA can leave an optimal solution and not find it again (so try to remember the best solution found so far) ●Proven to find the optimum under certain conditions;
- 22. Thank You!