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![Metropolis Criterion
• Let
• x0 be the current solution and x1 be the new solution
• E(x0), E(x1) be the energy state (cost) of x0 and x1 .
• K=1 (Boltzmann constant)
• Probability Paccept = exp [-(E(x1)-E(x0))/ KT]
• Let r=Random (0,1)
• Unconditional accepted if
• E(x1) < E(x0), the new solution is better
• Probably accepted if
• E(x1) >= E(x0), the new solution is worse . Accepted only
when r < Paccept](https://image.slidesharecdn.com/simulatedannealing-231123144331-37c78299/75/Simulated_Annealing-pptx-36-2048.jpg)
Simulated_Annealing.pptx
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- 2. Simulated Annealing • Itis a very general search technique. • Try to avoid being trapped in local minimum by making probabilistic moves. • The method was first proposed by Metropolis (1953). • Kirkpatrick et al. (1982) later improved the SA method applied optimization problems
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- 32. Basic Idea ofSimulated Annealing • Inspired by the Annealing Process • The process of carefully cooling molten metals in order to obtain a good crystal structure. • First, metal is heated to a very high temperature, the atoms have a higher energy state and a high possibility to re-arrange the crystal structure. • Then slowly cooled, the atoms have a lower and lower energy state and a smaller and smaller possibility to re- arrange the crystal structure
- 33. Simulated Annealing • Analogy •Metal Optimization Problem • Energy State Cost Function • Temperature Control Parameter • A completely ordered crystal structure the optimal solution for the problem Global optimal solution can be achieved as long as the cooling process is slow enough.
- 34. Metropolis Loop • Theessential characteristic of simulated annealing • Determining how to randomly explore new solution, reject or accept the new solution at a constant temperature T. • Finished until equilibrium is achieved.
- 35. Algorithm Initialize initial solutionx0 , highest temperature Th, and coolest temperature Tl T= Th When the temperature is higher than Tl While not in equilibrium Search for the new solution x1 Accept or reject x1 according to Metropolis Criterion End Decrease the temperature T, according to cooling schedule End
- 36. Metropolis Criterion • Let •x0 be the current solution and x1 be the new solution • E(x0), E(x1) be the energy state (cost) of x0 and x1 . • K=1 (Boltzmann constant) • Probability Paccept = exp [-(E(x1)-E(x0))/ KT] • Let r=Random (0,1) • Unconditional accepted if • E(x1) < E(x0), the new solution is better • Probably accepted if • E(x1) >= E(x0), the new solution is worse . Accepted only when r < Paccept
- 37. Control Parameters • Definitionof equilibrium • Cannot yield any significant improvement after certain number of loops • A constant number of loops • Annealing schedule (i.e. How to reduce the temperature) • A constant value, T’ = T - Td • A constant scale factor, T’= T * Rd x A scale factor usually can achieve better performance
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- 41. SA Advantages/Disadvantages • Advantages •Guaranteed to find optimum • Avoids being trapped at local minimums • Disadvantages • No time constraints • No past records
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