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Global+Optimization+Algorithms+Theory+and+Application_Part14

# Global+Optimization+Algorithms+Theory+and+Application_Part14...

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11.2 General Information 261 Application References Medicine [2255, 2256] Biology and Medicine [558] Machine Learning [1249] Function Optimization [1277, 1917]

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12 Simulated Annealing 12.1 Introduction In 1953, Metropolis et al. [1396] developed a Monte Carlo method for “calculating the properties of any substance which may be considered as composed of interacting individual molecules”. With this so-called “Metropolis” procedure stemming from statistical mechan- ics, the manner in which metal crystals reconfigure and reach equilibria in the process of annealing can be simulated. This inspired Kirkpatrick et al. [1142] to develop the Simulated Annealing 1 (SA) algorithm for global optimization in the early 1980s and to apply it to var- ious combinatorial optimization problems. Independently, ˇ Cern´y [363] employed a similar approach to the travelling salesman problem [1263, 78]. Simulated Annealing is an optimiza- tion method that can be applied to arbitrary search and problem spaces. Like simple hill climbing algorithms, Simulated Annealing only needs a single initial individual as starting point and a unary search operation. In metallurgy and material science, annealing 2 is a heat treatment of material with the goal of altering its properties such as hardness. Metal crystals have small defects, dislocations of ions which weaken the overall structure. By heating the metal, the energy of the ions and, thus, their diffusion rate is increased. Then, the dislocations can be destroyed and the structure of the crystal is reformed as the material cools down and approaches its equilibrium state. When annealing metal, the initial temperature must not be too low and the cooling must be done suﬃciently slowly so as to avoid the system getting stuck in a meta-stable, non-crystalline, state representing a local minimum of energy. In physics, each set of positions of all atoms of a system pos is weighted by its Boltzmann probability factor e E ( pos ) k B T where E ( pos ) is the energy of the configuration pos , T is the temperature measured in Kelvin, and k B is the Boltzmann’s constant 3 k B = 1 . 380 650 524 · 10 23 J/K. The Metropolis procedure was an exact copy of this physical process which could be used to simulate a collection of atoms in thermodynamic equilibrium at a given temperature. A new nearby geometry pos i +1 was generated as a random displacement from the current geometry pos i of an atom in each iteration. The energy of the resulting new geometry is computed and ΔE , the energetic difference between the current and the new geometry, was determined. The probability that this new geometry is accepted, P ( ΔE ) is defined in Equation 12.2 . 1 http://en.wikipedia.org/wiki/Simulated_annealing [accessed 2007-07-03] 2 http://en.wikipedia.org/wiki/Annealing_(metallurgy) [accessed 2008-09-19] 3 http://en.wikipedia.org/wiki/Boltzmann%27s_constant [accessed 2007-07-03]

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264 12 Simulated Annealing ΔE = E ( pos i +1 ) E ( pos i ) (12.1) P ( ΔE ) = braceleftBigg e ΔE k B T if ΔE> 0 1 otherwise (12.2) Thus, if the new nearby geometry has a lower energy level, the transition is accepted.
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