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SIMULATED ANNEALINGOverviewPhysical annealing is a three stage process that has been known and used for shaping metals since about 5000B.C. Any annealing process consists of three stages: heating to the desired temperature, holding at thattemperature, and cooling, usually to room temperature. Glass, crystals, and other materials are also annealed torender them less brittle and more workable. Sequences of times and temperatures, called `annealing' or`cooling' schedules, are critical in two ways: If the rate of the temperature change between the outside andinside of the piece is too great, temperature gradients and internal stresses may be induced that may lead to

warping or cracking. Also, the actual annealing time must be long enough to allow for any necessarytransformations to take place.

Using Monte Carlo sampling techniques, the physical annealing process has been modeled successfully by computersimulation methods. The analogy between a physical many-particle system and a combinatorial optimizationproblem is based on the following equivalences:

Solutions in a combinatorial optimization problem are equivalent to states of a physical system. The cost of a solution is equivalent to the energy of a state. A control parameter plays the role of temperature, such that: At large values, changes in energy are

accepted; as it is reduced, only decreases or smaller increases are accepted, and as it approaches zero, noincreases are accepted at all. Furthermore, there is no limitation on the acceptable size of an energy

increase, a characteristic feature of simulated annealing.

The Metropolis algorithm for Monte Carlo is the grandfather of simulated annealing algorithms, proposed in1953. Its significance is highlighted by its selection as the top algorithm with the greatest influence on thedevelopment of science and engineering in the 20th Century for offering ``an efficient way to stumble towardsanswers to problems too complicated to solve exactly.Kirkpatrick and Gelatt first tested simulated annealing on the travelling salesman problem, finding locallyoptimal solutions for up to 6000 sites. At that time the exact solution had been obtained for 318 sites.Annealing was also used to optimize the design of complex integrated circuits by arranging hundreds ofthousands of circuit elements to minimize chip space requirements and to reduce interference among theirconnecting wires.

Simulated annealing is a generalization of a Monte Carlo method for examining the equations of state andfrozen states of n-body systems [Metropolis et al. 1953]. The concept is based on the manner in which liquidsfreeze or metals recrystalize in the process of annealing. In an annealing process a melt, initially at hightemperature and disordered, is slowly cooled so that the system at any time is approximately inthermodynamic equilibrium. As cooling proceeds, the system becomes more ordered and approaches a "frozen"ground state at T=0. Hence the process can be thought of as an adiabatic approach to the lowest energy state. Ifthe initial temperature of the system is too low or cooling is done insufficiently slowly the system may becomequenched forming defects or freezing out in metastable states (ie. trapped in a local minimum energy state).

The original Metropolis scheme was that an initial state of a thermodynamic system was chosen at energy E

and temperature T, holding T constant the initial configuration is perturbed and the change in energy dE iscomputed. If the change in energy is negative the new configuration is accepted. If the change in energy ispositive it is accepted with a probability given by the Boltzmann factor exp -(dE/T). This processes is thenrepeated sufficient times to give good sampling statistics for the current temperature, and then the temperatureis decremented and the entire process repeated until a frozen state is achieved at T=0.

By analogy the generalization of this Monte Carlo approach to combinatorial problems is straight forward[Kirkpatrick et al. 1983, Cerny 1985]. The current state of the thermodynamic system is analogous to thecurrent solution to the combinatorial problem, the energy equation for the thermodynamic system is analogousto at the objective function, and ground state is analogous to the global minimum. The major difficulty (art) inimplementation of the algorithm is that there is no obvious analogy for the temperature T with respect to a free

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parameter in the combinatorial problem. Furthermore, avoidance of entrainment in local minima (quenching)is dependent on the "annealing schedule", the choice of initial temperature, how many iterations are performedat each temperature, and how much the temperature is decremented at each step as cooling proceeds.

Application Domains

Simulated annealing has been used in various combinatorial optimization problems and has been particularlysuccessful in circuit design problems (see Kirkpatrick et al. 1983).

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