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 is computed. If the change in energy is negative the new configuration is accepted. If the change in energy is positive it is accepted with a probability given by the Boltzmann factor exp -(dE/T). This processes is then repeated sufficient times to give good sampling statistics for the current temperature, and then the temperature is 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 the current solution to the combinatorial problem, the energy equation for the thermodynamic system is analogous to at the objective function, and ground state is analogous to the global minimum. The major
difficulty (art) in implementation of the algorithm is that there is no obvious analogy for the temperature T with respect to a free 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 performed at each temperature, and how much the temperature is decremented at each step as cooling proceeds.
Simulated annealing has been used in various combinatorial optimization problems and has been particularly successful in circuit design problems (see Kirkpatrick et al. 1983).
Implementation of simulated annealing applied to the traveling salesman problem can be found in Numerical Recipes section 10.9.
William Goffe fortran source for SA
Taygeta Scientific Inc. C++ C and Ada demo source.
CalTech ASA (adaptive simulated annealing) C code
Cerny, V., "Thermodynamical Approach to the Traveling Salesman Problem: An Efficient Simulation Algorithm", J. Opt. Theory Appl., 45, 1, 41-51, 1985
Kirkpatrick, S., C. D. Gelatt Jr., M. P. Vecchi, "Optimization by Simulated Annealing",Science, 220, 4598, 671-680, 1983.
Metropolis,N., A. Rosenbluth, M. Rosenbluth, A. Teller, E. Teller, "Equation of State Calculations by Fast Computing Machines", J. Chem. Phys.,21, 6, 1087-1092, 1953.
Press, Wm. H., B. Flannery, S. Teukolsky, Wm. Vettering, Numerical Recipes, 326-334, Cambridge University Press, New York, NY, 1986.
Taygeta Scientific Inc.additional references and demo source.
Last modified: March 10, 1997