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In interesting combinatorial settings the state space is usually exponential in the instance size. So constant-temperature SA would work in polynomial time if the relaxation time were polylogarithmic in the size of the state space "rapid mixing". There are a number of available positive results on rapid mixing in Markov chains, but they deal mostly with the case where , corresponding to a random walk.

This is a much simpler special case, because the cost function J has no effect. Unfortunately, simulated annealing becomes interesting at the opposite end, when T is very small. Proving rapid mixing for large SA Markov chains at small temperatures is a challenging task. One class of problems for which there is some hope of obtaining positive complexity results arises in the context of image processing. To each gridpoint i, j , we associate a variable s ij taking. We thus obtain a configuration space.

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Many image processing and pattern recognition problems lead to a cost function of the form. Starting with Geman and Geman , simulated annealing has become a very popular method for such problems. Here one defines two states configurations to be neighbors if they differ only at a single grid-point. Note that when a configuration change is contemplated that is, a change of some s ij , the cost difference which determines the probability of accepting the change depends only on the gridpoints neighboring i, j. For this reason, the evolution of the configuration can be viewed as the time evolution of a Markov random field.

The relaxation times of Markov random fields have been extensively studied see, e. Thus, the available results are not yet applicable to the cost functions that arise in image processing. As far as theory is concerned, there is at present a definite lack of rigorous results justifying the use of simulated annealing. Even if SA is accepted, there are no convincing theoretical arguments favoring the use of time-varying decreasing cooling schedules, as opposed to the use of a constant temperature. This latter question is partially addressed in Hajek and Sasaki Despite the lack of a rigorous theoretical justification of its speed of convergence, researchers have used SA extensively in the last decade.

There are numerous papers discussing applications of SA to various problems. We have already mentioned that SA is extensively used in image processing. In order to give an indication of its performance, we will review some of the work concerning the application of SA to combinatorial optimization problems. In a comprehensive study of SA, Johnson et al. Johnson et al. In general, the performance of SA was mixed: in some problems, it outperformed the best known heuristics for these problems, and, in other cases, specialized heuristics performed better. More specifically:.

For the GPP, SA obtains final solutions that are at best some 5 percent better than those obtained by the best of the more traditional algorithms e. For sparse graphs, SA was better than repeated applications of the Kernighan-Lin heuristic, which is based on ideas of local optimization, whereas for some structured graphs the Kernighan-Lin heuristic was better.

For the graph coloring problem, SA produces final solutions that are competitive with those obtained by a tailored heuristic the one by Johri and Matula , which is considered the best one for this problem. However, computation times for SA are considerably longer than those of the specialized heuristic.

For the traveling salesman problem, SA consistently outperforms solutions found by repeated application of iterative improvement, based on 2-opt or 3-opt transitions, but it is a consistent loser when compared with the well-known algorithm of Lin and Kernighan The latter is based on k -opt transitions, and at each iteration it decides dynamically the value of k. Another interesting point is that the choice of the cooling schedule influences the quality of solution obtained.

In Laarhoven and Aarts Another observation is that the computation times can be excessive for some problems. In addition to the above mentioned developments in image processing, SA and various alternative versions based roughly on it have been used in statistical applications. Bohachevsky et al. Many researchers have considered SA as a tool in the development of optimal experimental designs. Recent examples include Currin et al. Variants of SA based on Bayesian ideas have been proposed by Laud et al.

Overall, SA is a generally applicable and easy-to-implement probabilistic approximation algorithm that is able to produce good solutions for an optimization problem, even if we do not understand the structure of the problem well. We believe, however, that more research, both theoretical and experimental, is needed to assess further the potential of the method. Bohachevsky, I. Johnson, and M. Stein , Generalized simulated annealing for function optimization, Technometrics 28 , Cerny, V.

Theory Appl. Chiang, T. Chow , On eigenvalues and optimal annealing rate, Math. Connors, D.

Simulated annealing

Kumar , Balance of recurrence order in time-inhomogeneous Markov chains with applications to simulated annealing,. Currin, C. Mitchell, M. Morris, and D. Ylvisaker , Bayesian prediction of deterministic functions, with applications to the design and analysis of computer experiments, J. Faigle, U. Geman, S.

What Is Simulated Annealing?

Pattern Anal. Gidas, B. Hajek, B. Sasaki , Simulated annealing—to cool or not, Syst. Holley, R. Stroock , Simulated annealing via Sobolev inequalities, Commun. Kusuoka, and D. Stroock , Asymptotics of the spectral gap with applications to the theory of simulated annealing, J. Jeng, F.

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    McGeoch, and C. Schevon , Optimization by simulated annealing: An experimental evaluation, Part II: Graph coloring and number partitioning , Oper. Schevon , Optimization by simulated annealing: An experimental evaluation, Part III: The traveling salesman problem, in preparation. Johri, A. Kernighan, B.

    Simulated Annealing -- from Wolfram MathWorld

    Lin , An efficient heuristic procedure for partioning graphs, Bell Syst. By examining this equation we should note two things: the probability is proportional to temperature--as the solid cools, the probability gets smaller; and inversely proportional to --as the change in energy is larger the probability of accepting the change gets smaller.

    When applied to engineering design, an analogy is made between energy and the objective function. Random perturbations are then made to the design. If the objective is lower, the new design is made the current design; if it is higher, it may still be accepted according the probability given by the Boltzmann factor.

    Global optimization and simulated annealing

    The Boltzmann probability is compared to a random number drawn from a uniform distribution between 0 and 1; if the random number is smaller than the Boltzmann probability, the configuration is accepted. This allows the algorithm to escape local minima. As the temperature is gradually lowered, the probability that a worse design is accepted becomes smaller. Typically at high temperatures the gross structure of the design emerges which is then refined at lower temperatures. Although it can be used for continuous problems, simulated annealing is especially effective when applied to combinatorial or discrete problems.

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