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probabilistic heuristic approach for the global optimization problem, namely locating a good approximation to the global optimum of a given function in a large search space. It was independently invented by S. Kirkpatrick, C. D. Gelatt and M. P. Vecchi in 1983, and by V. Cerny in 1985.
The name and inspiration come from annealing in metallurgy, a technique involving heating and controlled cooling of a material to increase the size of its crystals and reduce their defects. The heat causes the atoms to become unstuck from their initial positions (a local minimum of the internal energy) and wander randomly through states of higher energy; the slow cooling gives them more chances of finding configurations with lower internal energy than the initial one.
In the simulated annealing (SA) method, each point s of the search space is compared to a state of some physical system, and the function E(s) to be minimized is interpreted as the internal energy of the system in that state. Therefore the goal is to bring the system, from an arbitrary initial state, to a state with the minimum possible energy.
At each step, the SA heuristic considers some neighbor s' of the current state s, and probabilistically decides between moving the system to state s' or staying put in state s. The probabilities are chosen so that the system ultimately tends to move to states of lower energy. Typically this step is repeated until the system reaches a state which is good enough for the application, or until a given computation budget has been exhausted.
The neighbors of each state are specified by the user, usually in an application-specific way. For example, in the traveling salesman problem, each state is typically defined as a particular tour (a permutation of the cities to be visited); then one could define two tours to be neighbors if and only if one can be converted to the other by interchanging a pair of adjacent cities.
The probability of making the transition to the new state s' is a function P(δE, T) of the energy difference δE = E(s' ) - E(s) between the two states, and of a global time-varying parameter T called the temperature.
One essential feature of the SA method is that the transition probability P is defined to be nonzero even when δE is positive, meaning that the system may move to the new state even when it is worse (has a higher energy) than the current one. It is this feature that prevents the method from becoming stuck in a false minimum — a state whose energy is far from being minimum, but is still less than that of any neighbor.
On the other hand, the function P is such that, as the temperature T goes to zero, P(δE, T) tends to 1 if δE is negative, and to 0 if δE is positive. Therefore, for sufficiently small values T, the system will increasingly favor moves that go "downhill" (to lower energy values), and avoid those that go "uphill". In particular, when T is 0, the procedure reduces to the greedy heuristic — which makes the move if and only if it goes downhill.
Obviously, the effect of the state energies on the system's evolution depend crucially on the temperature. Roughly speaking, the evolution is sensitive only to coarser energy variations when T is large, and to finer variations then T is small.
Another essential feature of the SA method is that the temperature is gradually reduced as the simulation proceeds. Initially, T is set to a high value (or infinity), and it is decreased at each step according to some annealing schedule — which may be specified by the user, but must end with T=0 towards the end of the allotted time budget. In this way, the system is expected to wander initially towards a broad region of the search space containing good solutions, ignoring small featues of the energy function; then drift towards low-energy regions that become narrower and narrower; and finally move downhill according to the graph — where the states are vertices, and there is an edge from each state to each of its neighbors. Roughly speaking, it must be possible to go from the initial state to a "good enough" state by a relatively short path on this graph, and such a path must be as likely as possible to be followed by the SA iteration.
In practice, one tries to achieve this criterion by using a search graph where the neighbors of s are expected to have about the same energy as s. Thus, in the traveling salesman problem above, generating the neighbor by swapping two adjacent cities is expected to be more effective than swapping two arbitrary cities. It is true that reaching the goal can always be done with only n-1 general swaps, while it may take as many as n(n-1)/2 adjacent swaps. However, if one were to apply a random general swap to a fairly good solution, one would almost certainly get a large energy increase; whereas swapping two adjacent cities is likely to have a smaller effect on the energy.
The transition probability function P is not as critical as the neighborhood graph, provided that it follows the general requirements of the SA method stated before. Since the probabilities depend on the temperature T, in practice the same probability function is used for all problems, and the annealing schedule is adjusted accordingly.
In the original formulation of the method by Kirkpatric et. al, the transition probability P(δE, T) was defined as 1 if δE < 0 (i.e., downhill moves were always performed); otherwise, the probability would be e-δE/T. This formula was inspired on that of the Maxwell-Boltzmann distribution governing the distribution of molecular energies, and is still commonly used.
However, the resemblance is quite superficial, and the analogy is false. The Maxwell-Boltzmann formula gives the equilibrium distribution over all states, not the transition probability between two states. Another difference is that in a physical system the probability is less than 1 even when δE < 0. In fact, there is no evidence or argument indicating that the "classical" formula is better than any other.
The annealing schedule is less critical than the neighborhood function, but still must be chosen with care. The initial temperature must be large enough to make the uphill and downhill transition probabilities about the same. To do that, one must have some estimate of the value of δE for a random state and its neighbors.
The temperature must then decrease so that it is zero, or nearly zero, at the end of the alloted time. A popular choice is the exponential schedule, where the temperature decreases by a fixed factor α < 1 at each step.
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