Simulated Annealing can be used to solve combinatorial problems. Here it is applied to the to minimize the length of a route that connects all 125 points.Simulated annealing ( SA) is a for approximating the of a given.
Specifically, it is a to approximate in a large for an. It is often used when the search space is discrete (e.g., the ).
For problems where finding an approximate global optimum is more important than finding a precise local optimum in a fixed amount of time, simulated annealing may be preferable to alternatives such as.The name and inspiration come from, a technique involving heating and controlled cooling of a material to increase the size of its crystals and reduce their defects. Both are attributes of the material that depend on its thermodynamic free energy. Heating and cooling the material affects both the temperature and the thermodynamic free energy.The simulation of annealing can be used to find an approximation of a global minimum for a function with a large number of variables.This notion of slow cooling implemented in the simulated annealing algorithm is interpreted as a slow decrease in the probability of accepting worse solutions as the solution space is explored. Accepting worse solutions is a fundamental property of metaheuristics because it allows for a more extensive search for the global optimal solution. In general, the simulated annealing algorithms work as follows.
@acceptancesa (default) — Simulated annealing acceptance function. The probability of acceptance is The probability of acceptance is 1 1 + exp ( Δ max ( T ) ). Simulated annealing improves this strategy through the introduction of two tricks. Are accepted, as are any bad trades that raise the cost function by less than a.
At each time step, the algorithm randomly selects a solution close to the current one, measures its quality, and then decides to move to it or to stay with the current solution based on either one of two probabilities between which it chooses on the basis of the fact that the new solution is better or worse than the current one. During the search, the temperature is progressively decreased from an initial positive value to zero and affects the two probabilities: at each step, the probability of moving to a better new solution is either kept to 1 or is changed towards a positive value; on the other hand, the probability of moving to a worse new solution is progressively changed towards zero.The simulation can be performed either by a solution of kinetic equations for density functions or by using the stochastic sampling method. The method is an adaptation of the, a to generate sample states of a thermodynamic system, published by et al. SlowExample illustrating the effect of cooling schedule on the performance of simulated annealing.
The problem is to rearrange the of an image so as to minimize a certain function, which causes similar to attract at short range and repel at a slightly larger distance. The elementary moves swap two adjacent pixels. These images were obtained with a fast cooling schedule (left) and a slow cooling schedule (right), producing results similar to and, respectively.For any given finite problem, the probability that the simulated annealing algorithm terminates with a solution approaches 1 as the annealing schedule is extended. This theoretical result, however, is not particularly helpful, since the time required to ensure a significant probability of success will usually exceed the time required for a of the.
Pseudocode The following pseudocode presents the simulated annealing heuristic as described above. It starts from a state s 0 and continues until a maximum of k max steps have been taken. In the process, the call neighbour( s) should generate a randomly chosen neighbour of a given state s; the call random(0, 1) should pick and return a value in the range 0, 1,. The annealing schedule is defined by the call temperature( r), which should yield the temperature to use, given the fraction r of the time budget that has been expended so far.