annealing

Simulated Annealing

Simulated annealing (SA) is a probabilistic technique for approximating the global optimum of a given function. Specifically, it is a metaheuristic to approximate global optimization in a large search space. From a practical perspective, simulated annealing is useful for combinatorial optimization, that is, the problem of finding an optimal configuration of discrete choices.

The concept of simulated annealing is based on 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 and wander randomly throughout the solid. As the material cools, the atoms settle into a more orderly and lower-energy configuration, which depends on the chemical composition of the material and the temperature. The cooling is analogous to a simulated annealing process, in which the temperature is slowly lowered and the system is expected to reach a stable low-energy state.

In simulated annealing, the system is slowly cooled down from an initial high temperature (Tmax) to a final resting temperature (Tmin). At each stage of the cooling process, a transition occurs from a randomly generated state to a more ordered, low-energy state. This transition is analogous to a process of minimizing a function of many variables.

During the transition period, states are accepted with a probability that is a decreasing function of the transition energy, determined by the Boltzmann probability. More precisely, the probability of a transition is:

P(E*→E) = exp(-(E*-E)/kT),

where E* is the value of the energy at the initial state and E is the value of the energy at the new state.

The Boltzmann probability implies that lower energy states are accepted with a higher probability than higher energy states. This encourages the system to move towards a minimal energy state. Furthermore, the temperature decreases as the system moves towards the final state, leading to an orderly evolution towards the desired low-energy state.

The advantages of simulated annealing compared to other optimization techniques include its ability to escape local minima, its ability to adapt to different types of problems, and its relative speed and simplicity. Furthermore, simulated annealing is an effective optimization technique for complex, large-scale problems where the number of variables is too large to completely search the entire state space.

Simulated annealing has been applied to a variety of different problems, including integer programming, combinatorial optimization, network control, simulation optimization, and many others. Due to its successes, simulated annealing has become popular in many fields, including mathematics, operations research, and engineering. In addition, simulated annealing has been used to solve many practical problems, such as design optimization, scheduling, resource allocation, and logistics.

Simulated annealing is a general-purpose optimization method that works on the principle of “annealing” in metallurgy, a process whereby a material is heated to a high temperature, cooled, heated again and again and eventually cooled to an optimal temperature, producing an end product with desirable properties. In the context of computer science, simulated annealing is a metaheuristic which is used to find approximate solutions to optimization problems with many possible variables.

The workings of simulated annealing are simple but highly effective. A set of parameters is defined which constitute the problem space, such as the variables and cost functions. A simulated temperature is then set and a starting solution, or set of parameters, is chosen. This solution is then evaluated and compared to its immediate neighbours to determine its cost. If any of the neighbouring solutions have a better cost, then that solution is accepted and the current solution discarded. This process is then repeated until no better solution can be found.

When the temperature reaches the ultimate low, the best solution so far is accepted as the optimal solution to the given problem. This process is based on the idea that, at higher temperatures, the system is more likely to accept solutions which may be slightly worse than the current one, as it is more likely to jump to a different region of the parameter space. In other words, at higher temperatures, the algorithm is more likely to explore the landscape of possible solutions, while at lower temperatures it is more likely to stay in the same region of the solution space.

As can be seen, simulated annealing is a powerful algorithm which can be used to find approximate solutions to a variety of optimization problems. One of its main advantages over mathematical optimization methods is that it is less prone to becoming stuck in a local optimum, due to its random walk behavior. Furthermore, it can provide good solutions without having to search the entire problem space, as it relies on evaluating only a few neighbouring solutions in order to ascertain the best one.