Simulated annealing (SA) is often used when the search space is clear. In addition, simulated annealing may be better than exact algorithms like gradient descent or branch and bound when finding an approximate global optimum is more important than a precise local optimum in a set amount of time.

The method is named after the metallurgical process of annealing, which involves heating and carefully controlling the cooling of a material to change its physical characteristics. Both are characteristics of the substance that rely on its thermodynamic free energy. The material's temperature and thermodynamic free energy, also known as Gibbs energy, are both impacted by heating and cooling. Even though simulated annealing typically only yields an approximation of the global minimum, it may be sufficient for many real-world issues when applied to highly challenging computational optimization situations where exact techniques fall short.

An objective function of numerous variables, subject to several restrictions, formulates the issues that SA presently solves. In actuality, the objective function may include a penalty for the constraint.

As the solution space is searched, this concept of slow cooling employed in the simulated annealing process is understood as a gradual decrease in the probability of accepting inferior solutions. Accepting low alternatives permits a more broad search for the ideal solution on a global scale. In general, simulated annealing techniques function as described below: The temperature gradually falls from a positive initial value to zero. 

At each time step, the algorithm evaluates its quality and advances based on the temperature-dependent probability of selecting better or worse solutions, which remain at 1 (or are positive) and decline towards 0 during the search. We can use a solution of kinetic equations for density functions or stochastic sampling to perform the simulation. The approach is an adaptation of the Metropolis-Hastings algorithm, published by N. Metropolis et al. in 1953.

Conclusion

SA is one of the most popular heuristic solutions for optimization issues. The researchers introduced SA by emulating the metalworking annealing process. The annealing technique identifies the ideal molecular configurations of metal particles in which the potential energy of the mass is minimised and refers to the progressive cooling of metals following exposure to high heat. The SA method generally employs an iterative movement based on a changeable temperature parameter that simulates the metals' annealing process.

A simple optimization technique successively compares the outputs of the objective functions running with the current position and the nearby point in the domain. If the neighbouring end produces a better result than the current point, it is saved as the base solution for the next iteration. If not, the algorithm terminates the procedure without examining the broader domain for better outcomes. Thus, the algorithm can become stuck in local minima or maxima. Instead, SA provides a method that incorporates two iterative loops, namely the cooling procedure for the annealing process and the Metropolis criterion. The Metropolis criterion is based on the premise that it should be applied arbitrarily to further search the neighbourhood of the candidate solution to avoid being trapped by local extremes.