Simulated Annealing
Simulated Annealing (SA) is a probabilistic technique used for finding the global optimum of a given function. It is particularly useful for optimization problems with a large search space. The method is inspired by the annealing process in metallurgy, a technique involving heating and controlled cooling of material to increase the size of its crystals and reduce their defects.
Overview
Simulated Annealing is a metaheuristic approach to solve global optimization problems that may have many local optima. It’s often used when the search space is discrete (e.g., all tours that visit a given set of cities). The main idea is to simulate the physical process of heating a material and then slowly lowering the temperature to decrease defects, thus minimizing the system energy.
At each step, the SA 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 a probability that decreases as the temperature goes down. In this way, this method avoids being trapped in local optima and is able to explore globally for more possible solutions.
Algorithm
The algorithm starts with an initial solution at a high initial temperature and then the temperature ‘cools’ over time. At each iteration, a neighbor solution is randomly generated. If the neighbor solution is better, it is accepted. However, if it is not, it can still be accepted with a certain probability, which is determined by the current temperature and the difference in quality between the current solution and the neighbor solution. This allows for a better exploration of the solution space.
The cooling schedule, which determines how the temperature decreases over time, is a critical component of the simulated annealing algorithm. It should be slow enough to allow a good exploration of the solution space, but not so slow that the algorithm takes too long to converge.
Applications
Simulated Annealing has been successfully applied to a wide range of optimization problems, including:
- Traveling Salesman Problem (TSP)
- Vehicle Routing Problem (VRP)
- Scheduling Problems
- Machine Learning Hyperparameter Tuning
Advantages and Disadvantages
Advantages of Simulated Annealing include:
- It’s a flexible method that can handle a wide range of optimization problems.
- It can escape local optima and has a good chance of finding the global optimum.
Disadvantages include:
- The cooling schedule needs to be carefully set. If it’s too fast, the algorithm may converge to a local optimum; if it’s too slow, the algorithm may take too long to converge.
- It may require a large number of iterations to find a good solution, especially for complex problems.
Further Reading
- Kirkpatrick, S., Gelatt, C.D., Vecchi, M.P., 1983. Optimization by Simulated Annealing. Science 220, 671–680.
- Cerny, V. (1985). Thermodynamical approach to the traveling salesman problem: An efficient simulation algorithm. Journal of Optimization Theory and Applications, 45(1), 41-51.
Simulated Annealing is a powerful optimization technique that can be applied to a wide range of problems. Its ability to avoid local optima and explore the solution space makes it a valuable tool for many optimization tasks.
