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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 for an optimization problem. It is often used when the search space is discrete (e.g., the traveling salesman problem). 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 gradient descent.
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. 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. Simulated annealing can be used to approximate the global minimum for a function with many variables.
A procedure that is called now Simulated Annealing was developed by M. Pincus in 1970  who proposed to employ the Monte Carlo sampling based on Metropolis et al. algorithm (1953)  to find the global minimum of a function of many variables. The Metropolis algorithm was developed to determine the equilibrium state of multi-atomic systems— the Markov chain generated by their sampling method converges to the average over the Gibbs ensemble providing the free energy minimum at the given temperature. Pincus emphasized an analogy of his optimization method to the statistical mechanics that directly follows from the very nature of the Metropolis algorithm.
A generalization of the statistical mechanics of a non-ideal gas by extending its application to non-physical multi-variant systems, which is not related to the Metropolis algorithm, was proposed by Khachaturyan et al ( 1979,1981)  . The authors introduced definitions of the non-physical thermodynamic functions, partition sum, free energy, temperature, and equilibrium densities as averages over the Gibbs ensemble of these systems. It was shown that equilibration of the system followed by slow cooling results in optimization of the appropriate thermodynamic functions of multiple variables describing the system. As illustration of this thermodynamic approach to the optimization, determination of atomic structure of an amino acid (L-proline) from x-ray diffraction intensities was performed. A deviation of the calculated intensities of diffraction spots from the observed ones was considered as artificial “Hamiltonian” of the system.
Similarly to Pincus (1970), Kirkpatrick et al ( 1983) used the Metropolis algorithm (1953) The authors solved the traveling salesman problem. They also emphasized an analogue with the statistical mechanics and proposed the currently accepted term Simulation Annealing
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 allows for a more extensive search for the global optimal solution. In general, simulated annealing algorithms work as follows. The temperature progressively decreases from an initial positive value to zero. At each time step, the algorithm randomly selects a solution close to the current one, measures its quality, and moves to it according to the temperature-dependent probabilities of selecting better or worse solutions, which during the search respectively remain at 1 (or positive) and decrease 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 Metropolis–Hastings algorithm, a Monte Carlo method to generate sample states of a thermodynamic system, published by N. Metropolis et al. in 1953.
The state of some physical systems, and the function E(s) to be minimized, is analogous to the internal energy of the system in that state. The goal is to bring the system, from an arbitrary initial state, to a state with the minimum possible energy.
The basic iteration
At each step, the simulated annealing heuristic considers some neighboring state s* of the current state s, and probabilistically decides between moving the system to state s* or staying in-state s. These probabilities ultimately lead the system to move to states of lower energy. Typically this step is repeated until the system reaches a state that is good enough for the application, or until a given computation budget has been exhausted.
The neighbours of a state
Optimization of a solution involves evaluating the neighbours of a state of the problem, which are new states produced through conservatively altering a given state. For example, in the travelling salesman problem each state is typically defined as a permutation of the cities to be visited, and its neighbours are the set of permutations produced by swapping any two cities. The well-defined way in which the states are altered to produce neighbouring states is called a "move", and different moves give different sets of neighbouring states. These moves usually result in minimal alterations of the last state, in an attempt to progressively improve the solution through iteratively improving its parts (such as the city connections in the traveling salesman problem).
Simple heuristics like hill climbing, which move by finding better neighbour after better neighbour and stop when they have reached a solution which has no neighbours that are better solutions, cannot guarantee to lead to any of the existing better solutions – their outcome may easily be just a local optimum, while the actual best solution would be a global optimum that could be different. Metaheuristics use the neighbours of a solution as a way to explore the solutions space, and although they prefer better neighbours, they also accept worse neighbours in order to avoid getting stuck in local optima; they can find the global optimum if run for a long enough amount of time.
The probability of making the transition from the current state to a candidate new state is specified by an acceptance probability function , that depends on the energies and of the two states, and on a global time-varying parameter called the temperature. States with a smaller energy are better than those with a greater energy. The probability function must be positive even when is greater than . This feature prevents the method from becoming stuck at a local minimum that is worse than the global one.
When tends to zero, the probability must tend to zero if and to a positive value otherwise. For sufficiently small values of , the system will then increasingly favor moves that go "downhill" (i.e., to lower energy values), and avoid those that go "uphill." With the procedure reduces to the greedy algorithm, which makes only the downhill transitions.
In the original description of simulated annealing, the probability was equal to 1 when —i.e., the procedure always moved downhill when it found a way to do so, irrespective of the temperature. Many descriptions and implementations of simulated annealing still take this condition as part of the method's definition. However, this condition is not essential for the method to work.
The function is usually chosen so that the probability of accepting a move decreases when the difference increases—that is, small uphill moves are more likely than large ones. However, this requirement is not strictly necessary, provided that the above requirements are met.
Given these properties, the temperature plays a crucial role in controlling the evolution of the state of the system with regard to its sensitivity to the variations of system energies. To be precise, for a large , the evolution of is sensitive to coarser energy variations, while it is sensitive to finer energy variations when is small.
The annealing schedule
The name and inspiration of the algorithm demand an interesting feature related to the temperature variation to be embedded in the operational characteristics of the algorithm. This necessitates a gradual reduction of the temperature as the simulation proceeds. The algorithm starts initially with set to a high value (or infinity), and then it is decreased at each step following some annealing schedule—which may be specified by the user, but must end with 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 features of the energy function; then drift towards low-energy regions that become narrower and narrower; and finally move downhill according to the steepest descent heuristic.
For any given finite problem, the probability that the simulated annealing algorithm terminates with a global optimal 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 complete search of the solution space.
The following pseudocode presents the simulated annealing heuristic as described above. It starts from a state s0 and continues until a maximum of kmax 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], uniformly at random. 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.
Selecting the parameters
In order to apply the simulated annealing method to a specific problem, one must specify the following parameters: the state space, the energy (goal) function
E(), the candidate generator procedure
neighbour(), the acceptance probability function
P(), and the annealing schedule
temperature() AND initial temperature <init temp>. These choices can have a significant impact on the method's effectiveness. Unfortunately, there are no choices of these parameters that will be good for all problems, and there is no general way to find the best choices for a given problem. The following sections give some general guidelines.
Sufficiently near neighbour
Simulated annealing may be modeled as a random walk on a search graph, whose vertices are all possible states, and whose edges are the candidate moves. An essential requirement for the
neighbour() function is that it must provide a sufficiently short path on this graph from the initial state to any state which may be the global optimum – the diameter of the search graph must be small. In the traveling salesman example above, for instance, the search space for n = 20 cities has n! = 2,432,902,008,176,640,000 (2.4 quintillion) states; yet the number of neighbors of each vertex is edges, and the diameter of the graph is .
To investigate the behavior of simulated annealing on a particular problem, it can be useful to consider the transition probabilities that result from the various design choices made in the implementation of the algorithm. For each edge of the search graph, the transition probability is defined as the probability that the simulated annealing algorithm will move to state when its current state is . This probability depends on the current temperature as specified by
temperature(), on the order in which the candidate moves are generated by the
neighbour() function, and on the acceptance probability function
P(). (Note that the transition probability is not simply , because the candidates are tested serially.)
The specification of
temperature() is partially redundant. In practice, it's common to use the same acceptance function
P() for many problems, and adjust the other two functions according to the specific problem.
In the formulation of the method by Kirkpatrick et al., the acceptance probability function was defined as 1 if , and otherwise. This formula was superficially justified by analogy with the transitions of a physical system; it corresponds to the Metropolis–Hastings algorithm, in the case where T=1 and the proposal distribution of Metropolis–Hastings is symmetric. However, this acceptance probability is often used for simulated annealing even when the
neighbour() function, which is analogous to the proposal distribution in Metropolis–Hastings, is not symmetric, or not probabilistic at all. As a result, the transition probabilities of the simulated annealing algorithm do not correspond to the transitions of the analogous physical system, and the long-term distribution of states at a constant temperature need not bear any resemblance to the thermodynamic equilibrium distribution over states of that physical system, at any temperature. Nevertheless, most descriptions of simulated annealing assume the original acceptance function, which is probably hard-coded in many implementations of SA.
In 1990, Moscato and Fontanari , and independently Dueck and Scheuer , proposed that a deterministic update (i.e. one that is not based on the probabilistic acceptance rule) could speed-up the optimization process without impacting on the final quality. Moscato and Fontanari conclude from observing the analogous of the "specific heat" curve of the "threshold updating" annealing originating from their study that "the stochasticity of the Metropolis updating in the simulated annealing algorithm does not play a major role in the search of near-optimal minima". Instead, they proposed that "the smoothening of the cost function landscape at high temperature and the gradual definition of the minima during the cooling process are the fundamental ingredients for the success of simulated annealing." The method subsequently popularized under the denomination of "threshold accepting" due to Dueck and Scheuer's denomination. In 2001, Franz, Hoffmann and Salamon showed that the deterministic update strategy is indeed the optimal one within the large class of algorithms that simulate a random walk on the cost/energy landscape .
Efficient candidate generation
When choosing the candidate generator
neighbour(), one must consider that after a few iterations of the simulated annealing algorithm, the current state is expected to have much lower energy than a random state. Therefore, as a general rule, one should skew the generator towards candidate moves where the energy of the destination state is likely to be similar to that of the current state. This heuristic (which is the main principle of the Metropolis–Hastings algorithm) tends to exclude "very good" candidate moves as well as "very bad" ones; however, the former are usually much less common than the latter, so the heuristic is generally quite effective.
In the traveling salesman problem above, for example, swapping two consecutive cities in a low-energy tour is expected to have a modest effect on its energy (length); whereas swapping two arbitrary cities is far more likely to increase its length than to decrease it. Thus, the consecutive-swap neighbour generator is expected to perform better than the arbitrary-swap one, even though the latter could provide a somewhat shorter path to the optimum (with swaps, instead of ).
A more precise statement of the heuristic is that one should try first candidate states for which is large. For the "standard" acceptance function above, it means that is on the order of or less. Thus, in the traveling salesman example above, one could use a
neighbour() function that swaps two random cities, where the probability of choosing a city-pair vanishes as their distance increases beyond .
When choosing the candidate generator
neighbour() one must also try to reduce the number of "deep" local minima—states (or sets of connected states) that have much lower energy than all its neighbouring states. Such "closed catchment basins" of the energy function may trap the simulated annealing algorithm with high probability (roughly proportional to the number of states in the basin) and for a very long time (roughly exponential on the energy difference between the surrounding states and the bottom of the basin).
As a rule, it is impossible to design a candidate generator that will satisfy this goal and also prioritize candidates with similar energy. On the other hand, one can often vastly improve the efficiency of simulated annealing by relatively simple changes to the generator. In the traveling salesman problem, for instance, it is not hard to exhibit two tours , , with nearly equal lengths, such that (1) is optimal, (2) every sequence of city-pair swaps that converts to goes through tours that are much longer than both, and (3) can be transformed into by flipping (reversing the order of) a set of consecutive cities. In this example, and lie in different "deep basins" if the generator performs only random pair-swaps; but they will be in the same basin if the generator performs random segment-flips.
The physical analogy that is used to justify simulated annealing assumes that the cooling rate is low enough for the probability distribution of the current state to be near thermodynamic equilibrium at all times. Unfortunately, the relaxation time—the time one must wait for the equilibrium to be restored after a change in temperature—strongly depends on the "topography" of the energy function and on the current temperature. In the simulated annealing algorithm, the relaxation time also depends on the candidate generator, in a very complicated way. Note that all these parameters are usually provided as black box functions to the simulated annealing algorithm. Therefore, the ideal cooling rate cannot be determined beforehand, and should be empirically adjusted for each problem. Adaptive simulated annealing algorithms address this problem by connecting the cooling schedule to the search progress. Other adaptive approach as Thermodynamic Simulated Annealing, automatically adjusts the temperature at each step based on the energy difference between the two states, according to the laws of thermodynamics.
Sometimes it is better to move back to a solution that was significantly better rather than always moving from the current state. This process is called restarting of simulated annealing. To do this we set
ebest and perhaps restart the annealing schedule. The decision to restart could be based on several criteria. Notable among these include restarting based on a fixed number of steps, based on whether the current energy is too high compared to the best energy obtained so far, restarting randomly, etc.
- Interacting Metropolis–Hasting algorithms (a.k.a. Sequential Monte Carlo) combined simulated annealing moves with an acceptance-rejection of the best fitted individuals equipped with an interacting recycling mechanism.
- Quantum annealing uses "quantum fluctuations" instead of thermal fluctuations to get through high but thin barriers in the target function.
- Stochastic tunneling attempts to overcome the increasing difficulty simulated annealing runs have in escaping from local minima as the temperature decreases, by 'tunneling' through barriers.
- Tabu search normally moves to neighbouring states of lower energy, but will take uphill moves when it finds itself stuck in a local minimum; and avoids cycles by keeping a "taboo list" of solutions already seen.
- Dual-phase evolution is a family of algorithms and processes (to which simulated annealing belongs) that mediate between local and global search by exploiting phase changes in the search space.
- Reactive search optimization focuses on combining machine learning with optimization, by adding an internal feedback loop to self-tune the free parameters of an algorithm to the characteristics of the problem, of the instance, and of the local situation around the current solution.
- Genetic algorithms maintain a pool of solutions rather than just one. New candidate solutions are generated not only by "mutation" (as in SA), but also by "recombination" of two solutions from the pool. Probabilistic criteria, similar to those used in SA, are used to select the candidates for mutation or combination, and for discarding excess solutions from the pool.
- Memetic algorithms search for solutions by employing a set of agents that both cooperate and compete in the process; sometimes the agents' strategies involve simulated annealing procedures for obtaining high quality solutions before recombining them. Annealing has also been suggested as a mechanism for increasing the diversity of the search .
- Graduated optimization digressively "smooths" the target function while optimizing.
- Ant colony optimization (ACO) uses many ants (or agents) to traverse the solution space and find locally productive areas.
- The cross-entropy method (CE) generates candidates solutions via a parameterized probability distribution. The parameters are updated via cross-entropy minimization, so as to generate better samples in the next iteration.
- Harmony search mimics musicians in improvisation process where each musician plays a note for finding a best harmony all together.
- Stochastic optimization is an umbrella set of methods that includes simulated annealing and numerous other approaches.
- Particle swarm optimization is an algorithm modelled on swarm intelligence that finds a solution to an optimization problem in a search space, or model and predict social behavior in the presence of objectives.
- The runner-root algorithm (RRA) is a meta-heuristic optimization algorithm for solving unimodal and multimodal problems inspired by the runners and roots of plants in nature.
- Intelligent water drops algorithm (IWD) which mimics the behavior of natural water drops to solve optimization problems
- Parallel tempering is a simulation of model copies at different temperatures (or Hamiltonians) to overcome the potential barriers.
- Adaptive simulated annealing
- Markov chain
- Combinatorial optimization
- Dual-phase evolution
- Automatic label placement
- Multidisciplinary optimization
- Place and route
- Molecular dynamics
- Traveling salesman problem
- Graph cuts in computer vision
- Particle swarm optimization
- Intelligent water drops algorithm
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