Simulated annealing

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.

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 the neighbors of any state are the set of permutations produced by swapping any two of these cities. The well-defined way in which the states are altered to produce neighboring states is called a "move", and different moves give different sets of neighboring 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 solution 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 ${\displaystyle s}$ to a candidate new state ${\displaystyle s_{\mathrm {new} }}$ is specified by an acceptance probability function ${\displaystyle P(e,e_{\mathrm {new} },T)}$, that depends on the energies ${\displaystyle e=E(s)}$ and ${\displaystyle e_{\mathrm {new} }=E(s_{\mathrm {new} })}$ of the two states, and on a global time-varying parameter ${\displaystyle T}$ called the temperature. States with a smaller energy are better than those with a greater energy. The probability function ${\displaystyle P}$ must be positive even when ${\displaystyle e_{\mathrm {new} }}$ is greater than ${\displaystyle e}$. This feature prevents the method from becoming stuck at a local minimum that is worse than the global one.

When ${\displaystyle T}$ tends to zero, the probability ${\displaystyle P(e,e_{\mathrm {new} },T)}$ must tend to zero if ${\displaystyle e_{\mathrm {new} }>e}$ and to a positive value otherwise. For sufficiently small values of ${\displaystyle T}$, the system will then increasingly favor moves that go "downhill" (i.e., to lower energy values), and avoid those that go "uphill." With ${\displaystyle T=0}$ the procedure reduces to the greedy algorithm, which makes only the downhill transitions.

In the original description of simulated annealing, the probability ${\displaystyle P(e,e_{\mathrm {new} },T)}$ was equal to 1 when ${\displaystyle e_{\mathrm {new} }<e}$—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 ${\displaystyle P}$ function is usually chosen so that the probability of accepting a move decreases when the difference ${\displaystyle e_{\mathrm {new} }-e}$ 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 ${\displaystyle T}$ plays a crucial role in controlling the evolution of the state ${\displaystyle s}$ of the system with regard to its sensitivity to the variations of system energies. To be precise, for a large ${\displaystyle T}$, the evolution of ${\displaystyle s}$ is sensitive to coarser energy variations, while it is sensitive to finer energy variations when ${\displaystyle T}$ is small.

Fast

Slow

Example illustrating the effect of cooling schedule on the performance of simulated annealing. The problem is to rearrange the pixels of an image so as to minimize a certain potential energy function, which causes similar colors 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 amorphous and crystalline solids, respectively.

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 ${\displaystyle T}$ 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 ${\displaystyle T=0}$ 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.[10] 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.

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 ${\displaystyle \sum _{k=1}^{n-1}k={\frac {n(n-1)}{2}}=190}$ edges (coming from n choose 2), and the diameter of the graph is ${\displaystyle n-1}$.

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 ${\displaystyle (s,s')}$ of the search graph, the transition probability is defined as the probability that the simulated annealing algorithm will move to state ${\displaystyle s'}$ when its current state is ${\displaystyle s}$. 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 ${\displaystyle P(e,e',T)}$, because the candidates are tested serially.)

The specification of neighbour(), P(), and 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 ${\displaystyle P(e,e',T)}$ was defined as 1 if ${\displaystyle e'<e}$, and ${\displaystyle \exp(-(e'-e)/T)}$ 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 ${\displaystyle T}$ 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,[11] and independently Dueck and Scheuer,[12] 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.[13]

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 ${\displaystyle s'}$ 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 ${\displaystyle n-1}$ swaps, instead of ${\displaystyle n(n-1)/2}$).

A more precise statement of the heuristic is that one should try first candidate states ${\displaystyle s'}$ for which ${\displaystyle P(E(s),E(s'),T)}$ is large. For the "standard" acceptance function ${\displaystyle P}$ above, it means that ${\displaystyle E(s')-E(s)}$ is on the order of ${\displaystyle T}$ 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 ${\displaystyle T}$.

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 ${\displaystyle A}$, ${\displaystyle B}$, with nearly equal lengths, such that (1) ${\displaystyle A}$ is optimal, (2) every sequence of city-pair swaps that converts ${\displaystyle A}$ to ${\displaystyle B}$ goes through tours that are much longer than both, and (3) ${\displaystyle A}$ can be transformed into ${\displaystyle B}$ by flipping (reversing the order of) a set of consecutive cities. In this example, ${\displaystyle A}$ and ${\displaystyle B}$ 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,[14] automatically adjusts the temperature at each step based on the energy difference between the two states, according to the laws of thermodynamics.