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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.
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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 toward zero.
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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). It is even better to reverse the order of an interval of cities. This is a smaller move since swapping two cities can be achieved by twice reversing an interval.
2552:āthe time one must wait for the equilibrium to be restored after a change in temperatureāstrongly 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
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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
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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
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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
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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,
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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.
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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,
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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
1520:. 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.
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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
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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.
1970:
function, which is analogous to the proposal distribution in
MetropolisāHastings, 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
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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
993:āi.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.
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algorithms address this problem by connecting the cooling schedule to the search progress. Other adaptive approaches such 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
2008:, 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
1991:
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.
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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).
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Similar techniques have been independently introduced on several occasions, including Pincus (1970), Khachaturyan et al (1979, 1981), Kirkpatrick, Gelatt and Vecchi (1983), and Cerny (1985). In 1983, this approach was used by
Kirkpatrick, Gelatt Jr., Vecchi, for a solution of the
385:, which move by finding better neighbor after better neighbor and stop when they have reached a solution which has no neighbors that are better solutions, cannot guarantee to lead to any of the existing better solutions – their outcome may easily be just a
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use the neighbors of a solution as a way to explore the solution space, and although they prefer better neighbors, they also accept worse neighbors in order to avoid getting stuck in local optima; they can find the global optimum if run for a long enough amount of time.
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cities is far more likely to increase its length than to decrease it. Thus, the consecutive-swap neighbor 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
357:. 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.
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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
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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
228:. Simulated annealing can be used for very hard computational optimization problems where exact algorithms fail; even though it usually achieves an approximate solution to the global minimum, it could be enough for many practical problems.
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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
201:). 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 exact algorithms such as
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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.
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2670:(CE) generates candidate solutions via a parameterized probability distribution. The parameters are updated via cross-entropy minimization, so as to generate better samples in the next iteration.
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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
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Optimization of a solution involves evaluating the neighbors of a state of the problem, which are new states produced through conservatively altering a given state. For example, in the
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is an algorithm modeled on swarm intelligence that finds a solution to an optimization problem in a search space, or models and predicts social behavior in the presence of objectives.
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1966:, in the case where T=1 and the proposal distribution of MetropolisāHastings is symmetric. However, this acceptance probability is often used for simulated annealing even when the
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Metropolis, Nicholas; Rosenbluth, Arianna W.; Rosenbluth, Marshall N.; Teller, Augusta H.; Teller, Edward (1953). "Equation of State
Calculations by Fast Computing Machines".
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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
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one must also try to reduce the number of "deep" local minimaāstates (or sets of connected states) that have much lower energy than all its neighboring states. Such "closed
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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.
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increasesāthat is, small uphill moves are more likely than large ones. However, this requirement is not strictly necessary, provided that the above requirements are met.
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V.Vassilev, A.Prahova: "The Use of
Simulated Annealing in the Control of Flexible Manufacturing Systems", International Journal INFORMATION THEORIES & APPLICATIONS,
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attempts to overcome the increasing difficulty simulated annealing runs have in escaping from local minima as the temperature decreases, by 'tunneling' through barriers.
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Moscato, Pablo (June 1993). "An introduction to population approaches for optimization and hierarchical objective functions: A discussion on the role of tabu search".
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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.
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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
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to the simulated annealing algorithm. Therefore, the ideal cooling rate cannot be determined beforehand and should be empirically adjusted for each problem.
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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
255:
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Khachaturyan, A.: Semenovskaya, S.: Vainshtein B., Armen (1979). "Statistical-Thermodynamic
Approach to Determination of Structure Amplitude Phases".
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Ars
Technica discusses the possibility that the D-Wave computer being used by Google may, in fact, be an efficient simulated annealing co-processor.
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2606:) combines simulated annealing moves with an acceptance-rejection of the best-fitted individuals equipped with an interacting recycling mechanism.
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Simulated annealing searching for a maximum. The objective here is to get to the highest point. In this example, it is not enough to use a simple
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Khachaturyan, A.; Semenovskaya, S.; Vainshtein, B. (1979). "Statistical-Thermodynamic
Approach to Determination of Structure Amplitude Phases".
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3843:
863:, the system will then increasingly favor moves that go "downhill" (i.e., to lower energy values), and avoid those that go "uphill." With
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Dueck, G.; Scheuer, T. (1990), "Threshold accepting: A general purpose optimization algorithm appearing superior to simulated annealing",
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Pincus, Martin (NovāDec 1970). "A Monte-Carlo Method for the Approximate Solution of Certain Types of Constrained Optimization Problems".
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of the search graph, the transition probability is defined as the probability that the simulated annealing algorithm will move to state
185:. For large numbers of local optima, SA can find the global optimum. It is often used when the search space is discrete (for example the
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function that swaps two random cities, where the probability of choosing a city-pair vanishes as their distance increases beyond
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otherwise. This formula was superficially justified by analogy with the transitions of a physical system; it corresponds to the
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Example illustrating the effect of cooling schedule on the performance of simulated annealing. The problem is to rearrange the
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Khachaturyan, A.; Semenovskaya, S.; Vainshtein, B. (1981). "The Thermodynamic Approach to the Structure Analysis of Crystals".
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562:
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Moscato, P. (1989). "On Evolution, Search, Optimization, Genetic Algorithms and Martial Arts: Towards Memetic Algorithms".
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optimization algorithm for solving unimodal and multimodal problems inspired by the runners and roots of plants in nature.
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uses "quantum fluctuations" instead of thermal fluctuations to get through high but thin barriers in the target function.
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ones; however, the former are usually much less common than the latter, so the heuristic is generally quite effective.
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3112:ÄernĆ½, V. (1985). "Thermodynamical approach to the traveling salesman problem: An efficient simulation algorithm".
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The following pseudocode presents the simulated annealing heuristic as described above. It starts from a state
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of the system with regard to its sensitivity to the variations of system energies. To be precise, for a large
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cities in a low-energy tour is expected to have a modest effect on its energy (length); whereas swapping two
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To investigate the behavior of simulated annealing on a particular problem, it can be useful to consider the
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Deb, Bandyopadhyay (June 2008). "A Simulated Annealing-Based Multiobjective Optimization Algorithm: AMOSA".
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De Vicente, Juan; Lanchares, Juan; Hermida, RomƔn (2003). "Placement by thermodynamic simulated annealing".
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135:
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Weinberger, E. (1990). "Correlated and uncorrelated fitness landscapes and how to tell the difference".
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that result from the various design choices made in the implementation of the algorithm. For each edge
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n! = 2,432,902,008,176,640,000 (2.4 quintillion) states; yet the number of neighbors of each vertex is
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For any given finite problem, the probability that the simulated annealing algorithm terminates with a
648:. States with a smaller energy are better than those with a greater energy. The probability function
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20:
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Moscato, P.; Fontanari, J.F. (1990), "Stochastic versus deterministic update in simulated annealing",
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mimics musicians in improvisation where each musician plays a note to find the best harmony together.
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function is usually chosen so that the probability of accepting a move decreases when the difference
224:. Heating and cooling the material affects both the temperature and the thermodynamic free energy or
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Granville, V.; Krivanek, M.; Rasson, J.-P. (1994). "Simulated annealing: A proof of convergence".
2663:(ACO) uses many ants (or agents) to traverse the solution space and find locally productive areas.
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Franz, A.; Hoffmann, K.H.; Salamon, P (2001), "Best optimal strategy for finding ground states",
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is sensitive to coarser energy variations, while it is sensitive to finer energy variations when
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Kirkpatrick, S.; Gelatt Jr, C. D.; Vecchi, M. P. (1983). "Optimization by Simulated Annealing".
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A Javascript app that allows you to experiment with simulated annealing. Source code included.
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is an umbrella set of methods that includes simulated annealing and numerous other approaches.
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A more precise statement of the heuristic is that one should try the first candidate states
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In the formulation of the method by Kirkpatrick et al., the acceptance probability function
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Simulated annealing can be used to solve combinatorial problems. Here it is applied to the
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Del Moral, Pierre; Doucet, Arnaud; Jasra, Ajay (2006). "Sequential Monte Carlo samplers".
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for many problems and adjust the other two functions according to the specific problem.
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set to a high value (or infinity), and then it is decreased at each step following some
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is partially redundant. In practice, it's common to use the same acceptance function
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2705:(IWD) which mimics the behavior of natural water drops to solve optimization problems
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3233:, vol. 1996, Berlin, Heidelberg: Springer Berlin Heidelberg, pp. 175ā180,
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by flipping (reversing the order of) a set of consecutive cities. In this example,
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239:. In practice, the constraint can be penalized as part of the objective function.
216:, a technique involving heating and controlled cooling of a material to alter its
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At each step, the simulated annealing heuristic considers some neighboring state
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Travelling salesman problem in 3D for 120 points solved with simulated annealing.
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of the system in that state. The goal is to bring the system, from an arbitrary
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A Simulated Annealing-Based Multiobjective Optimization Algorithm: AMOSA.
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An open-source MATLAB program for general simulated annealing exercises.
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or less. Thus, in the traveling salesman example above, one could use a
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1736:. This probability depends on the current temperature as specified by
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3618:"On simulated annealing phase transitions in phylogeny reconstruction"
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In the original description of simulated annealing, the probability
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and to a positive value otherwise. For sufficiently small values of
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In the traveling salesman problem above, for example, swapping two
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3057:
2900:"The Thermodynamic Approach to the Structure Analysis of Crystals"
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edges (coming from n choose 20), and the diameter of the graph is
274:
to generate sample states of a thermodynamic system, published by
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Multi-objective simulated annealing algorithms have been used in
1740:, on the order in which the candidate moves are generated by the
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is optimal, (2) every sequence of city-pair swaps that converts
1350:, which should yield the temperature to use, given the fraction
328:. By cooling the temperature slowly the global maximum is found.
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Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007).
1080:
plays a crucial role in controlling the evolution of the state
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to minimize the length of a route that connects all 125 points.
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Lecture Note in Physics, Vol. 679, Springer, Heidelberg (2005)
3200:
IEEE Transactions on Pattern Analysis and Machine Intelligence
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is a simulation of model copies at different temperatures (or
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should generate a randomly chosen neighbour of a given state
1206:
1198:
3705:
Google in superposition of using, not using quantum computer
3227:"A Note on the Finite Time Behaviour of Simulated Annealing"
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digressively "smooths" the target function while optimizing.
247:. They also proposed its current name, simulated annealing.
2898:
Khachaturyan, A.; Semenovskaya, S.; Vainshtein, B. (1981).
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goes through tours that are much longer than both, and (3)
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problem, for instance, it is not hard to exhibit two tours
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is likely to be similar to that of the current state. This
220:. Both are attributes of the material that depend on their
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1609:{\displaystyle \sum _{k=1}^{n-1}k={\frac {n(n-1)}{2}}=190}
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of the two states, and on a global time-varying parameter
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The problems solved by SA are currently formulated by an
617:{\displaystyle e_{\mathrm {new} }=E(s_{\mathrm {new} })}
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The simulation can be performed either by a solution of
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2957:. Aarts, E. H. L. (Emile H. L.). Dordrecht: D. Reidel.
3590:(3rd ed.). New York: Cambridge University Press.
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Probabilistic optimization technique and metaheuristic
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Journal of the Operations Research Society of America
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function, and on the acceptance probability function
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sampling method. The method is an adaptation of the
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Quantum Annealing and Related Optimization Methods,
56:. Unsourced material may be challenged and removed.
3588:Numerical Recipes: The Art of Scientific Computing
3414:Journal of the Royal Statistical Society, Series B
2954:Simulated annealing : theory and applications
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2222:is large. For the "standard" acceptance function
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1971:distribution of states at a constant temperature
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1342:. The annealing schedule is defined by the call
1322:steps have been taken. In the process, the call
3114:Journal of Optimization Theory and Applications
1798:, because the candidates are tested serially.)
313:, to a state with the minimum possible energy.
2951:Laarhoven, P. J. M. van (Peter J. M.) (1987).
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1338:should pick and return a value in the range ,
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3509:IEEE Transactions on Evolutionary Computation
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3584:"Section 10.12. Simulated Annealing Methods"
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2936:: CS1 maint: multiple names: authors list (
2883:: CS1 maint: multiple names: authors list (
389:, while the actual best solution would be a
2573:of simulated annealing. To do this we set
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349:decides between moving the system to state
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3699:Self-Guided Lesson on Simulated Annealing
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3225:Nolte, Andreas; Schrader, Rainer (1997),
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2600:Interacting MetropolisāHasting algorithms
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945:{\displaystyle P(e,e_{\mathrm {new} },T)}
795:{\displaystyle P(e,e_{\mathrm {new} },T)}
517:{\displaystyle P(e,e_{\mathrm {new} },T)}
116:Learn how and when to remove this message
1201:of an image so as to minimize a certain
1060:Given these properties, the temperature
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129:
3686:"General Simulated Annealing Algorithm"
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1276:finally move downhill according to the
986:{\displaystyle e_{\mathrm {new} }<e}
836:{\displaystyle e_{\mathrm {new} }>e}
305:) to be minimized, is analogous to the
3831:
3494:Caltech Concurrent Computation Program
2837:
2343:When choosing the candidate generator
2004:When choosing the candidate generator
1508:, the acceptance probability function
332:
235:of many variables, subject to several
3719:
3622:Molecular Phylogenetics and Evolution
3545:A. Das and B. K. Chakrabarti (Eds.),
3111:
2715:) to overcome the potential barriers.
369:each state is typically defined as a
212:The name of the algorithm comes from
3231:Operations Research Proceedings 1996
2984:
2982:
2338:
2037:(which is the main principle of the
1504:, the candidate generator procedure
1050:{\displaystyle e_{\mathrm {new} }-e}
54:adding citations to reliable sources
25:
3844:Optimization algorithms and methods
3616:Strobl, M.A.R.; Barker, D. (2016).
3506:
2539:
13:
3539:
3405:
3140:
2593:
2548:at all times. Unfortunately, the
1035:
1032:
1029:
971:
968:
965:
927:
924:
921:
821:
818:
815:
777:
774:
771:
696:{\displaystyle e_{\mathrm {new} }}
687:
684:
681:
605:
602:
599:
578:
575:
572:
499:
496:
493:
458:{\displaystyle s_{\mathrm {new} }}
449:
446:
443:
14:
3860:
3798:Infinite-dimensional optimization
3673:
3070:
3051:
2979:
2759:Intelligent water drops algorithm
2703:Intelligent water drops algorithm
1313:and continues until a maximum of
3436:10.1111/j.1467-9868.2006.00553.x
3303:Journal of Computational Physics
1184:
1172:
30:
3747:Major subfields of optimization
3500:
3450:
3370:
3336:
3294:
3260:
3218:
3191:
3149:The Journal of Chemical Physics
2344:
2312:
2215:{\displaystyle P(E(s),E(s'),T)}
2005:
1967:
1955:{\displaystyle \exp(-(e'-e)/T)}
1819:
1815:
1811:
1807:
1745:
1741:
1737:
1529:
1517:
1513:
1509:
1505:
1501:
1205:function, which causes similar
746:tends to zero, the probability
524:, that depends on the energies
467:acceptance probability function
41:needs additional citations for
3399:10.1016/j.physleta.2003.08.070
3105:
2944:
2891:
2867:Soviet Physics Crystallography
2858:
2831:
2815:"What is Simulated Annealing?"
2807:
2774:Multidisciplinary optimization
2278:
2272:
2263:
2252:
2209:
2200:
2189:
2180:
2174:
2168:
2109:
2097:
2000:Efficient candidate generation
1949:
1938:
1921:
1915:
1858:
1835:
1785:
1762:
1678:
1661:
1591:
1579:
939:
906:
789:
756:
611:
590:
546:
540:
511:
478:
406:The probability of making the
191:boolean satisfiability problem
1:
3459:Annals of Operations Research
2800:
2754:Graph cuts in computer vision
2039:MetropolisāHastings algorithm
1964:MetropolisāHastings algorithm
1512:, and the annealing schedule
1299:
889:the procedure reduces to the
268:MetropolisāHastings algorithm
260:probability density functions
3323:10.1016/0021-9991(90)90201-B
3289:10.1016/0375-9601(90)90166-L
3239:10.1007/978-3-642-60744-8_32
3021:10.1126/science.220.4598.671
2734:Adaptive simulated annealing
2720:multi-objective optimization
2634:Reactive search optimization
2558:Adaptive simulated annealing
195:protein structure prediction
7:
3813:Multiobjective optimization
3642:10.1016/j.ympev.2016.05.001
3357:10.1103/PhysRevLett.86.5219
2779:Particle swarm optimization
2726:
2686:Particle swarm optimization
2564:
2045:candidate moves as well as
1524:Sufficiently near neighbour
668:must be positive even when
281:
136:travelling salesman problem
10:
3865:
3793:Combinatorial optimization
2794:Traveling salesman problem
2744:Combinatorial optimization
2284:{\displaystyle E(s')-E(s)}
1716:when its current state is
367:traveling salesman problem
245:traveling salesman problem
187:traveling salesman problem
21:Annealing (disambiguation)
18:
3753:
3099:10.1107/S0567739481001630
3060:Sov.Phys. Crystallography
2924:10.1107/S0567739481001630
2739:Automatic label placement
2546:thermodynamic equilibrium
1864:{\displaystyle P(e,e',T)}
1791:{\displaystyle P(e,e',T)}
1418:Pick a random neighbour,
430:to a candidate new state
393:that could be different.
222:thermodynamic free energy
3521:10.1109/TEVC.2007.900837
2651:diversity of the search.
2476:can be transformed into
2123:{\displaystyle n(n-1)/2}
1802:Acceptance probabilities
1652:transition probabilities
1646:Transition probabilities
1516:AND initial temperature
1496:Selecting the parameters
1485:Output: the final state
402:Acceptance probabilities
361:The neighbors of a state
237:mathematical constraints
169:. Specifically, it is a
3808:Constraint satisfaction
3345:Physical Review Letters
2680:Stochastic optimization
2661:Ant colony optimization
1895:{\displaystyle e'<e}
410:from the current state
214:annealing in metallurgy
159:probabilistic technique
3783:Stochastic programming
3763:Fractional programming
3701:A Wikiversity project.
3555:Biological Cybernetics
3079:Acta Crystallographica
2904:Acta Crystallographica
2852:10.1287/opre.18.6.1225
2655:Graduated optimization
2604:sequential Monte Carlo
2530:
2510:
2490:
2470:
2450:
2430:
2410:
2390:
2370:
2329:
2305:
2285:
2236:
2216:
2152:
2124:
2081:
2027:
1985:
1956:
1896:
1865:
1792:
1730:
1710:
1685:
1684:{\displaystyle (s,s')}
1636:
1610:
1566:
1269:
1239:
1164:The annealing schedule
1154:
1134:
1114:
1094:
1074:
1051:
1010:
987:
946:
883:
857:
837:
796:
740:
717:
697:
662:
638:
618:
553:
552:{\displaystyle e=E(s)}
518:
459:
424:
329:
161:for approximating the
147:
139:
3778:Nonlinear programming
3773:Quadratic programming
2693:runner-root algorithm
2531:
2511:
2491:
2471:
2451:
2431:
2411:
2391:
2371:
2330:
2306:
2286:
2242:above, it means that
2237:
2217:
2153:
2125:
2082:
2028:
1986:
1957:
1897:
1866:
1806:The specification of
1793:
1731:
1711:
1686:
1637:
1611:
1540:
1270:
1240:
1155:
1135:
1115:
1095:
1075:
1052:
1011:
988:
947:
884:
858:
838:
802:must tend to zero if
797:
741:
718:
698:
663:
639:
619:
554:
519:
460:
425:
341:of the current state
319:
145:
133:
65:"Simulated annealing"
2749:Dual-phase evolution
2668:cross-entropy method
2628:Dual-phase evolution
2616:Stochastic tunneling
2520:
2500:
2480:
2460:
2440:
2420:
2400:
2380:
2360:
2319:
2295:
2246:
2226:
2162:
2137:
2091:
2065:
2012:
1975:
1906:
1875:
1871:was defined as 1 if
1829:
1756:
1720:
1695:
1658:
1620:
1537:
1253:
1229:
1144:
1124:
1104:
1084:
1064:
1020:
1000:
956:
952:was equal to 1 when
900:
867:
847:
806:
750:
730:
707:
672:
652:
628:
563:
528:
472:
434:
414:
353:or staying in state
324:, as there are many
322:hill climb algorithm
183:optimization problem
50:improve this article
19:For other uses, see
3849:Monte Carlo methods
3818:Simulated annealing
3788:Robust optimization
3768:Integer programming
3680:Simulated Annealing
3634:2016MolPE.101...46S
3391:2003PhLA..317..415D
3315:1990JCoPh..90..161D
3281:1990PhLA..146..204M
3161:1953JChPh..21.1087M
3091:1981AcCrA..37..742K
3003:1983Sci...220..671K
2916:1981AcCrA..37..742K
2554:black box functions
2291:is on the order of
2080:{\displaystyle n-1}
2041:) tends to exclude
1635:{\displaystyle n-1}
1399:ā temperature( 1 -
1340:uniformly at random
1268:{\displaystyle T=0}
1120:, the evolution of
882:{\displaystyle T=0}
465:is specified by an
333:The basic iteration
297:, and the function
218:physical properties
199:job-shop scheduling
175:global optimization
151:Simulated annealing
3758:Convex programming
3691:2008-09-23 at the
3567:10.1007/BF00202749
3471:10.1007/BF02022564
3126:10.1007/BF00940812
2769:Molecular dynamics
2709:Parallel tempering
2648:Memetic algorithms
2641:Genetic algorithms
2526:
2506:
2486:
2466:
2446:
2426:
2406:
2386:
2366:
2325:
2301:
2281:
2232:
2212:
2151:{\displaystyle s'}
2148:
2120:
2087:swaps, instead of
2077:
2026:{\displaystyle s'}
2023:
1981:
1952:
1892:
1861:
1788:
1726:
1709:{\displaystyle s'}
1706:
1681:
1632:
1606:
1265:
1247:annealing schedule
1235:
1216:crystalline solids
1150:
1130:
1110:
1090:
1070:
1047:
1006:
983:
942:
879:
853:
833:
792:
736:
713:
693:
658:
634:
614:
549:
514:
455:
420:
330:
272:Monte Carlo method
233:objective function
148:
140:
3826:
3825:
3597:978-0-521-88068-8
3379:Physics Letters A
3269:Physics Letters A
3248:978-3-540-62630-5
3212:10.1109/34.295910
3169:10.1063/1.1699114
2997:(4598): 671ā680.
2789:Quantum annealing
2610:Quantum annealing
2529:{\displaystyle B}
2509:{\displaystyle A}
2489:{\displaystyle B}
2469:{\displaystyle A}
2449:{\displaystyle B}
2429:{\displaystyle A}
2409:{\displaystyle A}
2389:{\displaystyle B}
2369:{\displaystyle A}
2339:Barrier avoidance
2328:{\displaystyle T}
2304:{\displaystyle T}
2235:{\displaystyle P}
1984:{\displaystyle T}
1729:{\displaystyle s}
1598:
1238:{\displaystyle T}
1153:{\displaystyle T}
1133:{\displaystyle s}
1113:{\displaystyle T}
1093:{\displaystyle s}
1073:{\displaystyle T}
1009:{\displaystyle P}
856:{\displaystyle T}
739:{\displaystyle T}
716:{\displaystyle e}
661:{\displaystyle P}
637:{\displaystyle T}
423:{\displaystyle s}
347:probabilistically
256:kinetic equations
126:
125:
118:
100:
3856:
3740:
3733:
3726:
3717:
3716:
3663:
3653:
3612:
3610:
3609:
3600:. Archived from
3578:
3533:
3532:
3504:
3498:
3497:
3489:
3483:
3482:
3454:
3448:
3447:
3429:
3427:cond-mat/0212648
3409:
3403:
3402:
3385:(5ā6): 415ā423.
3374:
3368:
3367:
3351:(3): 5219ā5222,
3340:
3334:
3333:
3298:
3292:
3291:
3264:
3258:
3257:
3256:
3255:
3222:
3216:
3215:
3195:
3189:
3188:
3144:
3138:
3137:
3109:
3103:
3102:
3085:(A37): 742ā754.
3074:
3068:
3067:
3055:
3049:
3048:
3014:
2986:
2977:
2976:
2948:
2942:
2941:
2935:
2927:
2895:
2889:
2888:
2882:
2874:
2862:
2856:
2855:
2835:
2829:
2828:
2826:
2825:
2811:
2588:
2584:
2580:
2576:
2561:thermodynamics.
2540:Cooling schedule
2535:
2533:
2532:
2527:
2515:
2513:
2512:
2507:
2495:
2493:
2492:
2487:
2475:
2473:
2472:
2467:
2455:
2453:
2452:
2447:
2435:
2433:
2432:
2427:
2415:
2413:
2412:
2407:
2395:
2393:
2392:
2387:
2375:
2373:
2372:
2367:
2346:
2334:
2332:
2331:
2326:
2314:
2310:
2308:
2307:
2302:
2290:
2288:
2287:
2282:
2262:
2241:
2239:
2238:
2233:
2221:
2219:
2218:
2213:
2199:
2157:
2155:
2154:
2149:
2147:
2129:
2127:
2126:
2121:
2116:
2086:
2084:
2083:
2078:
2032:
2030:
2029:
2024:
2022:
2007:
1990:
1988:
1987:
1982:
1969:
1961:
1959:
1958:
1953:
1945:
1931:
1901:
1899:
1898:
1893:
1885:
1870:
1868:
1867:
1862:
1851:
1821:
1817:
1813:
1809:
1797:
1795:
1794:
1789:
1778:
1747:
1743:
1739:
1735:
1733:
1732:
1727:
1715:
1713:
1712:
1707:
1705:
1690:
1688:
1687:
1682:
1677:
1641:
1639:
1638:
1633:
1615:
1613:
1612:
1607:
1599:
1594:
1574:
1565:
1554:
1531:
1519:
1515:
1511:
1507:
1503:
1488:
1478:
1463:
1462:) ā„ random(0, 1)
1431:
1415:
1391:
1382:
1373:
1353:
1349:
1337:
1333:
1329:
1321:
1312:
1278:steepest descent
1274:
1272:
1271:
1266:
1244:
1242:
1241:
1236:
1203:potential energy
1188:
1176:
1159:
1157:
1156:
1151:
1139:
1137:
1136:
1131:
1119:
1117:
1116:
1111:
1099:
1097:
1096:
1091:
1079:
1077:
1076:
1071:
1056:
1054:
1053:
1048:
1040:
1039:
1038:
1015:
1013:
1012:
1007:
992:
990:
989:
984:
976:
975:
974:
951:
949:
948:
943:
932:
931:
930:
891:greedy algorithm
888:
886:
885:
880:
862:
860:
859:
854:
842:
840:
839:
834:
826:
825:
824:
801:
799:
798:
793:
782:
781:
780:
745:
743:
742:
737:
722:
720:
719:
714:
703:is greater than
702:
700:
699:
694:
692:
691:
690:
667:
665:
664:
659:
643:
641:
640:
635:
623:
621:
620:
615:
610:
609:
608:
583:
582:
581:
558:
556:
555:
550:
523:
521:
520:
515:
504:
503:
502:
464:
462:
461:
456:
454:
453:
452:
429:
427:
426:
421:
295:physical systems
278:et al. in 1953.
262:, or by using a
207:branch and bound
203:gradient descent
121:
114:
110:
107:
101:
99:
58:
34:
26:
3864:
3863:
3859:
3858:
3857:
3855:
3854:
3853:
3829:
3828:
3827:
3822:
3749:
3744:
3693:Wayback Machine
3676:
3607:
3605:
3598:
3542:
3540:Further reading
3537:
3536:
3505:
3501:
3490:
3486:
3455:
3451:
3410:
3406:
3375:
3371:
3341:
3337:
3299:
3295:
3265:
3261:
3253:
3251:
3249:
3223:
3219:
3196:
3192:
3145:
3141:
3110:
3106:
3075:
3071:
3056:
3052:
3012:10.1.1.123.7607
2987:
2980:
2965:
2949:
2945:
2929:
2928:
2896:
2892:
2876:
2875:
2863:
2859:
2846:(6): 967ā1235.
2836:
2832:
2823:
2821:
2813:
2812:
2808:
2803:
2798:
2784:Place and route
2729:
2596:
2594:Related methods
2586:
2582:
2578:
2574:
2567:
2550:relaxation time
2542:
2521:
2518:
2517:
2501:
2498:
2497:
2481:
2478:
2477:
2461:
2458:
2457:
2441:
2438:
2437:
2421:
2418:
2417:
2401:
2398:
2397:
2381:
2378:
2377:
2361:
2358:
2357:
2341:
2320:
2317:
2316:
2296:
2293:
2292:
2255:
2247:
2244:
2243:
2227:
2224:
2223:
2192:
2163:
2160:
2159:
2140:
2138:
2135:
2134:
2112:
2092:
2089:
2088:
2066:
2063:
2062:
2015:
2013:
2010:
2009:
2002:
1976:
1973:
1972:
1941:
1924:
1907:
1904:
1903:
1878:
1876:
1873:
1872:
1844:
1830:
1827:
1826:
1804:
1771:
1757:
1754:
1753:
1721:
1718:
1717:
1698:
1696:
1693:
1692:
1670:
1659:
1656:
1655:
1648:
1621:
1618:
1617:
1575:
1573:
1555:
1544:
1538:
1535:
1534:
1526:
1498:
1493:
1492:
1491:
1486:
1477:
1467:
1457:
1435:
1425:
1419:
1413:
1395:
1390:
1384:
1377:
1372:
1362:
1351:
1343:
1335:
1331:
1323:
1320:
1314:
1311:
1305:
1302:
1290:complete search
1254:
1251:
1250:
1230:
1227:
1226:
1222:
1221:
1220:
1219:
1218:, respectively.
1194:
1193:
1192:
1189:
1181:
1180:
1177:
1166:
1145:
1142:
1141:
1125:
1122:
1121:
1105:
1102:
1101:
1085:
1082:
1081:
1065:
1062:
1061:
1028:
1027:
1023:
1021:
1018:
1017:
1001:
998:
997:
964:
963:
959:
957:
954:
953:
920:
919:
915:
901:
898:
897:
868:
865:
864:
848:
845:
844:
814:
813:
809:
807:
804:
803:
770:
769:
765:
751:
748:
747:
731:
728:
727:
708:
705:
704:
680:
679:
675:
673:
670:
669:
653:
650:
649:
629:
626:
625:
598:
597:
593:
571:
570:
566:
564:
561:
560:
529:
526:
525:
492:
491:
487:
473:
470:
469:
442:
441:
437:
435:
432:
431:
415:
412:
411:
404:
363:
335:
307:internal energy
284:
173:to approximate
122:
111:
105:
102:
59:
57:
47:
35:
24:
17:
12:
11:
5:
3862:
3852:
3851:
3846:
3841:
3839:Metaheuristics
3824:
3823:
3821:
3820:
3815:
3810:
3805:
3803:Metaheuristics
3800:
3795:
3790:
3785:
3780:
3775:
3770:
3765:
3760:
3754:
3751:
3750:
3743:
3742:
3735:
3728:
3720:
3714:
3713:
3708:
3702:
3696:
3683:
3675:
3674:External links
3672:
3671:
3670:
3664:
3613:
3596:
3579:
3561:(5): 325ā336.
3550:
3541:
3538:
3535:
3534:
3515:(3): 269ā283.
3499:
3484:
3449:
3420:(3): 411ā436.
3404:
3369:
3335:
3309:(1): 161ā175,
3293:
3275:(4): 204ā208,
3259:
3247:
3217:
3206:(6): 652ā656.
3190:
3139:
3104:
3069:
3050:
2978:
2963:
2943:
2910:(5): 742ā754.
2890:
2857:
2830:
2819:www.cs.cmu.edu
2805:
2804:
2802:
2799:
2797:
2796:
2791:
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2771:
2766:
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2756:
2751:
2746:
2741:
2736:
2730:
2728:
2725:
2724:
2723:
2716:
2706:
2700:
2697:meta-heuristic
2689:
2683:
2677:
2674:Harmony search
2671:
2664:
2658:
2652:
2645:
2638:
2631:
2625:
2619:
2613:
2607:
2595:
2592:
2566:
2563:
2541:
2538:
2525:
2505:
2485:
2465:
2445:
2425:
2405:
2385:
2365:
2340:
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2324:
2300:
2280:
2277:
2274:
2271:
2268:
2265:
2261:
2258:
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2251:
2231:
2211:
2208:
2205:
2202:
2198:
2195:
2191:
2188:
2185:
2182:
2179:
2176:
2173:
2170:
2167:
2146:
2143:
2119:
2115:
2111:
2108:
2105:
2102:
2099:
2096:
2076:
2073:
2070:
2021:
2018:
2001:
1998:
1980:
1951:
1948:
1944:
1940:
1937:
1934:
1930:
1927:
1923:
1920:
1917:
1914:
1911:
1891:
1888:
1884:
1881:
1860:
1857:
1854:
1850:
1847:
1843:
1840:
1837:
1834:
1803:
1800:
1787:
1784:
1781:
1777:
1774:
1770:
1767:
1764:
1761:
1725:
1704:
1701:
1680:
1676:
1673:
1669:
1666:
1663:
1647:
1644:
1631:
1628:
1625:
1605:
1602:
1597:
1593:
1590:
1587:
1584:
1581:
1578:
1572:
1569:
1564:
1561:
1558:
1553:
1550:
1547:
1543:
1525:
1522:
1497:
1494:
1490:
1489:
1483:
1482:
1481:
1480:
1479:
1475:
1455:
1432:
1423:
1416:
1411:
1388:
1374:
1370:
1358:
1357:
1356:
1318:
1309:
1301:
1298:
1294:solution space
1285:global optimal
1264:
1261:
1258:
1234:
1196:
1195:
1190:
1183:
1182:
1178:
1171:
1170:
1169:
1168:
1167:
1165:
1162:
1149:
1129:
1109:
1089:
1069:
1046:
1043:
1037:
1034:
1031:
1026:
1005:
982:
979:
973:
970:
967:
962:
941:
938:
935:
929:
926:
923:
918:
914:
911:
908:
905:
878:
875:
872:
852:
832:
829:
823:
820:
817:
812:
791:
788:
785:
779:
776:
773:
768:
764:
761:
758:
755:
735:
712:
689:
686:
683:
678:
657:
633:
613:
607:
604:
601:
596:
592:
589:
586:
580:
577:
574:
569:
548:
545:
542:
539:
536:
533:
513:
510:
507:
501:
498:
495:
490:
486:
483:
480:
477:
451:
448:
445:
440:
419:
403:
400:
395:Metaheuristics
391:global optimum
362:
359:
334:
331:
283:
280:
163:global optimum
124:
123:
38:
36:
29:
15:
9:
6:
4:
3:
2:
3861:
3850:
3847:
3845:
3842:
3840:
3837:
3836:
3834:
3819:
3816:
3814:
3811:
3809:
3806:
3804:
3801:
3799:
3796:
3794:
3791:
3789:
3786:
3784:
3781:
3779:
3776:
3774:
3771:
3769:
3766:
3764:
3761:
3759:
3756:
3755:
3752:
3748:
3741:
3736:
3734:
3729:
3727:
3722:
3721:
3718:
3711:
3709:
3706:
3703:
3700:
3697:
3694:
3690:
3687:
3684:
3681:
3678:
3677:
3669:
3668:VOLUME 6/1999
3665:
3661:
3657:
3652:
3647:
3643:
3639:
3635:
3631:
3627:
3623:
3619:
3614:
3604:on 2011-08-11
3603:
3599:
3593:
3589:
3585:
3580:
3576:
3572:
3568:
3564:
3560:
3556:
3551:
3548:
3544:
3543:
3530:
3526:
3522:
3518:
3514:
3510:
3503:
3496:(report 826).
3495:
3488:
3480:
3476:
3472:
3468:
3465:(2): 85ā121.
3464:
3460:
3453:
3445:
3441:
3437:
3433:
3428:
3423:
3419:
3415:
3408:
3400:
3396:
3392:
3388:
3384:
3380:
3373:
3366:
3362:
3358:
3354:
3350:
3346:
3339:
3332:
3328:
3324:
3320:
3316:
3312:
3308:
3304:
3297:
3290:
3286:
3282:
3278:
3274:
3270:
3263:
3250:
3244:
3240:
3236:
3232:
3228:
3221:
3213:
3209:
3205:
3201:
3194:
3186:
3182:
3178:
3174:
3170:
3166:
3162:
3158:
3154:
3150:
3143:
3135:
3131:
3127:
3123:
3119:
3115:
3108:
3100:
3096:
3092:
3088:
3084:
3080:
3073:
3066:(5): 519ā524.
3065:
3061:
3054:
3046:
3042:
3038:
3034:
3030:
3026:
3022:
3018:
3013:
3008:
3004:
3000:
2996:
2992:
2985:
2983:
2974:
2970:
2966:
2964:90-277-2513-6
2960:
2956:
2955:
2947:
2939:
2933:
2925:
2921:
2917:
2913:
2909:
2905:
2901:
2894:
2886:
2880:
2873:(5): 519ā524.
2872:
2868:
2861:
2853:
2849:
2845:
2841:
2834:
2820:
2816:
2810:
2806:
2795:
2792:
2790:
2787:
2785:
2782:
2780:
2777:
2775:
2772:
2770:
2767:
2765:
2762:
2760:
2757:
2755:
2752:
2750:
2747:
2745:
2742:
2740:
2737:
2735:
2732:
2731:
2721:
2717:
2714:
2710:
2707:
2704:
2701:
2698:
2694:
2690:
2687:
2684:
2681:
2678:
2675:
2672:
2669:
2665:
2662:
2659:
2656:
2653:
2649:
2646:
2642:
2639:
2635:
2632:
2629:
2626:
2623:
2620:
2617:
2614:
2611:
2608:
2605:
2601:
2598:
2597:
2591:
2572:
2562:
2559:
2555:
2551:
2547:
2537:
2523:
2503:
2483:
2463:
2443:
2423:
2403:
2383:
2363:
2353:
2350:
2336:
2322:
2298:
2275:
2269:
2266:
2259:
2256:
2249:
2229:
2206:
2203:
2196:
2193:
2186:
2183:
2177:
2171:
2165:
2144:
2141:
2131:
2117:
2113:
2106:
2103:
2100:
2094:
2074:
2071:
2068:
2059:
2055:
2050:
2048:
2044:
2040:
2036:
2019:
2016:
1997:
1993:
1978:
1965:
1946:
1942:
1935:
1932:
1928:
1925:
1918:
1912:
1909:
1889:
1886:
1882:
1879:
1855:
1852:
1848:
1845:
1841:
1838:
1832:
1823:
1816:temperature()
1799:
1782:
1779:
1775:
1772:
1768:
1765:
1759:
1751:
1738:temperature()
1723:
1702:
1699:
1674:
1671:
1667:
1664:
1653:
1643:
1629:
1626:
1623:
1603:
1600:
1595:
1588:
1585:
1582:
1576:
1570:
1567:
1562:
1559:
1556:
1551:
1548:
1545:
1541:
1521:
1514:temperature()
1484:
1474:
1470:
1466:
1465:
1461:
1454:
1450:
1446:
1442:
1438:
1433:
1429:
1422:
1417:
1410:
1406:
1402:
1398:
1394:
1393:
1392:(exclusive):
1387:
1380:
1375:
1369:
1365:
1360:
1359:
1355:
1347:
1341:
1327:
1317:
1308:
1297:
1295:
1291:
1286:
1281:
1279:
1262:
1259:
1256:
1248:
1232:
1217:
1213:
1208:
1204:
1200:
1187:
1175:
1161:
1147:
1127:
1107:
1087:
1067:
1058:
1044:
1041:
1024:
1003:
994:
980:
977:
960:
936:
933:
916:
912:
909:
903:
894:
892:
876:
873:
870:
850:
830:
827:
810:
786:
783:
766:
762:
759:
753:
733:
724:
710:
676:
655:
647:
631:
594:
587:
584:
567:
543:
537:
534:
531:
508:
505:
488:
484:
481:
475:
468:
438:
417:
409:
399:
396:
392:
388:
387:local optimum
384:
383:hill climbing
380:
375:
372:
368:
358:
356:
352:
348:
344:
340:
327:
323:
318:
314:
312:
311:initial state
308:
304:
300:
296:
292:
289:
279:
277:
276:N. Metropolis
273:
269:
265:
261:
257:
252:
248:
246:
240:
238:
234:
229:
227:
223:
219:
215:
210:
208:
204:
200:
196:
192:
188:
184:
180:
176:
172:
171:metaheuristic
168:
164:
160:
156:
152:
144:
137:
132:
128:
120:
117:
109:
106:December 2009
98:
95:
91:
88:
84:
81:
77:
74:
70:
67: ā
66:
62:
61:Find sources:
55:
51:
45:
44:
39:This article
37:
33:
28:
27:
22:
3817:
3625:
3621:
3606:. Retrieved
3602:the original
3587:
3558:
3554:
3546:
3512:
3508:
3502:
3493:
3487:
3462:
3458:
3452:
3417:
3413:
3407:
3382:
3378:
3372:
3348:
3344:
3338:
3306:
3302:
3296:
3272:
3268:
3262:
3252:, retrieved
3230:
3220:
3203:
3199:
3193:
3152:
3148:
3142:
3117:
3113:
3107:
3082:
3078:
3072:
3063:
3059:
3053:
2994:
2990:
2953:
2946:
2932:cite journal
2907:
2903:
2893:
2879:cite journal
2870:
2866:
2860:
2843:
2839:
2833:
2822:. Retrieved
2818:
2809:
2764:Markov chain
2713:Hamiltonians
2570:
2568:
2549:
2543:
2354:
2342:
2132:
2057:
2053:
2051:
2046:
2042:
2003:
1994:
1824:
1805:
1749:
1651:
1649:
1527:
1499:
1472:
1468:
1459:
1452:
1448:
1444:
1440:
1436:
1427:
1426:ā neighbour(
1420:
1408:
1404:
1400:
1396:
1385:
1378:
1367:
1363:
1345:
1344:temperature(
1336:random(0, 1)
1325:
1315:
1306:
1303:
1282:
1246:
1223:
1059:
995:
895:
725:
645:
466:
405:
376:
364:
354:
350:
342:
338:
336:
326:local maxima
310:
302:
298:
290:
285:
253:
249:
241:
230:
226:Gibbs energy
211:
179:search space
154:
150:
149:
127:
112:
103:
93:
86:
79:
72:
60:
48:Please help
43:verification
40:
3155:(6): 1087.
2695:(RRA) is a
2622:Tabu search
2345:neighbor ()
2313:neighbor ()
2054:consecutive
2006:neighbor ()
1968:neighbor ()
1808:neighbour()
1742:neighbor ()
1530:neighbor ()
1506:neighbor ()
1334:; the call
1280:heuristic.
646:temperature
644:called the
371:permutation
177:in a large
165:of a given
3833:Categories
3608:2011-08-13
3254:2023-02-06
2824:2023-05-13
2801:References
2571:restarting
2158:for which
1324:neighbour(
1300:Pseudocode
1160:is small.
408:transition
379:heuristics
264:stochastic
76:newspapers
3628:: 46ā55.
3331:0021-9991
3134:122729427
3120:: 41ā51.
3007:CiteSeerX
2637:solution.
2349:catchment
2267:−
2104:−
2072:−
2058:arbitrary
2043:very good
2035:heuristic
1933:−
1919:−
1913:
1627:−
1586:−
1560:−
1542:∑
1518:init_temp
1212:amorphous
1042:−
3689:Archived
3660:27150349
3529:12107321
3479:35382644
3444:12074789
3365:11384462
3037:17813860
2973:15548651
2727:See also
2602:(a.k.a.
2565:Restarts
2260:′
2197:′
2145:′
2047:very bad
2020:′
1929:′
1883:′
1849:′
1776:′
1703:′
1675:′
1383:through
293:of some
282:Overview
167:function
3651:4912009
3630:Bibcode
3387:Bibcode
3311:Bibcode
3277:Bibcode
3185:1046577
3177:4390578
3157:Bibcode
3087:Bibcode
3029:1690046
2999:Bibcode
2991:Science
2912:Bibcode
1752:simply
1292:of the
377:Simple
181:for an
157:) is a
90:scholar
3658:
3648:
3594:
3575:851736
3573:
3527:
3477:
3442:
3363:
3329:
3245:
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3175:
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3045:205939
3043:
3035:
3027:
3009:
2971:
2961:
1902:, and
1814:, and
1207:colors
1199:pixels
345:, and
197:, and
189:, the
92:
85:
78:
71:
63:
3571:S2CID
3525:S2CID
3475:S2CID
3440:S2CID
3422:arXiv
3181:S2CID
3130:S2CID
3041:S2CID
3025:JSTOR
2587:ebest
2583:sbest
726:When
381:like
288:state
97:JSTOR
83:books
3656:PMID
3592:ISBN
3361:PMID
3327:ISSN
3243:ISBN
3173:OSTI
3033:PMID
2969:OCLC
2959:ISBN
2938:link
2885:link
2691:The
2666:The
2585:and
2577:and
2516:and
1887:<
1376:For
1361:Let
1214:and
1191:Slow
1179:Fast
996:The
978:<
828:>
559:and
286:The
270:, a
258:for
69:news
3646:PMC
3638:doi
3626:101
3563:doi
3517:doi
3467:doi
3432:doi
3395:doi
3383:317
3353:doi
3319:doi
3285:doi
3273:146
3235:doi
3208:doi
3165:doi
3122:doi
3095:doi
3017:doi
2995:220
2920:doi
2908:A37
2848:doi
2581:to
2436:to
2130:).
1910:exp
1820:P()
1812:P()
1750:not
1746:P()
1604:190
1510:P()
1502:E()
1476:new
1458:),
1456:new
1447:),
1434:If
1424:new
1412:max
1401:(k+
1389:max
1381:= 0
1319:max
209:.
205:or
52:by
3835::
3654:.
3644:.
3636:.
3624:.
3620:.
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3463:41
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3438:.
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3325:,
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3307:90
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3204:16
3202:.
3179:.
3171:.
3163:.
3153:21
3151:.
3128:.
3118:45
3116:.
3093:.
3083:37
3081:.
3064:24
3062:.
3039:.
3031:.
3023:.
3015:.
3005:.
2993:.
2981:^
2967:.
2934:}}
2930:{{
2918:.
2906:.
2902:.
2881:}}
2877:{{
2871:24
2869:.
2844:18
2842:.
2817:.
2376:,
2335:.
1810:,
1642:.
1471:ā
1464::
1366:=
1296:.
351:s*
339:s*
193:,
155:SA
3739:e
3732:t
3725:v
3662:.
3640::
3632::
3611:.
3577:.
3565::
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3481:.
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2887:)
2854:.
2850::
2827:.
2722:.
2579:e
2575:s
2524:B
2504:A
2484:B
2464:A
2444:B
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2404:A
2384:B
2364:A
2323:T
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2276:s
2273:(
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2257:s
2253:(
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2194:s
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2178:s
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2172:E
2169:(
2166:P
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2118:2
2114:/
2110:)
2107:1
2101:n
2098:(
2095:n
2075:1
2069:n
2017:s
1979:T
1950:)
1947:T
1943:/
1939:)
1936:e
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1856:T
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1839:e
1836:(
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1786:)
1783:T
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1773:e
1769:,
1766:e
1763:(
1760:P
1724:s
1700:s
1679:)
1672:s
1668:,
1665:s
1662:(
1630:1
1624:n
1601:=
1596:2
1592:)
1589:1
1583:n
1580:(
1577:n
1571:=
1568:k
1563:1
1557:n
1552:1
1549:=
1546:k
1487:s
1473:s
1469:s
1460:T
1453:s
1451:(
1449:E
1445:s
1443:(
1441:E
1439:(
1437:P
1430:)
1428:s
1421:s
1414:)
1409:k
1407:/
1405:)
1403:1
1397:T
1386:k
1379:k
1371:0
1368:s
1364:s
1352:r
1348:)
1346:r
1332:s
1328:)
1326:s
1316:k
1310:0
1307:s
1263:0
1260:=
1257:T
1233:T
1148:T
1128:s
1108:T
1088:s
1068:T
1045:e
1036:w
1033:e
1030:n
1025:e
1004:P
981:e
972:w
969:e
966:n
961:e
940:)
937:T
934:,
928:w
925:e
922:n
917:e
913:,
910:e
907:(
904:P
877:0
874:=
871:T
851:T
831:e
822:w
819:e
816:n
811:e
790:)
787:T
784:,
778:w
775:e
772:n
767:e
763:,
760:e
757:(
754:P
734:T
711:e
688:w
685:e
682:n
677:e
656:P
632:T
612:)
606:w
603:e
600:n
595:s
591:(
588:E
585:=
579:w
576:e
573:n
568:e
547:)
544:s
541:(
538:E
535:=
532:e
512:)
509:T
506:,
500:w
497:e
494:n
489:e
485:,
482:e
479:(
476:P
450:w
447:e
444:n
439:s
418:s
355:s
343:s
303:s
301:(
299:E
291:s
153:(
119:)
113:(
108:)
104:(
94:Ā·
87:Ā·
80:Ā·
73:Ā·
46:.
23:.
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