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lundi 17 novembre 2008

More on "stochastic local search" definition

Petr Pošík let an interesting comment on a previous post : "Metaheuristic" or "Stochastic Local Search"?. As bloggers always love worthwhile comments from insightful readers, I copy it here, as an article, along with my answer.

I would like to present here an idea that will take the "stochastic" and "local" idea even more further than Thomas and Dirk. In that view, even GAs are local search techniques! Why?

It has a lot to do with the definition of neighborhood of the current state of the algorithm. The state may be the position of an algorithm in the search space. In classical local search, the state is given by mere 1 point. In population-based techniques, the state is the whole population. New solutions are usually generated only by local modifications of the state, i.e. the search takes place only in the local neighborhood of the algorithm state.

Yes, of course, the local neighborhood of the population as a whole is very broad and such a local search is much less prone to premature convergence and exhibits a kind of global behaviour. But, IMHO, it is still a local search...

I completely understand your view: if there is a non-zero probability of generating any solution in the search space, then the search must eventually find the global optimum, and thus performs a global search.

This is the case for many stochastic local search algorithm in continuous domain which use e.g. normal distribution. But if you set the variance too high, you will get very slow convergence to that global optimum which is in practice not desirable. If you set variance too low, you will quickly find local optimum, but you will have to wait for the global one virtually infinitely long - again undesirable. Or, put it in another way, would you call a "global optimization" technique still "global" if it takes zilions of years to find that global solution?

In practice, I think, for continuous spaces we have to resort to algorithms that exhibit local behavior since we always have some hard constraints on the length of the optimization run. In my eyes, this perfectly justifies the name "stochastic local search".

In fact, as far as I comprehend them, both Dick Thierens and Thomas Stützle shares your point of view. For them, GA are a kind of stochastic local search.

I find the classification in "descent" algorithm and "population" ones quite artificial. For example, simulated annealing is implemented as a descent algorithm, but does actually perform a sampling of the objective function, which is commonly seen as a population algorithm characteristic.

More rigourously, I think that every stochastic metaheuristic does try to avoid local optima. Stochastic processes are always here to do so[1].

Maybe the problem is the definition of local search. In my humble opinion, a local search is a method that search for a local optimum. This can be opposed to global search, where one want to find a global optimum.

You are right when you point out that this is related to ergodicity, as only an ergodic[2] algorithm may converge[3], and thus be able to reliably find a global optimum. Thus, yes, I will say that a true and rigorous optimization method is global if, and only if, it is at least quasi-ergodic[4]. A pure random search is the basic global optimization method, that a good metaheuristic should at least outperforms.

Here, it is the stochastic operators that manipulates the solutions that permits the global behaviour. This is not related to the encoding of the solutions/the neighbourhood structure, even if it is a really crucial part of the algorithm.

Thus, a "stochastic local search" may be defined as a local search seeking a global optimum, which is a paradoxical definition. I'd rather prefer that we keep the implementation and the mathematical bases separated, and thus talk of "stochastic search", or "stochastic metaheuristic".

Or else, one may want to use a paradoxical definition, as a matter of fun, which is also a good reason to do so :-)


[1] There may be probabilistic choices that are not related to such tasks, but they are not linked to the iterative aspects of the search algorithms, thus not being stochastics

[2] i.e. that can evaluate any solution

[3] i.e. that have a non-null probability of finding the global optimum in a finite time, or to say it differently, that can find the global optimum after a time that may tends towards infinity (for continuous problems, for discrete ones it is bounded by the size of the instance)

[4] i.e. that can reach any solution in a finite number of iterations

lundi 10 novembre 2008

Multi-criteria meta-parameter tuning for mono-objective stochastic metaheuristics

Here are some explanations about the work I have presented at the META'08 conference. This post is based on the notes I used for my presentation.

In all the automatic parameter setting methods, the problem of finding the better parameter set is considered as an optimization problem, with only one objective, generally finding the best optimum, or reducing the uncertainty of the results. Sometimes, one try to improve the speed. More rarely, speed, precision or robustness are aggregated in one criterion, with an adhoc formula. In fact, one can set parameters according to several objectives (improve speed, improve robustness, etc.). One cannot find a set of parameters fitting all the potential uses of a single algorithm on a single problem instance. Thus, parameter setting is a multi-objective problem.

The key point here is that it is easier to set the parameters of a solver than to solve the problem directly. The simpler example of this idea is when you want to solve a continuous optimization problem with hundreds of variables, with a metaheuristic that have 3 parameters. Moreover, you only have to tune your parameters once, even if you will solve many problems instances later.

In this work, I only consider speed and precision, although the method may handle any performance metrics.

What is crucial in our method is that we do not want to aggregates the criterions, instead, we want the Pareto front corresponding to all the non-dominated parameters set. I use plots representing the Pareto front, that I will sometimes call the « performance » front, or performance profile.

The idea is that one can then compare more rigorously several algorithms, by comparing their respective performance fronts. We can also benefits from having a cursor, scaling from a behaviour oriented towards speed, at one extreme, or precision, at the other side. Even more interesting is the performance profile projected on the parameters space. One can see that every algorithm has its very own profile, that tells a lot on how it behaves.

Performance profiles of 4 metaheuristics The figure above shows performance profiles of 4 metaheuristics: a Simulated Annealing, a Genetic Algorithm, two Estimation of Distribution Algorithms (produced by NSGA-II, with median estimation, on the Rosenbrock-2D problem, using the parameter corresponding to the sampling density for each method with an absolute time stopping criterion).

Our results suggest that the choice of the stopping criterion has a drastic influence on the interest of the performance profile, it must be chosen carefully. Similarly, the method can not naturally find a unique profile for a set of problem instances, but is strictly valid only for an instance of a given problem. Finally, we note that the performance profiles are often convex in the objectives space, which could indicate that aggregation may be usefull.

The proposed method allows to aggregate all parameters into one, determining the position within the profile of performance, since behavior strongly oriented towards production (fast, unaccurate) to conception (slow, accurate). The projection of the profile in the space of parameters can also reflect the impact of parameters on performance, or dependence between parameters. Such information may be very relevant to better understand some complex metaheuristics. It also becomes possible to compare several metaheuristics, delaying the performance profiles on the same scale. The statistic validation also receives additional dimensions of discrimination.

In perspective, it remains to reduce the demand for calculations of the meta-optimizer, using dedicated methods (SPO, racing, etc.).. It is also possible to extend the method taking into account robustness as supplementary objectives and checking the possibility of rebuilding correlations on a set of instances.

Finally, here are the slides. I use light slides without a lot o text, so I suggest that you read the notes while looking at the presentation. You will find the abstract, the extended abstract and the slides on my professional website, at the corresponding publication page.

vendredi 7 novembre 2008

"Metaheuristic" or "Stochastic Local Search"?

During their excellent tutorials at META 2008, both Thomas and Dick talked about "stochastic local search" and seems to be rather uncomfortable with the "metaheuristic" term. They seems to reserve it for very high level well known algorithms.

I'm not sure that using the term "stochastic" along with "local" is a good idea. In all the algorithms, the use of probabilistic processes aims at avoiding local optima. Thus, stochastic algorithms are not "local search" anymore, but "global search". While I agree that it is a very good approach to start with local search techniques, I would say that when you introduce stochastic processes, then you enter the field of metaheuristics. On the other hand, the paradoxal use of "stochastic" along with "local" may be interesting from a marketing point of vue... but I like paradoxes.

Anyway, despite the fact that there would be a lot more to say about the problem of nomenclature in our field (who says "everything is evolutionary"?), this is not very important, I tink I will continue using "metaheuristics", until a common term establish itself in the litterature.

jeudi 6 novembre 2008

2nd International Conference on Metaheuristics and Nature Inspired Computing

I've just attend the META 2008 international conference on metaheuristics and nature inspired computing.

The weather was nice in Tunisia, we had a place to sleep, a restaurant and a swimming pool, the organization was just fine. The acceptance rate was of 60%, with 116 accepted papers, for 130 attendants and one short review by paper (at least for mine).

OK, now let's talk about what is really exciting: science.

I was more than pleased to attend to two tutorials, given by Dick Thierens and Thomas Stützle, that both were talking about the use of stochastic local search.

What was definitely interesting is that these two talented researchers were insisting a lot on the need of a rigorous experimental approach for the design and the validation of metaheuristics. That's good news for our research domain: the idea that metaheuristics should be employed in a scientific manner rather than in an artistic one gains more and more importance.

First, they both says that a good way to tackle a hard optimization problem is to employ a bottom-up approach: start first with a simple local search, then use metaheuristics operators to improve the results.

Thomas Stützle, particularly, insist on the crucial need of rigorous parameter setting and experimental validation with statistical tests. That's definitely a very important point.

Another good point made by Thomas is the use of the term "algorithm engineering" to describe a rigorous design and evaluation approach of optimization algorithms. I was searching a nice term to name it, I think this one is a good candidate. The bad news at this conference is that, despite these two incredible tutorials, there was very few peoples speaking about algorithm engineering. I was presenting a new method for parameter setting and behaviour understanding, but I was in a unrelated "metaheuristics for real-world problem" session. I haven't seen other works specifically dedicated to such subjects.

More badly, I have attend to several presentations with very bad experimental work. Some peoples keeps telling their stochastic algorithm is better only by showing the best result found. More often, their is a mean and a standard deviation, but without a statistical test. But there is hope as, since 2001 (when some works made by Éric D. Taillard definitely introduced experimental validation for metaheuristics, at least for me), I find that the proportion of better experimental plans is increasing in the literature.

Anyway, my wish is that there will be more and more special sessions on algorithm engineering in future conferences on metaheuristics. In the meantime, there is the 2nd "Engineering Stochastic local search algorithms" conference, in september 2009, in Brussels, that seems really interesting...

jeudi 11 septembre 2008

The ultimate metaheuristic?

There exists a lot of different algorithms families that can be called "metaheuristics", stricly speaking, there are a very, very, very large number of metaheuristics instances.

Defining what is a metaheuristic "family" is a difficult problem: when may I called this or this algorithm an evolutionary one? Is estimation of distribution a sub-family of genetic algorithms? What is the difference between ant colony optimization and stochastic gradient ascent? Etc.

Despite the difficulty of classifying metaheuristics, there is some interesting characteristics shared by stochastic metaheuristics. Indeed, they are all iteratively manipulating a sample of the objective function[1]

For example, simulated annealing is often depicted as a probabilistic descent algorithm, but it is more than that. Indeed, simulated annealing is based on the Metropolis-Hastings algorithm, which is a way of sampling any probability distributionn, as long as you can calculate its density at any point. Thus, simulated annealing use an approximation of the objective function as a probability density function to generate a sampling. It is even more obvious if you consider a step by step decrease of the temperature. Estimation of distribution are another obvious example: they are explicitly manipulating samplings, but one can also have the same thoughts about evolutionary algorithms, even if they are manipulating the sampling rather implicitely.

The diagram tries to illustrate this idea: (a) a descent algorithm can have the same sampling behaviour than an iteration of a (b) "population" method.

Given these common processes, is it possible to design a kind of "universal" metaheuristic ? Theoretically, the answer is yes. For example, in the continuous domain, consider an estimation of distribution algorithm, using a mixture of gaussian kernel: it can learn any probability density function (possibly needing an infinite number of kernels). Thus, carefully choosing the function to use at each iteration and the selection operator, one can reproduce the behaviour of any stochastic metaheuristic.

Of course, choosing the correct mixture (and the other parameters) is a very difficult problem in practice. But I find interesting the idea that the problem of designing a metaheuristic can be reduced to a configuration problem.


[1] Johann Dréo, Patrick Siarry, "Stochastic metaheuristics as sampling techniques using swarm intelligence. ", in "Swarm Intelligence: Focus on Ant and Particle Swarm Optimization", Felix T. S. Chan, Manoj Kumar Tiwari (Eds.), Advanced Robotic Systems International, I-Tech Education and Publishing, Vienna, Austria , ISBN 978-3-902613-09-7 - December 2008

mercredi 18 juin 2008

Metaheuristic validation in a nutshell

People using metaheuristics often forget that the price to pay for their ease of adaptation to a new problem is the hard validation work. There is several things to keep in mind when using a metaheuristic, especially when one want to prove that they work in practice.

This (kind of) mind map try to list what you should do, and a short set of main tools to do it. It is not always mandatory to use all the tools, sometimes it is just a matter of choice (like for the parameter setting), sometimes the more you do, the better it is (like for performance assessment).

The graphic has been drawn in SVG, and I have put some references in a very small font at the bottom of some boxes. Thus, it would be more confortable to view it in Firefox or in Inkscape, and to zoom where needed. Try the SVG version.

Metaheuristic design

lundi 3 mars 2008

A chronology of metaheuristics

At last, I found some time to translate my chronology of metaheuristics I have made for the french Wikipedia. I put it here for the records, but you will find a more up-to-date version on the english Wikipedia article about Metaheuristics.


Detailled chronology


  1. V. Angel, La rugosité des paysages : une théorie pour la difficulté des problèmes d’optimisation combinatoire relativement aux métaheuristiques, thèse de doctorat de l’université de Paris-Sud, Orsay, 1998.
  2. J. Dreo, J.-P. Aumasson, W. Tfaili, P. Siarry, Adaptive Learning Search, a new tool to help comprehending metaheuristics, to appear in the International Journal on Artificial Intelligence Tools.
  3. El-Ghazali Talbi, A taxonomy of hybrid metaheuristics, Journal of Heuristics, volume 8, no 2, pages 541-564, septembre 2002
  4. (en) exemples de fonctions de tests pour métaheuristiques d’optimisation de problèmes continus.
  5. W. Dullaert, M. Sevaux, K. Sörensen et J. Springael, Applications of metaheuristics, numéro spécial du European Journal of Operational Research, no 179, 2007.
  6. Robbins, H. and Monro, S., A Stochastic Approximation Method, Annals of Mathematical Statistics, vol. 22, pp. 400-407, 1951
  7. Barricelli, Nils Aall, Esempi numerici di processi di evoluzione, Methodos, pp. 45-68, 1954
  8. Rechenberg, I., Cybernetic Solution Path of an Experimental Problem, Royal Aircraft Establishment Library Translation, 1965
  9. Fogel, L., Owens, A.J., Walsh, M.J., Artificial Intelligence through Simulated Evolution, Wiley, 1966
  10. W.K. Hastings. Monte Carlo Sampling Methods Using Markov Chains and Their Applications, Biometrika, volume 57, no 1, pages 97-109, 1970
  11. Holland, John H., Adaptation in Natural and Artificial Systems, University of Michigan Press, Ann Arbor, 1975
  12. Smith, S.F., A Learning System Based on Genetic Adaptive Algorithms, PhD dissertation (University of Pittsburgh), 1980
  13. S. Kirkpatrick, C. D. Gelatt et M. P. Vecchi, Optimization by Simulated Annealing, Science, volume 220, no  4598, pages 671-680, 1983
  14. V. Cerny, A thermodynamical approach to the travelling salesman problem : an efficient simulation algorithm Journal of Optimization Theory and Applications, volume45, pages 41-51, 1985
  15. Fred GLover, Future Paths for Integer Programming and Links to Artificial Intelligence, Comput. & Ops. Res.Vol. 13, No.5, pp. 533-549, 1986
  16. J.D. Farmer, N. Packard and A. Perelson, The immune system, adaptation and machine learning, Physica D, vol. 22, pp. 187--204, 1986
  17. F. Moyson, B. Manderick, The collective behaviour of Ants : an Example of Self-Organization in Massive Parallelism, Actes de AAAI Spring Symposium on Parallel Models of Intelligence, Stanford, Californie, 1988
  18. Koza, John R. Non-Linear Genetic Algorithms for Solving Problems. United States Patent 4,935,877. Filed May 20, 1988. Issued June 19, 1990
  19. Goldberg, David E., Genetic Algorithms in Search, Optimization and Machine Learning, Kluwer Academic Publishers, Boston, MA., 1989
  20. P. Moscato, On Evolution, Search, Optimization, Genetic Algorithms and Martial Arts : Towards Memetic Algorithms, Caltech Concurrent Computation Program, C3P Report 826, 1989.
  21. M. Dorigo, Optimization, Learning and Natural Algorithms, Thèse de doctorat, Politecnico di Milano, Italie, 1992.
  22. Feo, T., Resende, M., Greedy randomized adaptive search procedure, Journal of Global Optimization, tome 42, page 32--37, 1992
  23. Eberhart, R. C. et Kennedy, J., A new optimizer using particle swarm theory, Proceedings of the Sixth International Symposium on Micromachine and Human Science, Nagoya, Japan. pp. 39-43, 1995
  24. Kennedy, J. et Eberhart, R. C., Particle swarm optimization, Proceedings of IEEE International Conference on Neural Networks, Piscataway, NJ. pp. 1942-1948, 1995
  25. Mülhenbein, H., Paaß, G., From recombination of genes to the estimation of distribution I. Binary parameters, Lectures Notes in Computer Science 1411: Parallel Problem Solving from Nature, tome PPSN IV, pages 178--187, 1996
  26. Rainer Storn, Kenneth Price, Differential Evolution – A Simple and Efficient Heuristic for global Optimization over Continuous Spaces, Journal of Global Optimization, volume 11, no 4, pages 341-359, 1997
  27. Rubinstein, R.Y., Optimization of Computer simulation Models with Rare Events, European Journal of Operations Research, 99, 89-112, 1997
  28. Stefan Boettcher, Allon G. Percus, "Extremal Optimization : Methods derived from Co-Evolution", Proceedings of the Genetic and Evolutionary Computation Conference (1999)
  29. Takagi, H., Active user intervention in an EC Search, Proceesings of the JCIS 2000

The problem with spreading new metaheuristics

Marcelo De Brito had interesting thoughts about what he call New Wave Of Genetic Algorithms. He is surprised that when "evolutionary computation" is applied to a new problem, the first algorithm used is the good old canonic genetic algorithm, despite that there exist active researchs on Estimation of Distribution Algorithms. Julian Togelius write that it may be because people does not understand other algorithms, or even know that anything else exists.

I think that is definitely true. This subject is a kind of hobby for me. Indeed, as I have came from ecology to applied mathematics, I feel like a kind of generalist researcher, not being able to be the best somewhere, but trying to be as good as possible on several fields. Concerning the field of what Marcelo called NWOGA, I would like to emphasize some other things.

As David E. Goldberg say in its courses, genetic algorithm is the term everybody use. For specialist, a GA is just a kind of "evolutionary algorithm", with specific rules, that are more defined by history than by anything else.

The litterature on evolutionary computation is quite big, the first algorithm being designed in 1965 (evolutionary strategies, followed by evolutionary programming in 1966), making it difficult to spread deep changes on basic concepts.

There exist a lot more stochastic algorithms for global optimization than just evolutionary ones. I prefer to call the stochastic metaheuristics, or simply metaheuristics, because this lead to far less bias than a metaphoric naming (cf. the previous post on classification of metaheuristics).

For example, during my PhD thesis, I was convinced that some Ant Colony Optimization algorithms were just equivalent to Estimation of Distribution Algorithms, when talking about continuous problems. Moreover, I'm now convinced that a lot of metaheuristics just shares some common stochastic sampling processes, that are not specifiquely related to evolution. For example, mathematically, Simulated Annealing is just a kind of EDA using an approximation of the objective function as a model (or inversely, of course).

As Julian says: I know roughly what an EDA does, but I couldn't sit down an implement one on the spot. This is, in my humble opinion, one of the more important thing to keep in mind. Indeed, there exist more and more papers claming that a correct parameter setting of a metaheuristic can lead to the performances of any competing metaheuristic.

Thus, the true discriminatory criterion is not the fantasised intrinsic capability, but the ease of implementation and parameter setting on a specific problem. In other words, choose the algorithm you like, but be aware that there exists a lot of other ones.

vendredi 12 octobre 2007

Classification of metaheuristics

I eventually find some time to try a graphical representation of how metaheuristics could be classified.

Here is a slidified version, that shows each classes independently:

And a static image version: Graphical classification of metaheuristics

Note that some metaheuristics are not completely contained in certains classes, this indicate that the method could be considered as part of the class or not, depending on your point of view.

I have reported the following metaheuristics:

  • genetic programming, genetic algorithms, differential evolution,
  • evolution strategies,
  • estimation of distribution algorithms,
  • particle swarm optimization,
  • ant colony optimization,
  • simulated annealing,
  • tabu search,
  • GRASP,
  • variable neighborhood search,
  • iterated, stochastic and guided local search.

And the following classes :

  • metaheurisitcs vs local search,
  • population metaheuristics vs trajectory-based ones,
  • evolutionary computation or not,
  • nature-inspired methods or not,
  • dynamic objective function vs static ones,
  • memory-based algorithms vs memory-less,
  • implicit, explicit or direct metaheuristics.

I proposed the last class, so that it may not be well-known. You will find more informations about it in the following paper: Adaptive Learning Search, a new tool to help comprehending metaheuristics, J. Dreo, J.-P. Aumasson, W. Tfaili, P. Siarry, International Journal on Artificial Intelligence Tools, Vol. 16, No. 3.. - 1 June 2007

I didn't placed a stochastic category, as it seems a bit difficult to represent graphically. Indeed, a lot of methods could be "stochasticized" or "derandomized" in several ways.

There is surely several lacks or errors, feel free to give your point of view with a trackback, an email or by modifying the SVG source version (comments are disabled due to spam that I didn't have time to fight accurately).

mercredi 23 août 2006

What are metaheuristics ?

Despite the title of this blog, the term metaheuristic is not really well defined.

One of the first occurence of the term can (of course) be found in a paper by Fred Glover[1]: Future Paths for Integer Programming and Links to Artificial Intelligence[2]. In the section concerning tabu search, he talks about meta-heuristic:

Tabu search may be viewed as a "meta-heuristic" superimposed on another heuristic. The approach undertakes to transcend local optimality by a strategy of forbidding (or, more broadly, penalizing) certain moves.

In the AI field, a heuristic is a specific method that help solving a problem (from the greek for to find), but how must we understand the meta word ? Well, in greek, it means "after", "beyond" (like in metaphysic) or "about" (like in metadata). Reading Glover, metaheuristics seems to be heuristics beyond heuristics, which seems to be a good old definition, but what is the definition nowadays ? The litterature is really prolific on this subject, and the definitions are numerous.

There is at least three tendencies :

  1. one that consider that the most important part of metaheuristcs is the gathering of several heuristics,
  2. one other that promotes the fact that metaheuristics are designed as generalistic methods, that can tackle several problems without major changes in their design,
  3. the last one that use the term only for evolutionnary algorithms when they are hybridicized with local searches (methods that are called memetic algorithms in the other points of vue).

The last one is quite minor in the generalistic litterature, it can mainly be found in the field of evolutionnary computation, separate out the two other tendencies is more difficult.

Here are some definitions gathered in more or less generalistic papers:

"iterative generation process which guides a subordinate heuristic by combining intelligently different concepts for exploring and exploiting the search space" (Osman and Laporte, 1996[3])

"(metaheuristics) combine basic heuristic methods in higher level frameworks aimed at efficiently and effectively exploring a search space" (Blum and Roli, 2003[4])

"a metaheuristic can be seen as a general-purpose heuristic method designed to guide an underlying problem-specific heuristic (...) A metaheuristic is therefore a general algorithmic framework which can be applied to different optimization problems with relative few modifications to make them adapted to a specific problem." (Dorigo and Stützle, 2004[5])

"(metaheuristics) apply to all kinds of problems (...) are, at least to some extent, stochastic (...) direct, i.e. they do not resort to the calculation of the gradients of the objective function (...) inspired by analogies: with physics, biology or ethology" (Dréo, Siarry, Petrowski and Taillard, 2006[6])

One can summarize by enumerating the expected characteristics:

  • optimization algorithms,
  • with an iterative design,
  • combining low level heuristics,
  • aiming to tackle a large scale of "hard" problems.

As it is pointed out by the last reference, a large majority of metaheuristics (well, not to say all) use at least one stochastic (probabilistic) process and does not use more information than the solution and the associated value(s) of the objective function.

Talking about combining heuristics seems to be appropriate for Ant Colony Optimization, that specifically needs one (following Dorigo's point of vue), it can be less obvious for Evolutionnary Algorithms. One can consider that mutation, or even the method's strategy itself, is a heuristic, but isn't it too generalistic to be called a heuristic ?

If we forget the difficulty to demarcate what can be called a heuristic and what is the scope of the term meta, one can simply look at the use of the term among specialists. Despite the fact that the definition can be used in several fields (data mining, machine learning, etc.), the term is used for optimization algorithms. This is perhaps the best reason among others: the term permits to separate a research field from others, thus adding a little bit of marketing...

I would thus use this definition:

Metaheuristics are algorithms designed to tackle "hard" optimization problems, with the help of iterative stochastic processes. These methods are manipulating direct samples of the objective function, and can be applied to several problems without major changes in their design.


[1] A recurrent joke says that whatever is your new idea, it has already be written down by Glover

[2] Comput. & Ops. Res.Vol. 13, No.5, pp. 533-549, 1986

[3] Metaheuristic: A bibliography, Annals of Operations Research, vol. 63, pp. 513-623, 1996

[4] Metaheuristics in combinatorial optimization: Overview and conceptual comparison, ACM Computing Surveys, vol. 35, issue 3, 2003

[5] Ant Colony Optimization, MIT Press, 2004

[6] Metaheuristics for Hard Optimization, Springer, 2006