Editing Metaheuristic

{{Short description|Optimization technique}}
{{Use American English|date=January 2019}}
In [[computer science]] and [[mathematical optimization]], a '''metaheuristic''' is a higher-level [[procedure (computer science)|procedure]] or [[Heuristic (computer science)|heuristic]] designed to find, generate, tune, or select a heuristic (partial [[search algorithm]]) that may provide a sufficiently good solution to an [[optimization problem]] or a [[machine learning]] problem, especially with incomplete or imperfect information or limited computation capacity.<ref name=glover03handbook/><ref name="Bala2015" /><ref name="Bianchi2009" /><ref name="blum03metaheuristics" /> Metaheuristics sample a subset of solutions which is otherwise too large to be completely enumerated or otherwise explored. Metaheuristics may make relatively few assumptions about the optimization problem being solved and so may be usable for a variety of problems.<ref name=glover03handbook/><ref name=":1" /><ref name=":2" /> Their use is always of interest when exact or other (approximate) methods are not available or are not expedient, either because the calculation time is too long or because, for example, the solution provided is too imprecise.

Compared to [[optimization algorithm]]s and [[iterative method]]s, metaheuristics do not guarantee that a [[global optimum|globally optimal solution]] can be found on some class of problems.<ref name=blum03metaheuristics /> Many metaheuristics implement some form of [[stochastic optimization]], so that the solution found is dependent on the set of [[random variable]]s generated.<ref name=Bianchi2009 /> In [[combinatorial optimization]], there are many problems that belong to the class of [[NP-completeness|NP-complete]] problems and thus can no longer be solved exactly in an acceptable time from a relatively low degree of complexity.<ref>{{Cite book |last1=Brucker |first1=Peter |url=https://link.springer.com/10.1007/978-3-642-23929-8 |title=Complex Scheduling |last2=Knust |first2=Sigrid |date=2012 |publisher=Springer |isbn=978-3-642-23928-1 |location=Berlin, Heidelberg |language=en |doi=10.1007/978-3-642-23929-8}}</ref><ref>{{Cite book |last1=Papadimitriou |first1=Christos H. |title=Combinatorial Optimization: Algorithms and Complexity |last2=Steiglitz |first2=Kenneth |date=1998 |publisher=Dover Publ., corrected, unabridged new edition of the work published by Prentice-Hall in 1982. |isbn=978-0-486-40258-1 |location=Mineola, N.Y}}</ref> Metaheuristics then often provide good solutions with less computational effort than approximation methods, iterative methods, or simple heuristics.<ref name=blum03metaheuristics /><ref name=glover03handbook/> This also applies in the field of continuous or mixed-integer optimization.<ref name=glover03handbook/><ref name=":5">{{Cite journal |last=Gad |first=Ahmed G. |date=2022 |title=Particle Swarm Optimization Algorithm and Its Applications: A Systematic Review |journal=Archives of Computational Methods in Engineering |language=en |volume=29 |issue=5 |pages=2531–2561 |doi=10.1007/s11831-021-09694-4 |issn=1134-3060|doi-access=free }}</ref><ref>{{Cite journal |last1=Li |first1=Zhenhua |last2=Lin |first2=Xi |last3=Zhang |first3=Qingfu |last4=Liu |first4=Hailin |date=2020 |title=Evolution strategies for continuous optimization: A survey of the state-of-the-art |url=https://linkinghub.elsevier.com/retrieve/pii/S221065021930584X |journal=Swarm and Evolutionary Computation |language=en |volume=56 |pages=100694 |doi=10.1016/j.swevo.2020.100694}}</ref> As such, metaheuristics are useful approaches for optimization problems.<ref name=Bianchi2009 /> Several books and survey papers have been published on the subject.<ref name=Bianchi2009 /><ref name=blum03metaheuristics/><ref name=glover03handbook/><ref name=goldberg89genetic/><ref name=talbi09metaheuristics/> Literature review on metaheuristic optimization,<ref>X. S. Yang, Metaheuristic optimization, Scholarpedia, 6(8):11472 (2011).</ref> suggested that it was Fred Glover who coined the word metaheuristics.<ref>{{cite journal|last1=Glover |first1=Fred |title=Future paths for integer programming and links to artificial intelligence |journal=Computers and Operations Research |date=January 1986 |volume=13 |issue=5 |pages=533–549 |doi=10.1016/0305-0548(86)90048-1 |issn=0305-0548 |url=https://leeds-faculty.colorado.edu/glover/fred%20pubs/174%20-%20Future%20Paths%20for%20Integer%20Programming%20TS.pdf}}</ref>

Most literature on metaheuristics is experimental in nature, describing empirical results based on [[computer experiment]]s with the algorithms. But some formal theoretical results are also available, often on [[convergence (mathematics)|convergence]] and the possibility of finding the global optimum.<ref name=blum03metaheuristics /><ref>{{Cite journal |last=Rudolph |first=Günter |date=2001 |title=Self-adaptive mutations may lead to premature convergence |url=https://ieeexplore.ieee.org/document/942534 |journal=IEEE Transactions on Evolutionary Computation |volume=5 |issue=4 |pages=410–414 |doi=10.1109/4235.942534|hdl=2003/5378 |hdl-access=free }}</ref> Also worth mentioning are the [[No free lunch theorem|no-free-lunch theorems]], which state that there can be no metaheuristic that is better than all others for any given problem.

Especially since the turn of the millennium, many metaheuristic methods have been published with claims of novelty and practical efficacy. While the field also features high-quality research, many of the more recent publications have been of poor quality; flaws include vagueness, lack of conceptual elaboration, poor experiments, and ignorance of previous literature.<ref name="Sörensen2013">{{cite journal |last=Sörensen |first=Kenneth |year=2015 |title=Metaheuristics—the metaphor exposed |url=https://onlinelibrary.wiley.com/doi/10.1111/itor.12001 |journal=International Transactions in Operational Research |language=en |volume=22 |issue=1 |pages=3–18 |citeseerx=10.1.1.470.3422 |doi=10.1111/itor.12001 |s2cid=14042315}}</ref><ref name=":3">{{Cite web |last1=Brownlee |first1=Alexander |last2=Woodward |first2=John R. |title=Why we fell out of love with algorithms inspired by nature |url=https://theconversation.com/why-we-fell-out-of-love-with-algorithms-inspired-by-nature-42718 |access-date=2024-08-30 |website=The Conversation (website)|date=3 June 2015 }}</ref>

==Properties==

These are properties that characterize most metaheuristics:<ref name="blum03metaheuristics" />

* Metaheuristics are strategies that guide the search process.
* The goal is to efficiently explore the search space in order to find optimal or near–optimal solutions.
* Techniques which constitute metaheuristic algorithms range from [[Local search (optimization)|simple local search]] procedures to complex learning processes.
* Metaheuristic algorithms are approximate and usually non-deterministic.
* Metaheuristics are not problem-specific. However, they were often developed in relation to a problem class such as continuous<ref>{{Cite book |last=Schwefel |first=Hans-Paul |title=Evolution and optimum seeking |date=1995 |publisher=Wiley |isbn=978-0-471-57148-3 |series=Sixth-generation computer technology series |location=New York}}</ref><ref>{{Citation |last1=Eberhart |first1=R. |last2=Kennedy |first2=J. |title=A new optimizer using particle swarm theory |date=1995 |url=https://ieeexplore.ieee.org/document/494215 |work=Conf. Proc. MHS'95 |pages=39–43 |place= |publisher=IEEE |doi=10.1109/MHS.1995.494215 |isbn=978-0-7803-2676-7 }}</ref> or combinatorial optimization<ref>{{Citation |last1=Colorni |first1=Alberto |last2=Dorigo |first2=Marco |last3=Maniezzo |first3=Vittorio |title=Distributed Optimization by Ant Colonies |date=1991 |url=https://www.researchgate.net/publication/216300484 |work=Conf. Proc. of ECAL91 - European Conference on Artificial Life |pages=134–142 |place=Amsterdam |publisher=Elsevier Publ. |doi= |isbn=9780262720199 |editor-last1=Varela |editor-first1=F. |editor2-last=Bourgine |editor2-first=P.}}</ref> and then generalized later in some cases.<ref>{{Cite journal |last1=Socha |first1=Krzysztof |last2=Dorigo |first2=Marco |date=2008 |title=Ant colony optimization for continuous domains |url=https://linkinghub.elsevier.com/retrieve/pii/S0377221706006333 |journal=European Journal of Operational Research |language=en |volume=185 |issue=3 |pages=1155–1173 |doi=10.1016/j.ejor.2006.06.046}}</ref><ref>{{Citation |last1=Nissen |first1=Volker |title=Constrained Combinatorial Optimization with an Evolution Strategy |date=1994 |work=Fuzzy Logik |pages=33–40 |editor-last=Reusch |editor-first=Bernd |url=http://link.springer.com/10.1007/978-3-642-79386-8_5 |access-date=2024-08-24 |place=Berlin, Heidelberg |publisher=Springer Berlin Heidelberg |doi=10.1007/978-3-642-79386-8_5 |isbn=978-3-540-58649-4 |last2=Krause |first2=Matthias}}</ref>
* They can draw on domain-specific knowledge in the form of heuristics that are controlled by a higher-level strategy of the metaheuristic.
* They can contain mechanisms that prevent them from getting stuck in certain areas of the search space.
* Modern metaheuristics often use the search history to control the search.

== Classification ==
[[Image:Metaheuristics classification.svg|thumb|[[Euler diagram]] of the different classifications of metaheuristics<ref name=nojhan07>[http://metah.nojhan.net/post/2007/10/12/Classification-of-metaheuristics Classification of metaheuristics]</ref>]]

There are a wide variety of metaheuristics<ref name=Bianchi2009 /><ref name="glover03handbook" /> and a number of properties with respect to which to classify them.<ref name=blum03metaheuristics /><ref>{{Citation |last=Raidl |first=Günther R. |title=A Unified View on Hybrid Metaheuristics |date=2006 |work=Hybrid Metaheuristics |series=Lecture Notes in Computer Science |volume=4030 |pages=1–12 |editor-last=Almeida |editor-first=Francisco |url=http://link.springer.com/10.1007/11890584_1 |access-date=2024-08-24 |place=Berlin, Heidelberg |publisher=Springer Berlin Heidelberg |doi=10.1007/11890584_1 |isbn=978-3-540-46384-9 |editor2-last=Blesa Aguilera |editor2-first=María J. |editor3-last=Blum |editor3-first=Christian |editor4-last=Moreno Vega |editor4-first=José Marcos}}</ref><ref name=":4">{{Cite journal |last1=Glover |first1=Fred |last2=Sörensen |first2=Kenneth |date=2015 |title=Metaheuristics |journal=Scholarpedia |language=en |volume=10 |issue=4 |pages=6532 |doi=10.4249/scholarpedia.6532 |doi-access=free |issn=1941-6016}}</ref><ref>{{Cite web |last1=Birattari |first1=Mauro |last2=Paquete |first2=Luis |last3=Stützle |first3=Thomas |last4=Varrentrapp |first4=Klaus |date=2001 |title=Classification of Metaheuristics and Design of Experiments for the Analysis of Components. |s2cid=18347906 |url=https://api.semanticscholar.org/CorpusID:18347906}}</ref> The following list is therefore to be understood as an example.

=== Local search vs. global search ===
One approach is to characterize the type of search strategy.<ref name="blum03metaheuristics" /> One type of search strategy is an improvement on simple local search algorithms. A well known local search algorithm is the [[hill climbing]] method which is used to find local optimums. However, hill climbing does not guarantee finding global optimum solutions.

Many metaheuristic ideas were proposed to improve local search heuristic in order to find better solutions. Such metaheuristics include [[simulated annealing]], [[tabu search]], [[iterated local search]], [[Variable Neighborhood Search|variable neighborhood search]], and [[Greedy randomized adaptive search procedure|GRASP]].<ref name="blum03metaheuristics" /> These metaheuristics can both be classified as local search-based or global search metaheuristics.

Other global search metaheuristic that are not local search-based are usually [[Population model (evolutionary algorithm)|population-based]] metaheuristics. Such metaheuristics include [[ant colony optimization]], [[evolutionary computation]] such as [[genetic algorithm]] or [[Evolution strategy|evolution strategies]], [[particle swarm optimization]], [[rider optimization algorithm]]<ref>{{cite journal |last1=D |first1=Binu |title=RideNN: A New Rider Optimization Algorithm-Based Neural Network for Fault Diagnosis in Analog Circuits |journal=IEEE Transactions on Instrumentation and Measurement |year=2019 |volume=68 |issue=1 |pages=2–26 |doi=10.1109/TIM.2018.2836058 |bibcode=2019ITIM...68....2B |s2cid=54459927 |url=https://ieeexplore.ieee.org/document/8370147}}</ref> and bacterial foraging algorithm.<ref name=":0">{{Cite journal |last1=Pang |first1=Shinsiong |last2=Chen |first2=Mu-Chen |date=2023-06-01 |title=Optimize railway crew scheduling by using modified bacterial foraging algorithm |url=https://www.sciencedirect.com/science/article/pii/S0360835223002425 |journal=Computers & Industrial Engineering |language=en |volume=180 |pages=109218 |doi=10.1016/j.cie.2023.109218 |s2cid=257990456 |issn=0360-8352}}</ref>

=== Single-solution vs. population-based ===
Another classification dimension is single solution vs [[Population model (evolutionary algorithm)|population-based]] searches.<ref name="blum03metaheuristics" /><ref name="talbi09metaheuristics" /> Single solution approaches focus on modifying and improving a single candidate solution; single solution metaheuristics include [[simulated annealing]], [[iterated local search]], [[Variable Neighborhood Search|variable neighborhood search]], and [[Guided Local Search|guided local search]].<ref name="talbi09metaheuristics" /> Population-based approaches maintain and improve multiple candidate solutions, often using population characteristics to guide the search; population based metaheuristics include [[evolutionary computation]] and [[particle swarm optimization]].<ref name="talbi09metaheuristics" /> Another category of metaheuristics  is [[Swarm intelligence]] which is a collective behavior of decentralized, self-organized agents in a population or swarm. [[Ant colony optimization]],<ref name="M. Dorigo, Optimization, Learning and Natural Algorithms">M. Dorigo, ''Optimization, Learning and Natural Algorithms'', PhD thesis, Politecnico di Milano, Italie, 1992.</ref> [[particle swarm optimization]],<ref name="talbi09metaheuristics" /> [[social cognitive optimization]] and bacterial foraging algorithm<ref name=":0" /> are examples of this category.

=== Hybridization and memetic algorithms ===
A hybrid metaheuristic is one that combines a metaheuristic with other optimization approaches, such as algorithms from [[mathematical programming]], [[constraint programming]], and [[machine learning]]. Both components of a hybrid metaheuristic may run concurrently and exchange information to guide the search.

On the other hand, [[Memetic algorithm]]s<ref name="moscato89evolution" /> represent the synergy of evolutionary or any population-based approach with separate individual learning or local improvement procedures for problem search. An example of memetic algorithm is the use of a local search algorithm instead of or in addition to a basic [[Mutation (genetic algorithm)|mutation operator]] in evolutionary algorithms.

=== Parallel metaheuristics ===
A [[parallel metaheuristic]] is one that uses the techniques of [[parallel programming]] to run multiple metaheuristic searches in parallel; these may range from simple [[distributed computing|distributed]] schemes to concurrent search runs that interact to improve the overall solution. 

With population-based metaheuristics, the population itself can be parallelized by either processing each individual or group with a separate thread or the metaheuristic itself runs on one computer and the offspring are evaluated in a distributed manner per iteration.<ref>{{Cite book |last=Cantú-Paz |first=Erick |url=http://link.springer.com/10.1007/978-1-4615-4369-5 |title=Efficient and Accurate Parallel Genetic Algorithms |date=2001 |publisher=Springer US |isbn=978-1-4613-6964-6 |series=Genetic Algorithms and Evolutionary Computation |volume=1 |location=Boston, MA |doi=10.1007/978-1-4615-4369-5}}</ref> The latter is particularly useful if the computational effort for the evaluation is considerably greater than that for the generation of descendants. This is the case in many practical applications, especially in simulation-based calculations of solution quality.<ref name=":6">{{Citation |last=Sudholt |first=Dirk |title=Parallel Evolutionary Algorithms |date=2015 |work=Springer Handbook of Computational Intelligence |pages=929–959 |editor-last=Kacprzyk |editor-first=Janusz |url=http://link.springer.com/10.1007/978-3-662-43505-2_46 |access-date=2024-09-04 |place=Berlin, Heidelberg |publisher=Springer |language=en |doi=10.1007/978-3-662-43505-2_46 |isbn=978-3-662-43504-5 |editor2-last=Pedrycz |editor2-first=Witold}}</ref><ref name=":7">{{Citation |last1=Khalloof |first1=Hatem |last2=Mohammad |first2=Mohammad |last3=Shahoud |first3=Shadi |last4=Duepmeier |first4=Clemens |last5=Hagenmeyer |first5=Veit |date=2020-11-02 |title=A Generic Flexible and Scalable Framework for Hierarchical Parallelization of Population-Based Metaheuristics |work=Proc. of the 12th Int. Conf. on Management of Digital EcoSystems (MEDES'20) |url=https://dl.acm.org/doi/10.1145/3415958.3433041 |language=en |publisher=ACM |pages=124–131 |doi=10.1145/3415958.3433041 |isbn=978-1-4503-8115-4}}</ref>

=== Nature-inspired and metaphor-based metaheuristics ===
{{main|Swarm intelligence|List of metaphor-inspired metaheuristics}}
A very active area of research is the design of nature-inspired metaheuristics. Many recent metaheuristics, especially evolutionary computation-based algorithms, are inspired by natural systems. Nature acts as a source of concepts, mechanisms and principles for designing of artificial computing systems to deal with complex computational problems. Such metaheuristics include [[simulated annealing]], [[evolutionary algorithms]], [[ant colony optimization]] and [[particle swarm optimization]]. 

A large number of more recent metaphor-inspired metaheuristics have started to [[List of metaphor-inspired metaheuristics#Criticism|attract criticism in the research community]] for hiding their lack of novelty behind an elaborate metaphor.<ref name="Sörensen2013" /><ref name=":3" /><ref name=":4" /><ref>{{Citation |last=Lones |first=Michael A. |title=Metaheuristics in nature-inspired algorithms |date=2014 |work=Proceedings of the Companion Publication of the 2014 Annual Conference on Genetic and Evolutionary Computation (GECCO'14) |publisher=ACM |isbn=978-1-4503-2881-4 |editor-last=Igel |editor-first=Christian |series=ACM Conferences |location=New York, NY |pages=1419–1422 |language=en |doi=10.1145/2598394.2609841}}</ref> As a result, a number of renowned scientists of the field have proposed a research agenda for the standardization of metaheuristics in order to make them more comparable, among other things.<ref>{{cite web |last1=Swan |first1=Jerry |last2=Adriaensen |first2=Steven |last3=Bishr |first3=Mohamed |last4=Burke |first4=Edmund K. |last5=Clark |first5=John A. |last6=De Causmaecke |first6=Patrick |last7=Durillo |first7=Juan José |last8=Hammond |first8=Kevin |last9=Hart |first9=Emma |last10=Johnson |first10=Colin G. |last11=Kocsis |first11=Zoltan A. |last12=Kovitz |first12=Ben |last13=Krawiec |first13=Krzysztof |last14=Martin |first14=Simon |last15=Merelo |first15=Juan J. |date=2015 |title=A Research Agenda for Metaheuristic Standardization |url=http://www.cs.nott.ac.uk/~exo/docs/publications/research-agenda-metaheuristic.pdf |access-date=2024-08-30 |website=Semantic Scholar |first16=Leandro L. |last16=Minku |first17=Ender |last17=Özcan |first18=Gisele Lobo |last18=Pappa |first19=Erwin |last19=Pesch |first20=Pablo |last20=García-Sánchez |first21=Andrea |last21=Schaerf |first22=Kevin |last22=Sim |first23=Jim |last23=Smith |first24=Thomas |last24=Stützle |first25=Stefan |last25=Wagner |s2cid=63728283}}</ref> Another consequence is that the publication guidelines of a number of scientific journals have been adapted accordingly.<ref>{{Cite web |title=Journal of Heuristic Policies on Heuristic Search Research |url=https://link.springer.com/journal/10732/submission-guidelines |archive-url=https://web.archive.org/web/20170709034524/https://www.springer.com/cda/content/document/cda_downloaddocument/Journal+of+Heuristic+Policies+on+Heuristic+Search.pdf?SGWID=0-0-45-1483502-p35487524 |archive-date=2017-07-09 |access-date=2024-09-01 |website=Journal of Heuristics - Submission guidelines}}</ref><ref>{{Cite web |title=Aims and scope |url=https://link.springer.com/journal/10288/aims-and-scope |access-date=2024-09-01 |website=4OR}}</ref><ref>{{Cite web |title=Aims and scope |url=https://link.springer.com/journal/12293/aims-and-scope |access-date=2024-09-01 |website=Memetic Computing}}</ref>

== Applications ==

Most metaheuristics are search methods and when using them, the evaluation function should be subject to greater demands than a mathematical optimization. Not only does the desired target state have to be formulated, but the evaluation should also reward improvements to a solution on the way to the target in order to support and accelerate the search process. The [[Fitness function|fitness functions]] of evolutionary or memetic algorithms can serve as an example. 

Metaheuristics are used for all types of optimization problems, ranging from [[Continuous optimization|continuous]] through mixed integer problems to [[combinatorial optimization]] or combinations thereof.<ref name=":5" /><ref>{{Cite book |url=https://link.springer.com/book/10.1007/11890584 |title=Hybrid Metaheuristics |date=2006 |publisher=Springer |isbn=978-3-540-46384-9 |editor-last=Almeida |editor-first=Francisco |series=Lecture Notes in Computer Science |volume=4030 |location=Berlin, Heidelberg |doi=10.1007/11890584 |editor-last2=Blesa Aguilera |editor-first2=María J. |editor-last3=Blum |editor-first3=Christian |editor-last4=Moreno Vega |editor-first4=José Marcos |editor-last5=Pérez Pérez |editor-first5=Melquíades |editor-last6=Roli |editor-first6=Andrea |editor-last7=Sampels |editor-first7=Michael}}</ref><ref>{{Cite book |url=http://link.springer.com/10.1007/978-3-642-23247-3 |title=Handbook of Memetic Algorithms |date=2012 |publisher=Springer Berlin Heidelberg |isbn=978-3-642-23246-6 |editor-last=Neri |editor-first=Ferrante |series=Studies in Computational Intelligence |volume=379 |location=Berlin, Heidelberg |language=en |doi=10.1007/978-3-642-23247-3 |editor-last2=Cotta |editor-first2=Carlos |editor-last3=Moscato |editor-first3=Pablo}}</ref> In combinatorial optimization, an optimal solution is sought over a [[discrete mathematics|discrete]] search-space. An example problem is the [[travelling salesman problem]] where the search-space of candidate solutions grows faster than [[exponential growth|exponentially]] as the size of the problem increases, which makes an [[exhaustive search]] for the optimal solution infeasible.<ref>{{Cite journal |last1=Dorigo |first1=M. |last2=Gambardella |first2=L.M. |date=April 1997 |title=Ant colony system: a cooperative learning approach to the traveling salesman problem |url=https://ieeexplore.ieee.org/document/585892 |journal=IEEE Transactions on Evolutionary Computation |volume=1 |issue=1 |pages=53–66 |doi=10.1109/4235.585892}}</ref><ref>{{Cite journal |last1=Merz |first1=Peter |last2=Freisleben |first2=Bernd |date=2002 |title=Memetic Algorithms for the Traveling Salesman Problem |url=https://www.researchgate.net/publication/2928026 |journal=Complex Systems |volume=13 |issue=4}}</ref> Additionally, multidimensional combinatorial problems, including most design problems in [[engineering]]<ref name=":2" /><ref>Tomoiagă B, Chindriş M, Sumper A, Sudria-Andreu A, Villafafila-Robles R. [http://www.mdpi.com/1996-1073/6/3/1439/pdf Pareto Optimal Reconfiguration of Power Distribution Systems Using a Genetic Algorithm Based on NSGA-II. ] Energies. 2013; 6(3):1439–1455.</ref><ref>{{Cite journal|title = Swarm intelligence and gravitational search algorithm for multi-objective optimization of synthesis gas production|journal = Applied Energy|date = 2013-03-01|pages = 368–374|volume = 103|doi= 10.1016/j.apenergy.2012.09.059|first1 = T.|last1 = Ganesan|first2 = I.|last2 = Elamvazuthi|first3 = Ku Zilati|last3 = Ku Shaari|first4 = P.|last4 = Vasant| bibcode=2013ApEn..103..368G }}</ref><ref>{{Cite book|date = 2011-11-01|pages = 86–91|doi = 10.1109/ICCSCE.2011.6190501|first1 = T.|last1 = Ganesan|first2 = I.|last2 = Elamvazuthi|first3 = P.|last3 = Vasant| title=2011 IEEE International Conference on Control System, Computing and Engineering | chapter=Evolutionary normal-boundary intersection (ENBI) method for multi-objective optimization of green sand mould system |isbn = 978-1-4577-1642-3|s2cid = 698459}}</ref> such as form-finding and behavior-finding, suffer from the [[curse of dimensionality]], which also makes them infeasible for exhaustive search or [[Analytical solution|analytical method]]s. 

Metaheuristics are also frequently applied to scheduling problems. A typical representative of this combinatorial task class is job shop scheduling, which involves assigning the work steps of jobs to processing stations in such a way that all jobs are completed on time and altogether in the shortest possible time.<ref name=":1">{{Cite book |url=https://www.wiley.com/en-dk/Metaheuristics+for+Production+Scheduling-p-9781848214972 |title=Metaheuristics for production scheduling |date=2013 |publisher=ISTE |isbn=978-1-84821-497-2 |editor-last=Jarboui |editor-first=Bassem |series=Automation - control and industrial engineering series |location=London |editor-last2=Siarry |editor-first2=Patrick |editor-last3=Teghem |editor-first3=Jacques}}</ref><ref>{{Cite book |url=http://link.springer.com/10.1007/978-3-540-78985-7 |title=Metaheuristics for Scheduling in Industrial and Manufacturing Applications |date=2008 |publisher=Springer Berlin Heidelberg |isbn=978-3-540-78984-0 |editor-last=Xhafa |editor-first=Fatos |series=Studies in Computational Intelligence |volume=128 |location=Berlin, Heidelberg |language=en |doi=10.1007/978-3-540-78985-7 |s2cid=42238720 |editor-last2=Abraham |editor-first2=Ajith}}</ref> In practice, restrictions often have to be observed, e.g. by limiting the permissible sequence of work steps of a job through predefined workflows<ref>{{Cite journal |last1=Jakob |first1=Wilfried |last2=Strack |first2=Sylvia |last3=Quinte |first3=Alexander |last4=Bengel |first4=Günther |last5=Stucky |first5=Karl-Uwe |last6=Süß |first6=Wolfgang |date=2013-04-22 |title=Fast Rescheduling of Multiple Workflows to Constrained Heterogeneous Resources Using Multi-Criteria Memetic Computing |journal=Algorithms |language=en |volume=6 |issue=2 |pages=245–277 |doi=10.3390/a6020245 |issn=1999-4893 |doi-access=free }}</ref> and/or with regard to resource utilisation, e.g. in the form of smoothing the energy demand.<ref>{{Cite journal |last1=Kizilay |first1=Damla |last2=Tasgetiren |first2=M. Fatih |last3=Pan |first3=Quan-Ke |last4=Süer |first4=Gürsel |date=2019 |title=An Ensemble of Meta-Heuristics for the Energy-Efficient Blocking Flowshop Scheduling Problem |journal=Procedia Manufacturing |language=en |volume=39 |pages=1177–1184 |doi=10.1016/j.promfg.2020.01.352|s2cid=213710393 |doi-access=free }}</ref><ref>{{Cite journal |last1=Grosch |first1=Benedikt |last2=Weitzel |first2=Timm |last3=Panten |first3=Niklas |last4=Abele |first4=Eberhard |date=2019 |title=A metaheuristic for energy adaptive production scheduling with multiple energy carriers and its implementation in a real production system |journal=Procedia CIRP |language=en |volume=80 |pages=203–208 |doi=10.1016/j.procir.2019.01.043|s2cid=164850023 |doi-access=free }}</ref> Popular metaheuristics for combinatorial problems include [[genetic algorithms]] by Holland et al.,<ref name="holland75adaptation" /> scatter search<ref name="glover77scattersearch" /> and [[tabu search]]<ref name="glover86future" /> by Glover. 

Another large field of application are optimization tasks in continuous or mixed-integer search spaces. This includes, e.g., design optimization<ref name=":2">{{Cite journal |last1=Gupta |first1=Shubham |last2=Abderazek |first2=Hammoudi |last3=Yıldız |first3=Betül Sultan |last4=Yildiz |first4=Ali Riza |last5=Mirjalili |first5=Seyedali |last6=Sait |first6=Sadiq M. |date=2021 |title=Comparison of metaheuristic optimization algorithms for solving constrained mechanical design optimization problems |url=https://linkinghub.elsevier.com/retrieve/pii/S095741742100779X |journal=Expert Systems with Applications |language=en |volume=183 |pages=115351 |doi=10.1016/j.eswa.2021.115351}}</ref><ref>{{Citation |last1=Quinte |first1=Alexander |title=Optimization of a Micro Actuator Plate Using Evolutionary Algorithms and Simulation Based on Discrete Element Methods |date=2002 |work=International Conference on Modeling and Simulation of Microsystems: MSM 2002 |pages=192–197 |editor-last=Laudon |editor-first=Matthew |location=Cambridge, Mass |publisher=Computational Publications |isbn=978-0-9708275-7-9 |last2=Jakob |first2=Wilfried |last3=Scherer |first3=Klaus-Peter |last4=Eggert |first4=Horst |url=https://www.researchgate.net/publication/228790476}}</ref><ref>{{Citation |last=Parmee |first=I. C. |title=Strategies for the Integration of Evolutionary/Adaptive Search with the Engineering Design Process |date=1997 |url=http://link.springer.com/10.1007/978-3-662-03423-1_25 |work=Evolutionary Algorithms in Engineering Applications |pages=453–477 |editor-last=Dasgupta |editor-first=Dipankar |access-date=2023-07-17 |place=Berlin, Heidelberg |publisher=Springer |language=en |doi=10.1007/978-3-662-03423-1_25 |isbn=978-3-642-08282-5 |editor2-last=Michalewicz |editor2-first=Zbigniew}}</ref> or various engineering tasks.<ref>{{Cite book |url=https://link.springer.com/10.1007/978-3-319-06508-3 |title=Applications of Metaheuristics in Process Engineering |date=2014 |publisher=Springer International Publishing |isbn=978-3-319-06507-6 |editor-last=Valadi |editor-first=Jayaraman |location=Cham |language=en |doi=10.1007/978-3-319-06508-3 |s2cid=40197553 |editor-last2=Siarry |editor-first2=Patrick}}</ref><ref>{{Cite book |last1=Sanchez |first1=Ernesto |url=http://link.springer.com/10.1007/978-3-642-27467-1 |title=Industrial Applications of Evolutionary Algorithms |last2=Squillero |first2=Giovanni |last3=Tonda |first3=Alberto |date=2012 |publisher=Springer |isbn=978-3-642-27466-4 |series=Intelligent Systems Reference Library |volume=34 |location=Berlin, Heidelberg |language=en |doi=10.1007/978-3-642-27467-1}}</ref><ref>{{Cite book |url=https://link.springer.com/10.1007/978-3-031-16832-1 |title=Engineering Applications of Modern Metaheuristics |date=2023 |publisher=Springer International Publishing |isbn=978-3-031-16831-4 |editor-last=Akan |editor-first=Taymaz |series=Studies in Computational Intelligence |volume=1069 |location=Cham |language=en |doi=10.1007/978-3-031-16832-1 |s2cid=254222401 |editor-last2=Anter |editor-first2=Ahmed M. |editor-last3=Etaner-Uyar |editor-first3=A. Şima |editor-last4=Oliva |editor-first4=Diego}}</ref> An example of the mixture of combinatorial and continuous optimization is the planning of favourable motion paths for industrial robots.<ref>{{Citation |last=Blume |first=Christian |title=Optimized Collision Free Robot Move Statement Generation by the Evolutionary Software GLEAM |date=2000 |url=http://link.springer.com/10.1007/3-540-45561-2_32 |work=Real-World Applications of Evolutionary Computing |series=Lecture Notes in Computer Science |volume=1803 |pages=330–341 |editor-last=Cagnoni |editor-first=Stefano |access-date=2023-07-17 |place=Berlin, Heidelberg |publisher=Springer |language=en |doi=10.1007/3-540-45561-2_32 |isbn=978-3-540-67353-8}}</ref><ref>{{Citation |last1=Pholdee |first1=Nantiwat |title=Multiobjective Trajectory Planning of a 6D Robot based on Multiobjective Meta Heuristic Search |date=2018-12-14 |url=https://dl.acm.org/doi/10.1145/3301326.3301356 |work=International Conference on Network, Communication and Computing (ICNCC 2018) |pages=352–356 |publisher=ACM |language=en |doi=10.1145/3301326.3301356 |isbn=978-1-4503-6553-6 |last2=Bureerat |first2=Sujin|s2cid=77394395 }}</ref>

== Metaheuristic Optimization Frameworks ==

A MOF can be defined as ‘‘a set of software tools that provide a correct and reusable implementation of a set of metaheuristics, and the basic mechanisms to accelerate the implementation of its partner subordinate heuristics (possibly including solution encodings and technique-specific operators), which are necessary to solve a particular problem instance using techniques provided’’.<ref name=":8">{{Cite journal |last1=Parejo |first1=José Antonio |last2=Ruiz-Cortés |first2=Antonio |last3=Lozano |first3=Sebastián |last4=Fernandez |first4=Pablo |date=March 2012 |title=Metaheuristic optimization frameworks: a survey and benchmarking |journal=Soft Computing |language=en |volume=16 |issue=3 |pages=527–561 |doi=10.1007/s00500-011-0754-8 |issn=1432-7643}}</ref>

There are many candidate optimization tools which can be considered as a MOF of varying feature. The following list of 33 MOFs is compared and evaluated in detail in:<ref name=":8" /> Comet, EvA2, evolvica, Evolutionary::Algorithm, GAPlayground, jaga, JCLEC, JGAP, jMetal, n-genes, Open Beagle, Opt4j, ParadisEO/EO, Pisa, Watchmaker, FOM, Hypercube, HotFrame, Templar, EasyLocal, iOpt, OptQuest, JDEAL, Optimization Algorithm Toolkit, HeuristicLab, MAFRA, Localizer, GALIB, DREAM, Discropt, MALLBA, MAGMA, and UOF. There have been a number of publications on the support of parallel implementations, which was missing in this comparative study, particularly from the late 10s onwards.<ref name=":6" /><ref name=":7" /><ref>{{Citation |last1=García-Valdez |first1=Mario |last2=Merelo |first2=J.J. |date=2017-07-15 |title=evospace-js: asynchronous pool-based execution of heterogeneous metaheuristics |work=GECCO '17: Proceedings of the Genetic and Evolutionary Computation Conference, Companion |language=en |publisher=ACM |place=New York |pages=1202–1208 |doi=10.1145/3067695.3082473 |isbn=978-1-4503-4939-0}}</ref><ref>{{Cite journal |last1=Lim |first1=Dudy |last2=Ong |first2=Yew-Soon |last3=Jin |first3=Yaochu |last4=Sendhoff |first4=Bernhard |last5=Lee |first5=Bu-Sung |date=May 2007 |title=Efficient Hierarchical Parallel Genetic Algorithms using Grid computing |journal=Future Generation Computer Systems |language=en |volume=23 |issue=4 |pages=658–670 |doi=10.1016/j.future.2006.10.008}}</ref><ref>{{Citation |last1=Nebro |first1=Antonio J. |last2=Barba-González |first2=Cristóbal |last3=Nieto |first3=José García |last4=Cordero |first4=José A. |last5=Montes |first5=José F. Aldana |date=2017-07-15 |title=Design and architecture of the jMetaISP framework |language=en |work=GECCO '17: Proceedings of the Genetic and Evolutionary Computation Conference, Companion |publisher=ACM |place=New York |pages=1239–1246 |doi=10.1145/3067695.3082466 |isbn=978-1-4503-4939-0}}</ref>

== Contributions ==
<!-- PLEASE ONLY ADD THE MOST SIGNIFICANT CONTRIBUTIONS!
     Entries that are too recent will be removed. Entries that do no have an article will be removed.
     Articles that do not establish the significance of a particular algorithm (as measured by substantial coverage in secondary and tertiary sources) will be deleted.
 -->

Many different metaheuristics are in existence and new variants are continually being proposed. Some of the most significant contributions to the field are:

* 1952: Robbins and Monro work on stochastic optimization methods.<ref name=robbins52stochastic/>
* 1954: [[Nils Aall Barricelli|Barricelli]] carries out the first simulations of the [[evolution]] process and uses them on general optimization problems.<ref name=barricelli54esempi/>
* 1963: Rastrigin proposes [[random search]].<ref name=rastrigin63convergence/>
* 1965: Matyas proposes [[random optimization]].<ref name=matyas65random/>
* 1965: [[John Nelder|Nelder]] and Mead propose a [[Nelder–Mead method|simplex heuristic]], which was shown by [[Michael J. D. Powell|Powell]] to converge to non-stationary points on some problems.<ref name=nelder65simplex/>
* 1965: [[Ingo Rechenberg]] discovers the first [[Evolution Strategies]] algorithm.<ref name=rechenberg65ES/>
* 1966: [[Lawrence J. Fogel|Fogel]] et al. propose [[evolutionary programming]].<ref name=fogel66artificial/>
* 1970: Hastings proposes the [[Metropolis–Hastings algorithm]].<ref name=hastings70monte/>
* 1970: Cavicchio proposes adaptation of control parameters for an optimizer.<ref name=cavicchio70adaptive/>
* 1970: Kernighan and Lin propose a graph partitioning method, related to variable-depth search and [[tabu search|prohibition-based (tabu) search]].<ref name=kernighan1970efficient/>
* 1975: [[John Henry Holland|Holland]] proposes the [[genetic algorithm]].<ref name=holland75adaptation/>
* 1977: [[Fred W. Glover|Glover]] proposes scatter search.<ref name=glover77scattersearch/>
* 1978: Mercer and Sampson propose a [[meta-optimization|metaplan]] for tuning an optimizer's parameters by using another optimizer.<ref name=mercer78adaptive/>
* 1980: Smith describes [[genetic programming]].<ref name=smith80learning/>
* 1983: Kirkpatrick et al. propose [[simulated annealing]].<ref name=kirkpatrick83optimization/>
* 1986: [[Fred W. Glover|Glover]] proposes [[tabu search]], first mention of the term ''metaheuristic''.<ref name=glover86future/>
* 1989: Moscato proposes [[memetic algorithms]].<ref name=moscato89evolution/>
* 1990: Moscato and Fontanari,<ref>{{citation |journal=Physics Letters A |pages= 204–208 |last1=Moscato |first1=P. |last2=Fontanari |first2=J.F.| title=Stochastic versus deterministic update in simulated annealing |volume=146 |issue=4 |year=1990|doi=10.1016/0375-9601(90)90166-L|bibcode= 1990PhLA..146..204M }}</ref> and Dueck and Scheuer,<ref>{{citation |journal=Journal of Computational Physics| pages=161–175 | last1=Dueck |first1=G.| last2=Scheuer| first2=T.| title=Threshold accepting: A general purpose optimization algorithm appearing superior to simulated annealing| volume=90 | issue=1 | year=1990 | issn=0021-9991 | 
doi=10.1016/0021-9991(90)90201-B| bibcode=1990JCoPh..90..161D }}</ref> independently proposed a deterministic update rule for [[simulated annealing]] which accelerated the search. This led to the [[threshold accepting]] metaheuristic.
* 1992: [[Marco Dorigo|Dorigo]] introduces [[ant colony optimization]] in his PhD thesis.<ref name="M. Dorigo, Optimization, Learning and Natural Algorithms"/>
* 1995: Wolpert and Macready prove the [[No free lunch in search and optimization|no free lunch]] theorems.<ref name=wolpert95nofreelunch/><ref name="Igel2003" /><ref name=Auger2010/><ref name=Droste2002/>

== See also ==
<!-- Please don't add a whole list of optimization algorithms, the categories serve that purpose. -->
<!-- 27/3/11 Updated with more meaningful category organisation – I have tried to keep the list as brief as possible, relying on sub-pages for further breakdown of categories. Please try to maintain this spirit on edits. -->

* [[Stochastic search]]
* [[Meta-optimization]]
* [[Matheuristics]]
* [[Hyper-heuristics]]
* [[Swarm intelligence]]
* [[Evolutionary algorithm|Evolutionary algorithms]] and in particular [[genetic algorithms]], [[genetic programming]], or [[Evolution strategy|evolution strategies]].
* [[Simulated annealing]]
* [[Workforce modeling]]

== References ==
<!-- {{reflist}} -->
<!-- Please provide complete references to journal / conference papers, tech reports, msc/phd theses, etc. and make sure the reference format is correct. Only include major research contributions on this page. Only add a reference when you use it in a good description in the main article. -->
{{reflist|2|refs=

<ref name=Bianchi2009>{{cite journal|last=Bianchi|first=Leonora|author2=Marco Dorigo |author3=Luca Maria Gambardella |author4=Walter J. Gutjahr |title=A survey on metaheuristics for stochastic combinatorial optimization|journal=Natural Computing|year=2009|volume=8|issue=2|pages=239–287|doi=10.1007/s11047-008-9098-4|s2cid=9141490|url=http://doc.rero.ch/record/319945/files/11047_2008_Article_9098.pdf}}</ref>
<ref name=Bala2015>
{{cite journal
  | title=Stellar-Mass Black Hole Optimization for Biclustering Microarray Gene Expression Data
  |author1=R. Balamurugan |author2=A.M. Natarajan|author3=K. Premalatha| journal=Applied Artificial Intelligence 
  | volume=29
  | number=4
  | pages=353–381
  | year=2015
  |doi=10.1080/08839514.2015.1016391 |s2cid=44624424 | doi-access=free}}</ref>
<ref name="Droste2002">{{cite journal|author1=Stefan Droste |author2=Thomas Jansen |author3=Ingo Wegener | title=Optimization with Randomized Search Heuristics – The (A)NFL Theorem, Realistic Scenarios, and Difficult Functions |journal=Theoretical Computer Science| year=2002| volume=287| number=1| pages=131–144| doi=10.1016/s0304-3975(02)00094-4|citeseerx=10.1.1.35.5850 }}</ref>

<ref name="Igel2003">{{cite journal| author=Igel, Christian, Toussaint, Marc| title=On classes of functions for which No Free Lunch results hold| journal=Information Processing Letters|date=Jun 2003| volume=86| number=6| pages=317–321| doi=10.1016/S0020-0190(03)00222-9|issn=0020-0190| arxiv=cs/0108011| s2cid=147624}}</ref>

<ref name="Auger2010">{{cite journal| author=[[Anne Auger|Auger, Anne]], Teytaud, Olivier| title=Continuous Lunches Are Free Plus the Design of Optimal Optimization Algorithms|journal=Algorithmica| year=2010| volume=57| issue=1| pages=121–146| doi=10.1007/s00453-008-9244-5|issn=0178-4617| citeseerx=10.1.1.186.6007| s2cid=1989533}}</ref>

<ref name=wolpert95nofreelunch>
{{Cite journal
|last1=Wolpert
|first1=D.H.
|last2=Macready
|first2=W.G.
|s2cid=12890367
|title=No free lunch theorems for search
|journal=Technical Report SFI-TR-95-02-010
|publisher=Santa Fe Institute
|year=1995
}}
</ref>

<ref name=kernighan1970efficient>
{{cite journal
|title=An efficient heuristic procedure for partitioning graphs
|author1=Kernighan, B.W. |author2=Lin, S. |journal=Bell System Technical Journal
|volume=49
|issue=2
|pages=291–307
|year=1970
|doi=10.1002/j.1538-7305.1970.tb01770.x
}}
</ref>

<ref name=talbi09metaheuristics>
{{cite book
|title=Metaheuristics: from design to implementation
|last1=Talbi
|first1=E-G.
|year=2009
|publisher=Wiley
|isbn=978-0-470-27858-1
}}
</ref>

<ref name=glover03handbook>
{{cite book
|title=Handbook of metaheuristics
|last1=Glover
|first1=F.
|last2=Kochenberger
|first2=G.A.
|year=2003
|publisher=Springer, International Series in Operations Research & Management Science
|volume=57
|isbn=978-1-4020-7263-5
}}
</ref>

<ref name=glover77scattersearch>
{{cite journal
|last1=Glover
|first1=Fred
|title=Heuristics for Integer programming Using Surrogate Constraints
|journal=Decision Sciences
|year=1977
|pages=156–166
|volume=8
|issue=1
|doi=10.1111/j.1540-5915.1977.tb01074.x
|citeseerx=10.1.1.302.4071
}}
</ref>

<ref name=glover86future>
{{cite journal
|doi=10.1016/0305-0548(86)90048-1
|last1=Glover
|first1=F.
|title=Future Paths for Integer Programming and Links to Artificial Intelligence
|journal=Computers and Operations Research
|year=1986
|pages=533–549
|volume=13
|issue=5
}}
</ref>

<ref name=blum03metaheuristics>
{{cite journal
|title=Metaheuristics in combinatorial optimization: Overview and conceptual comparison
|last1=Blum
|first1=Christian
|last2=Roli
|first2=Andrea
|year=2003
|publisher=ACM
|journal=ACM Computing Surveys
|volume=35
|pages=268–308
|issue=3
|doi=10.1145/937503.937505
|url=https://www.researchgate.net/publication/221900771
}}
</ref>

<ref name=holland75adaptation>
{{cite book
|title=Adaptation in Natural and Artificial Systems
|last1=Holland
|first1=J.H.
|year=1975
|publisher=University of Michigan Press
|isbn=978-0-262-08213-6
|url-access=registration
|url=https://archive.org/details/adaptationinnatu00holl
}}
</ref>

<ref name=kirkpatrick83optimization>
{{cite journal
|last1=Kirkpatrick
|first1=S.
|last2=Gelatt Jr.
|first2=C.D.
|last3=Vecchi
|first3=M.P.
|title=Optimization by Simulated Annealing
|journal=Science
|volume=220
|year=1983
|pages=671–680
|issue=4598
|pmid=17813860
|doi=10.1126/science.220.4598.671
|bibcode=1983Sci...220..671K
|citeseerx=10.1.1.123.7607
|s2cid=205939
}}
</ref>

<ref name=nelder65simplex>
{{cite journal
|last1=Nelder
|first1=J.A.
|last2=Mead
|first2=R.
|s2cid=2208295
|title=A simplex method for function minimization
|journal=Computer Journal
|year=1965
|pages=308–313
|volume=7
|issue=4
|doi=10.1093/comjnl/7.4.308
}}
</ref>

<ref name=robbins52stochastic>
{{cite journal
|doi=10.1214/aoms/1177729586
|last1=Robbins
|first1=H.
|last2=Monro
|first2=S.
|title=A Stochastic Approximation Method
|journal=Annals of Mathematical Statistics
|year=1951
|pages=400–407
|issue=3
|volume=22
|url=http://dml.cz/bitstream/handle/10338.dmlcz/100283/CzechMathJ_08-1958-1_10.pdf
|doi-access=free
}}
</ref>

<ref name=barricelli54esempi>
{{cite journal
|last1=Barricelli
|first1=N.A.
|title=Esempi numerici di processi di evoluzione
|journal=Methodos
|year=1954
|pages=45–68
}}
</ref>

<ref name=rastrigin63convergence>
{{cite journal
|last=Rastrigin
|first=L.A.
|title=The convergence of the random search method in the extremal control of a many parameter system
|journal=Automation and Remote Control
|year=1963
|volume=24
|pages=1337–1342
|issue=10
}}
</ref>

<ref name=matyas65random>
{{cite journal
|last=Matyas
|first=J.
|title=Random optimization
|journal=Automation and Remote Control
|year=1965
|volume=26
|pages=246–253
|issue=2
|url=http://www.mathnet.ru/eng/at11288
}}
</ref>

<ref name=fogel66artificial>
{{cite book
|title=Artificial Intelligence through Simulated Evolution
|last1=Fogel
|first1=L.
|last2=Owens
|first2=A.J.
|last3=Walsh
|first3=M.J.
|year=1966
|publisher=Wiley
|isbn=978-0-471-26516-0
}}
</ref>

<ref name=hastings70monte>
{{cite journal
|doi=10.1093/biomet/57.1.97
|last1=Hastings
|first1=W.K.
|s2cid=21204149
|title=Monte Carlo Sampling Methods Using Markov Chains and Their Applications
|journal=Biometrika
|year=1970
|pages=97–109
|volume=57
|issue=1
|bibcode=1970Bimka..57...97H
}}
</ref>

<ref name=cavicchio70adaptive>
{{Cite journal
|last1=Cavicchio
|first1=D.J.
|hdl=2027.42/4042 
|title=Adaptive search using simulated evolution
|journal=Technical Report 
|publisher=University of Michigan, Computer and Communication Sciences Department
|year=1970
}}
</ref>

<ref name=mercer78adaptive>
{{cite journal
|last=Mercer
|first=R.E.
|author2=Sampson, J.R.
|title=Adaptive search using a reproductive metaplan
|journal=Kybernetes
|year=1978
|volume=7
|issue=3
|pages=215&ndash;228
|doi=10.1108/eb005486
}}
</ref>

<ref name=goldberg89genetic>
{{cite book
|title=Genetic Algorithms in Search, Optimization and Machine Learning
|last1=Goldberg
|first1=D.E.
|year=1989
|publisher=Kluwer Academic Publishers
|isbn=978-0-201-15767-3
}}
</ref>

<ref name=smith80learning>
{{cite thesis
|type=PhD Thesis
|title=A Learning System Based on Genetic Adaptive Algorithms
|last=Smith
|first=S.F.
|year=1980
|publisher=University of Pittsburgh
|url=https://dl.acm.org/citation.cfm?id=909835
}}
</ref>

<ref name=rechenberg65ES>
{{cite journal
|last=Rechenberg
|first=Ingo
|title=Cybernetic Solution Path of an Experimental Problem
|journal=Royal Aircraft Establishment, Library Translation
|year=1965
}}
</ref>

<ref name=moscato89evolution>{{cite journal|last=Moscato|first=P.|year=1989|title=On Evolution, Search, Optimization, Genetic Algorithms and Martial Arts: Towards Memetic Algorithms|journal=Caltech Concurrent Computation Program|issue=report 826|url=https://www.researchgate.net/publication/2354457}}</ref>

}}

== Further reading ==
* {{cite book|first1=Kenneth|last1=Sörensen|first2=Marc|last2=Sevaux|first3=Fred|last3=Glover|editor-last1=Martí|editor-first1=Rafael|editor-last2=Panos|editor-first2=Pardalos|editor-last3=Resende|editor-first3=Mauricio|editor3-link=Mauricio Resende|isbn=978-3-319-07123-7|title=Handbook of Heuristics|chapter=A History of Metaheuristics|publisher=Springer|chapter-url=http://leeds-faculty.colorado.edu/glover/468%20-%20A%20History%20of%20Metaheuristics%20w%20Sorensen%20%26%20Sevaux.pdf|date=2017-01-16}}
* Ashish Sharma (2022), Nature Inspired Algorithms with Randomized Hypercomputational Perspective. ''Information Sciences.'' https://doi.org/10.1016/j.ins.2022.05.020.

== External links ==
<!-- Please don't add software-libraries here! -->
*{{scholarpedia|title=Metaheuristics|urlname=Metaheuristics|curator=[[Fred W. Glover|Fred Glover]] and Kenneth Sörensen}} 
*[http://www.metaheuristics.eu EU/ME] forum for researchers in the field.
*[https://jmejia8.github.io/Metaheuristics.jl/stable/algorithms/ Metaheuristics.jl] Source of some implementations.

{{Optimization algorithms|heuristic}}

[[Category:Metaheuristics| ]]