In computer science and mathematical optimization, a metaheuristic is a higher-level procedure or heuristic designed to find, generate, or select a heuristic (partial search algorithm) that may provide a sufficiently good solution to an optimization problem, especially with incomplete or imperfect information or limited computation capacity. 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.
Compared to optimization algorithms and iterative methods, metaheuristics do not guarantee that a globally optimal solution can be found on some class of problems. Many metaheuristics implement some form of stochastic optimization, so that the solution found is dependent on the set of random variables generated. In combinatorial optimization, by searching over a large set of feasible solutions, metaheuristics can often find good solutions with less computational effort than optimization algorithms, iterative methods, or simple heuristics. As such, they are useful approaches for optimization problems. Several books and survey papers have been published on the subject.
Most literature on metaheuristics is experimental in nature, describing empirical results based on computer experiments with the algorithms. But some formal theoretical results are also available, often on convergence and the possibility of finding the global optimum. Many metaheuristic methods have been published with claims of novelty and practical efficacy. While the field also features high-quality research, many of the publications have been of poor quality; flaws include vagueness, lack of conceptual elaboration, poor experiments, and ignorance of previous literature.
These are properties that characterize most metaheuristics:
- Metaheuristics are strategies that guide the search process.
- The goal is to efficiently explore the search space in order to find near–optimal solutions.
- Techniques which constitute metaheuristic algorithms range from simple local search procedures to complex learning processes.
- Metaheuristic algorithms are approximate and usually non-deterministic.
- Metaheuristics are not problem-specific.
Local search vs. global search
One approach is to characterize the type of search strategy. 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, and GRASP. 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-based metaheuristics. Such metaheuristics include ant colony optimization, evolutionary computation, particle swarm optimization, genetic algorithm, and rider optimization algorithm 
Single-solution vs. population-based
Another classification dimension is single solution vs population-based searches. Single solution approaches focus on modifying and improving a single candidate solution; single solution metaheuristics include simulated annealing, iterated local search, variable neighborhood search, and guided local search. Population-based approaches maintain and improve multiple candidate solutions, often using population characteristics to guide the search; population based metaheuristics include evolutionary computation, genetic algorithms, and particle swarm optimization. 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, particle swarm optimization, social cognitive optimization 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 algorithms 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 a basic mutation operator in evolutionary algorithms.
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 schemes to concurrent search runs that interact to improve the overall solution.
Nature-inspired and metaphor-based 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 attract criticism in the research community for hiding their lack of novelty behind an elaborate metaphor.
We seem to be facing a novel source of inspiration. This will make way towards many ancient-inspired algorithms. There have been numerous limitations in the ancient era, but various man-made structures indicate that limitations and lack of facilities have led to some sort of optimization. A closer look at these ancient relics shows that the methods, strategies, and technologies used in antiquity are far more advanced and optimized than we would have imagined. Ancient-inspired ideology observes and reflects on the features and seeks to understand ways of managing the project at that time.
Metaheuristics are used for combinatorial optimization in which an optimal solution is sought over a discrete search-space. An example problem is the travelling salesman problem where the search-space of candidate solutions grows faster than exponentially as the size of the problem increases, which makes an exhaustive search for the optimal solution infeasible. Additionally, multidimensional combinatorial problems, including most design problems in engineering such as form-finding and behavior-finding, suffer from the curse of dimensionality, which also makes them infeasible for exhaustive search or analytical methods. Metaheuristics are also widely used for jobshop scheduling and job selection problems. Popular metaheuristics for combinatorial problems include simulated annealing by Kirkpatrick et al., genetic algorithms by Holland et al., scatter search and tabu search by Glover. Literature review on metaheuristic optimization, suggested that it was Fred Glover who coined the word metaheuristics.
Metaheuristic Optimization Frameworks (MOFs)
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’’.
There are many candidate optimization tools which can be considered as a MOF of varying feature: 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, UOF and OptaPlanner.
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.
- 1954: Barricelli carry out the first simulations of the evolution process and use them on general optimization problems.
- 1963: Rastrigin proposes random search.
- 1965: Matyas proposes random optimization.
- 1965: Nelder and Mead propose a simplex heuristic, which was shown by Powell to converge to non-stationary points on some problems.
- 1965: Ingo Rechenberg discovers the first Evolution Strategies algorithm.
- 1966: Fogel et al. propose evolutionary programming.
- 1970: Hastings proposes the Metropolis–Hastings algorithm.
- 1970: Cavicchio proposes adaptation of control parameters for an optimizer.
- 1970: Kernighan and Lin propose a graph partitioning method, related to variable-depth search and prohibition-based (tabu) search.
- 1975: Holland proposes the genetic algorithm.
- 1977: Glover proposes scatter search.
- 1978: Mercer and Sampson propose a metaplan for tuning an optimizer's parameters by using another optimizer.
- 1980: Smith describes genetic programming.
- 1983: Kirkpatrick et al. propose simulated annealing.
- 1986: Glover proposes tabu search, first mention of the term metaheuristic.
- 1989: Moscato proposes memetic algorithms.
- 1990: Moscato and Fontanari, and Dueck and Scheuer, independently proposed a deterministic update rule for simulated annealing which accelerated the search. This led to the threshold accepting metaheuristic.
- 1992: Dorigo introduces ant colony optimization in his PhD thesis.
- 1995: Wolpert and Macready prove the no free lunch theorems.
- Stochastic search
- Swarm intelligence
- Genetic algorithms
- Simulated annealing
- Workforce modeling
- R. Balamurugan; A.M. Natarajan; K. Premalatha (2015). "Stellar-Mass Black Hole Optimization for Biclustering Microarray Gene Expression Data". Applied Artificial Intelligence. 29 (4): 353–381. doi:10.1080/08839514.2015.1016391. S2CID 44624424.
- Bianchi, Leonora; Marco Dorigo; Luca Maria Gambardella; Walter J. Gutjahr (2009). "A survey on metaheuristics for stochastic combinatorial optimization" (PDF). Natural Computing. 8 (2): 239–287. doi:10.1007/s11047-008-9098-4. S2CID 9141490.
Blum, C.; Roli, A. (2003). "Metaheuristics in combinatorial optimization: Overview and conceptual comparison". 35 (3). ACM Computing Surveys: 268–308. Cite journal requires
- Goldberg, D.E. (1989). Genetic Algorithms in Search, Optimization and Machine Learning. Kluwer Academic Publishers. ISBN 978-0-201-15767-3.
- Glover, F.; Kochenberger, G.A. (2003). Handbook of metaheuristics. 57. Springer, International Series in Operations Research & Management Science. ISBN 978-1-4020-7263-5.
- Talbi, E-G. (2009). Metaheuristics: from design to implementation. Wiley. ISBN 978-0-470-27858-1.
- Sörensen, Kenneth (2015). "Metaheuristics—the metaphor exposed" (PDF). International Transactions in Operational Research. 22: 3–18. CiteSeerX 10.1.1.470.3422. doi:10.1111/itor.12001. Archived from the original (PDF) on 2013-11-02.
- Classification of metaheuristics
- D, Binu (2019). "RideNN: A New Rider Optimization Algorithm-Based Neural Network for Fault Diagnosis in Analog Circuits". IEEE Transactions on Instrumentation and Measurement. 68 (1): 2–26. doi:10.1109/TIM.2018.2836058. S2CID 54459927.
- M. Dorigo, Optimization, Learning and Natural Algorithms, PhD thesis, Politecnico di Milano, Italie, 1992.
- Moscato, P. (1989). "On Evolution, Search, Optimization, Genetic Algorithms and Martial Arts: Towards Memetic Algorithms". Caltech Concurrent Computation Program (report 826).
- "Giza Pyramids Construction (GPC) algorithm". www.mathworks.com. Retrieved 2020-09-28.
- Tomoiagă B, Chindriş M, Sumper A, Sudria-Andreu A, Villafafila-Robles R. Pareto Optimal Reconfiguration of Power Distribution Systems Using a Genetic Algorithm Based on NSGA-II. Energies. 2013; 6(3):1439–1455.
- Ganesan, T.; Elamvazuthi, I.; Ku Shaari, Ku Zilati; Vasant, P. (2013-03-01). "Swarm intelligence and gravitational search algorithm for multi-objective optimization of synthesis gas production". Applied Energy. 103: 368–374. doi:10.1016/j.apenergy.2012.09.059.
- Ganesan, T.; Elamvazuthi, I.; Vasant, P. (2011-11-01). Evolutionary normal-boundary intersection (ENBI) method for multi-objective optimization of green sand mould system. 2011 IEEE International Conference on Control System, Computing and Engineering (ICCSCE). pp. 86–91. doi:10.1109/ICCSCE.2011.6190501. ISBN 978-1-4577-1642-3. S2CID 698459.
- Kirkpatrick, S.; Gelatt Jr., C.D.; Vecchi, M.P. (1983). "Optimization by Simulated Annealing". Science. 220 (4598): 671–680. Bibcode:1983Sci...220..671K. CiteSeerX 10.1.1.123.7607. doi:10.1126/science.220.4598.671. PMID 17813860. S2CID 205939.
- Holland, J.H. (1975). Adaptation in Natural and Artificial Systems. University of Michigan Press. ISBN 978-0-262-08213-6.
- Glover, Fred (1977). "Heuristics for Integer programming Using Surrogate Constraints". Decision Sciences. 8 (1): 156–166. CiteSeerX 10.1.1.302.4071. doi:10.1111/j.1540-5915.1977.tb01074.x.
- Glover, F. (1986). "Future Paths for Integer Programming and Links to Artificial Intelligence". Computers and Operations Research. 13 (5): 533–549. doi:10.1016/0305-0548(86)90048-1.
- X. S. Yang, Metaheuristic optimization, Scholarpedia, 6(8):11472 (2011).
- Glover F., (1986). Future paths for integer programming and links to artificial intelligence, Computers and Operations Research, 13, 533–549 (1986).
- Moscato, P. (2012). "Metaheuristic optimization frameworks a survey and benchmarking". Soft Comput (2012) 16, 527–561. 16 (3): 527–561. doi:10.1007/s00500-011-0754-8. hdl:11441/24597. S2CID 1497912.
- Robbins, H.; Monro, S. (1951). "A Stochastic Approximation Method" (PDF). Annals of Mathematical Statistics. 22 (3): 400–407. doi:10.1214/aoms/1177729586.
- Barricelli, N.A. (1954). "Esempi numerici di processi di evoluzione". Methodos: 45–68.
- Rastrigin, L.A. (1963). "The convergence of the random search method in the extremal control of a many parameter system". Automation and Remote Control. 24 (10): 1337–1342.
- Matyas, J. (1965). "Random optimization". Automation and Remote Control. 26 (2): 246–253.
- Nelder, J.A.; Mead, R. (1965). "A simplex method for function minimization". Computer Journal. 7 (4): 308–313. doi:10.1093/comjnl/7.4.308. S2CID 2208295.
- Rechenberg, Ingo (1965). "Cybernetic Solution Path of an Experimental Problem". Royal Aircraft Establishment, Library Translation.
- Fogel, L.; Owens, A.J.; Walsh, M.J. (1966). Artificial Intelligence through Simulated Evolution. Wiley. ISBN 978-0-471-26516-0.
- Hastings, W.K. (1970). "Monte Carlo Sampling Methods Using Markov Chains and Their Applications". Biometrika. 57 (1): 97–109. Bibcode:1970Bimka..57...97H. doi:10.1093/biomet/57.1.97. S2CID 21204149.
- Cavicchio, D.J. (1970). "Adaptive search using simulated evolution". Technical Report. University of Michigan, Computer and Communication Sciences Department. hdl:2027.42/4042.
- Kernighan, B.W.; Lin, S. (1970). "An efficient heuristic procedure for partitioning graphs". Bell System Technical Journal. 49 (2): 291–307. doi:10.1002/j.1538-7305.1970.tb01770.x.
- Mercer, R.E.; Sampson, J.R. (1978). "Adaptive search using a reproductive metaplan". Kybernetes. 7 (3): 215–228. doi:10.1108/eb005486.
- Smith, S.F. (1980). A Learning System Based on Genetic Adaptive Algorithms (PhD Thesis). University of Pittsburgh.
- Moscato, P.; Fontanari, J.F. (1990), "Stochastic versus deterministic update in simulated annealing", Physics Letters A, 146 (4): 204–208, Bibcode:1990PhLA..146..204M, doi:10.1016/0375-9601(90)90166-L
- Dueck, G.; Scheuer, T. (1990), "Threshold accepting: A general purpose optimization algorithm appearing superior to simulated annealing", Journal of Computational Physics, 90 (1): 161–175, Bibcode:1990JCoPh..90..161D, doi:10.1016/0021-9991(90)90201-B, ISSN 0021-9991
- Wolpert, D.H.; Macready, W.G. (1995). "No free lunch theorems for search". Technical Report SFI-TR-95-02-010. Santa Fe Institute. S2CID 12890367.
- Igel, Christian, Toussaint, Marc (Jun 2003). "On classes of functions for which No Free Lunch results hold". Information Processing Letters. 86 (6): 317–321. arXiv:cs/0108011. doi:10.1016/S0020-0190(03)00222-9. ISSN 0020-0190. S2CID 147624.CS1 maint: multiple names: authors list (link)
- Auger, Anne, Teytaud, Olivier (2010). "Continuous Lunches Are Free Plus the Design of Optimal Optimization Algorithms". Algorithmica. 57 (1): 121–146. CiteSeerX 10.1.1.186.6007. doi:10.1007/s00453-008-9244-5. ISSN 0178-4617. S2CID 1989533.CS1 maint: multiple names: authors list (link)
- Stefan Droste; Thomas Jansen; Ingo Wegener (2002). "Optimization with Randomized Search Heuristics – The (A)NFL Theorem, Realistic Scenarios, and Difficult Functions". Theoretical Computer Science. 287 (1): 131–144. CiteSeerX 10.1.1.35.5850. doi:10.1016/s0304-3975(02)00094-4.
- Sörensen, Kenneth; Sevaux, Marc; Glover, Fred (2017-01-16). "A History of Metaheuristics" (PDF). In Martí, Rafael; Panos, Pardalos; Resende, Mauricio (eds.). Handbook of Heuristics. Springer. ISBN 978-3-319-07123-7.