Particle swarm optimization: Difference between revisions

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{{Short description|Iterative simulation method}}

[[File:ParticleSwarmArrowsAnimation.gif|thumb|A particle swarm searching for the [[global minimum]] of a function]]

In [[computational science]], '''particle swarm optimization''' ('''PSO''')<ref name=bonyadi16survey/> is a computational method that [[Mathematical optimization|optimizes]] a problem by [[iterative method|iteratively]] trying to improve a [[candidate solution]] with regard to a given measure of quality. It solves a problem by having a population of candidate solutions, here dubbed [[Point particle|particle]]s, and moving these particles around in the [[Optimization (mathematics)#Concepts and notation|search-space]] according to simple [[formula|mathematical formulaformulae]] over the particle's [[Position (vector)|position]] and [[velocity]]. Each particle's movement is influenced by its local best known position, but is also guided toward the best known positions in the search-space, which are updated as better positions are found by other particles. This is expected to move the swarm toward the best solutions.

{{Evolutionary algorithms}}

In [[computational science]], '''particle swarm optimization''' ('''PSO''')<ref name=bonyadi16survey/> is a computational method that [[Mathematical optimization|optimizes]] a problem by [[iterative method|iteratively]] trying to improve a [[candidate solution]] with regard to a given measure of quality. It solves a problem by having a population of candidate solutions, here dubbed [[Point particle|particle]]s, and moving these particles around in the [[Optimization (mathematics)#Concepts and notation|search-space]] according to simple [[formula|mathematical formula]] over the particle's [[Position (vector)|position]] and [[velocity]]. Each particle's movement is influenced by its local best known position, but is also guided toward the best known positions in the search-space, which are updated as better positions are found by other particles. This is expected to move the swarm toward the best solutions.

PSO is originally attributed to [[James Kennedy (social psychologist)|Kennedy]], [[Russell C. Eberhart|Eberhart]] and [[Yuhui Shi|Shi]]<ref name=kennedy95particle/><ref name=shi98modified/> and was first intended for [[computer simulation|simulating]] [[social behaviour]],<ref name=kennedy97particle/> as a stylized representation of the movement of organisms in a bird [[Flocking (behavior)|flock]] or [[fish school]]. The algorithm was simplified and it was observed to be performing optimization. The book by Kennedy and Eberhart<ref name=kennedy01swarm/> describes many philosophical aspects of PSO and [[swarm intelligence]]. An extensive survey of PSO applications is made by [[Riccardo Poli|Poli]].<ref name=poli07analysis/><ref name=poli08analysis/> RecentlyIn 2017, a comprehensive review on theoretical and experimental works on PSO has been published by Bonyadi and Michalewicz.<ref name=bonyadi16survey/>

PSO is a [[metaheuristic]] as it makes few or no assumptions about the problem being optimized and can search very large spaces of candidate solutions. Also, PSO does not use the [[gradient]] of the problem being optimized, which means PSO does not require that the optimization problem be [[Differentiable function|differentiable]] as is required by classic optimization methods such as [[gradient descent]] and [[quasi-newton methods]]. However, metaheuristics such as PSO do not guarantee an optimal solution is ever found.

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Formally, let ''f'':&nbsp;ℝ^''n''&nbsp;→ ℝ be the cost function which must be minimized. The function takes a candidate solution as an argument in the form of a [[Row vector|vector]] of [[real number]]s and produces a real number as output which indicates the objective function value of the given candidate solution. The [[gradient]] of ''f'' is not known. The goal is to find a solution '''a''' for which ''f''('''a''')&nbsp;≤&nbsp;''f''('''b''') for all '''b''' in the search-space, which would mean '''a''' is the global minimum.

Let ''S'' be the number of particles in the swarm, each having a position '''x'''_i&nbsp;∈ ℝ^''n'' in the search-space and a velocity '''v'''_i&nbsp;∈ ℝ^''n''. Let '''p'''_i be the best known position of particle ''i'' and let '''g''' be the best known position of the entire swarm. A basic PSO algorithm to minimize the cost function is then:<ref name=clerc12spso/>

<!-- Please see discussion page why this particular PSO variant was chosen. -->

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Initialize the particle's position with a [[Uniform distribution (continuous)|uniformly distributed]] random vector: '''x'''_i&nbsp;~&nbsp;''U''('''b_lo''',&nbsp;'''b_up''')

Initialize the particle's best known position to its initial position: '''p'''_i&nbsp;←&nbsp;'''x'''_i

'''if''' ''f''('''p'''_i) >< ''f''('''g''') '''then'''

update the swarm's best known position: '''g'''&nbsp;←&nbsp;'''p'''_i

Initialize the particle's velocity: '''v'''_i&nbsp;~&nbsp;''U''(-|'''b_up'''-'''b_lo'''|,&nbsp;|'''b_up'''-'''b_lo'''|)

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Update the particle's velocity: '''v'''_i,d&nbsp;←&nbsp;w '''v'''_i,d + φ_p ''r''_p ('''p'''_i,d-'''x'''_i,d) + φ_g ''r''_g ('''g'''_d-'''x'''_i,d)

Update the particle's position: '''x'''_i&nbsp;←&nbsp;'''x'''_i + '''v'''_i

'''if''' ''f''('''x'''_i) >< ''f''('''p'''_i) '''then'''

Update the particle's best known position: '''p'''_i&nbsp;←&nbsp;'''x'''_i

'''if''' ''f''('''p'''_i) >< ''f''('''g''') '''then'''

Update the swarm's best known position: '''g'''&nbsp;←&nbsp;'''p'''_i

The values '''b_lo''' and '''b_up''' represent the lower and upper boundaries of the search-space respectively. The w parameter is the inertia weight. The parameters φ_p and φ_g are often called cognitive coefficient and social coefficient.

The termination criterion can be the number of iterations performed, or a solution where the adequate objective function value is found.<ref name="bratton2007" /> The parameters w, φ_p, and φ_g are selected by the practitioner and control the behaviour and efficacy of the PSO method ([[#Parameter selection|below]]).

== Parameter selection ==

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==Neighbourhoods and topologies==

The topology of the swarm defines the subset of particles with which each particle can exchange information.<ref name=kennedy2002population/> The basic version of the algorithm uses the global topology as the swarm communication structure.<ref name=bratton2007/> This topology allows all particles to communicate with all the other particles, thus the whole swarm share the same best position '''g''' from a single particle. However, this approach might lead the swarm to be trapped into a local minimum,<ref>Mendes, R. (2004). [https://pdfs.semanticscholar.org/d224/80b09d1f0759fb20e0fb0bd2de205457c8bc.pdf Population Topologies and Their Influence in Particle Swarm Performance]{{Dead link|date=August 2025 |bot=InternetArchiveBot |fix-attempted=yes }} (PhD thesis). Universidade do Minho.</ref> thus different topologies have been used to control the flow of information among particles. For instance, in local topologies, particles only share information with a subset of particles.<ref name=bratton2007/> This subset can be a geometrical one<ref>Suganthan, Ponnuthurai N. "[https://ieeexplore.ieee.org/abstract/document/785514/ Particle swarm optimiser with neighbourhood operator]." Evolutionary Computation, 1999. CEC 99. Proceedings of the 1999 Congress on. Vol. 3. IEEE, 1999.</ref> – for example "the ''m'' nearest particles" – or, more often, a social one, i.e. a set of particles that is not depending on any distance. In such cases, the PSO variant is said to be local best (vs global best for the basic PSO).

A commonly used swarm topology is the ring, in which each particle has just two neighbours, but there are many others.<ref name=bratton2007/> The topology is not necessarily static. In fact, since the topology is related to the diversity of communication of the particles,<ref name=oliveira2016communication/> some efforts have been done to create adaptive topologies (SPSO,<ref>SPSO [http://www.particleswarm.info Particle Swarm Central]</ref> APSO,<ref> Almasi, O. N. and Khooban, M. H. (2017). A parsimonious SVM model selection criterion for classification of real-world data sets via an adaptive population-based algorithm. Neural Computing and Applications, 1-9. [https://link.springer.com/article/10.1007/s00521-017-2930-y https://doi.org/10.1007/s00521-017-2930-y]</ref> stochastic star,<ref>Miranda, V., Keko, H. and Duque, Á. J. (2008). [https://repositorio.inesctec.pt/bitstream/123456789/1561/1/PS-05818.pdf Stochastic Star Communication Topology in Evolutionary Particle Swarms (EPSO)]. International Journal of Computational Intelligence Research (IJCIR), Volume 4, Number 2, pp. 105-116</ref> TRIBES,<ref>Clerc, M. (2006). Particle Swarm Optimization. ISTE (International Scientific and Technical Encyclopedia), 2006</ref> Cyber Swarm,<ref>Yin, P., Glover, F., Laguna, M., & Zhu, J. (2011). [http://leeds-faculty.colorado.edu/glover/fred%20pubs/428%20-%20A_complementary_cyber_swarm_algorithm_pub%20version%20w%20pen%20et%20al.pdf A Complementary Cyber Swarm Algorithm]. International Journal of Swarm Intelligence Research (IJSIR), 2(2), 22-41</ref> and C-PSO<ref name=elshamy07sis/>).

By using the ring topology, PSO can attain generation-level parallelism, significantly enhancing the evolutionary speed.<ref>{{cite journal |last1=Jian-Yu |first1=Li |title=Generation-Level Parallelism for Evolutionary Computation: A Pipeline-Based Parallel Particle Swarm Optimization |journal=IEEE Transactions on Cybernetics |date=2021 |volume=51 |issue=10 |pages=4848–4859 |doi=10.1109/TCYB.2020.3028070 |pmid=33147159 |bibcode=2021ITCyb..51.4848L }}</ref>

== Inner workings ==

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* Convergence to a local optimum where all personal bests '''p''' or, alternatively, the swarm's best known position '''g''', approaches a local optimum of the problem, regardless of how the swarm behaves.

Convergence of the sequence of solutions has been investigated for PSO.<ref name=bergh01thesis/><ref name=clerc02explosion/><ref name=trelea03particle/> These analyses have resulted in guidelines for selecting PSO parameters that are believed to cause convergence to a point and prevent divergence of the swarm's particles (particles do not move unboundedly and will converge to somewhere). However, the analyses were criticized by Pedersen<ref name=pedersen08simplifying/> for being oversimplified as they assume the swarm has only one particle, that it does not use stochastic variables and that the points of attraction, that is, the particle's best known position '''p''' and the swarm's best known position '''g''', remain constant throughout the optimization process. However, it was shown<ref>{{cite journalbook|last1=Cleghorn|first1=Christopher W|title=Swarm Intelligence |chapter=Particle Swarm Convergence: Standardized Analysis and Topological Influence|journal=Swarm Intelligence Conference|volume=8667|pages=134–145|date=2014|doi=10.1007/978-3-319-09952-1_12|series=Lecture Notes in Computer Science|isbn=978-3-319-09951-4}}</ref> that these simplifications do not affect the boundaries found by these studies for parameter where the swarm is convergent. Considerable effort has been made in recent years to weaken the modellingmodeling assumption utilized during the stability analysis of PSO,<ref name=Liu2015/> with the most recent generalized result applying to numerous PSO variants and utilized what was shown to be the minimal necessary modeling assumptions.<ref name=Cleghorn2018/>

Convergence to a local optimum has been analyzed for PSO in<ref>{{cite journal|last1=Van den Bergh|first1=F|title=A convergence proof for the particle swarm optimiseroptimizer |journal=Fundamenta Informaticae|url=https://repository.up.ac.za/bitstream/handle/2263/17262/VanDenBergh_Convergence(2010).pdf?sequence=1}}</ref> and.<ref name=Bonyadi2014/> It has been proven that PSO needs some modification to guarantee finding a local optimum.

This means that determining the convergence capabilities of different PSO algorithms and parameters still depends on [[empirical]] results. One attempt at addressing this issue is the development of an "orthogonal learning" strategy for an improved use of the information already existing in the relationship between '''p''' and '''g''', so as to form a leading converging exemplar and to be effective with any PSO topology. The aims are to improve the performance of PSO overall, including faster global convergence, higher solution quality, and stronger robustness.<ref name=zhan10OLPSO/> However, such studies do not provide theoretical evidence to actually prove their claims.

=== Adaptive mechanisms ===

Without the need for a trade-off between convergence ('exploitation') and divergence ('exploration'), an adaptive mechanism can be introduced. Adaptive particle swarm optimization (APSO) <ref name=zhan09adaptive/> features better search efficiency than standard PSO. APSO can perform global search over the entire search space with a higher convergence speed. It enables automatic control of the inertia weight, acceleration coefficients, and other algorithmic parameters at the run time, thereby improving the search effectiveness and efficiency at the same time. Also, APSO can act on the globally best particle to jump out of the likely local optima. However, APSO will introduce new algorithm parameters, it does not introduce additional design or implementation complexity nonetheless.

Besides, through the utilization of a scale-adaptive fitness evaluation mechanism, PSO can efficiently address computationally expensive optimization problems.<ref>{{cite journal |last1=Wang |first1=Ye-Qun |last2=Li |first2=Jian-Yu |last3=Chen |first3=Chun-Hua |last4=Zhang |first4=Jun |last5=Zhan |first5=Zhi-Hui |title=Scale adaptive fitness evaluation-based particle swarm optimisation for hyperparameter and architecture optimisation in neural networks and deep learning |journal=CAAI Transactions on Intelligence Technology |date=September 2023 |volume=8 |issue=3 |page=849-862 |doi=10.1049/cit2.12106 |doi-access=free }}</ref>

== Variants ==

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A series of standard implementations have been created by leading researchers, "intended for use both as a baseline for performance testing of improvements to the technique, as well as to represent PSO to the wider optimization community. Having a well-known, strictly-defined standard algorithm provides a valuable point of comparison which can be used throughout the field of research to better test new advances."<ref name=bratton2007/> The latest is Standard PSO 2011 (SPSO-2011).<ref name=Zambrano-Bigiarini2013/>

In addition, some PSO variants have been developed to solve large-scale global optimization (LSGO) problems with more than 1000 dimensions. Representative variants include competitive swarm optimizer (CSO) and level-based learning swarm optimizer (LLSO).<ref name=llso/> Recently, PSO has also been extended to solve multi-agent consensus-based distributed optimization problems, e.g., multi-agent consensus-based PSO with adaptive internal and external learning (MASOIE),<ref name=masoie/> etc.

=== Hybridization ===

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=== Alleviate premature convergence===

Another research trend is to try andto alleviate premature convergence (that is, optimization stagnation), e.g. by reversing or perturbing the movement of the PSO particles,<ref name=evers09thesis/><ref name=lovbjerg02extending/><ref name=xinchao10perturbed/><ref name=xzy02dpso/> another approach to deal with premature convergence is the use of multiple swarms<ref>{{cite journal | last1 = Cheung | first1 = N. J. | last2 = Ding | first2 = X.-M. | last3 = Shen | first3 = H.-B. | year = 2013 | title = OptiFel: A Convergent Heterogeneous Particle Sarm Optimization Algorithm for Takagi-Sugeno Fuzzy Modeling | journal = IEEE Transactions on Fuzzy Systems | volume = 22| issue = 4| pages = 919–933 | doi = 10.1109/TFUZZ.2013.2278972 | s2cid = 27974467 }}</ref> ([[multi-swarm optimization]]). The multi-swarm approach can also be used to implement multi-objective optimization.<ref name=nobile2012 /> Finally, there are developments in adapting the behavioural parameters of PSO during optimization.<ref name=zhan09adaptive/><ref name=nobile2017/>

=== Simplifications ===

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Another argument in favour of simplifying PSO is that [[metaheuristic]]s can only have their efficacy demonstrated [[empirical]]ly by doing computational experiments on a finite number of optimization problems. This means a metaheuristic such as PSO cannot be [[Program correctness|proven correct]] and this increases the risk of making errors in its description and implementation. A good example of this<ref name=tu04robust/> presented a promising variant of a [[genetic algorithm]] (another popular metaheuristic) but it was later found to be defective as it was strongly biased in its optimization search towards similar values for different dimensions in the search space, which happened to be the optimum of the benchmark problems considered. This bias was because of a programming error, and has now been fixed.<ref name=tu04corrections/>

==== Bare Bones PSO ====

Initialization of velocities may require extra inputs. The Bare Bones PSO variant<ref>{{Cite journalbook|last=Kennedy|first=James|date=2003|title=Bare Bones Particle Swarms|journal=Proceedings of the 2003 IEEE Swarm Intelligence Symposium. SIS'03 (Cat. No.03EX706) |chapter=Bare bones particle swarms |date=2003|pages=80–87|doi=10.1109/SIS.2003.1202251|isbn=0-7803-7914-4|s2cid=37185749}}</ref> has been proposed in 2003 by James Kennedy, and does not need to use velocity at all.

In this variant of PSO one dispenses with the velocity of the particles and instead updates the positions of the particles using the following simple rule,

:<math>

\vec x_i = G\left(\frac{\vec p_i+\vec g}{2},||\vec p_i-\vec g||\right) \,,

</math>

where <math>\vec x_i</math>, <math>\vec p_i</math> are the position and the best position of the particle <math>i</math>; <math>\vec g</math> is the global best position; <math>G(\vec x,\sigma)</math> is the [[normal distribution]] with the mean <math>\vec x</math> and standard deviation <math>\sigma</math>; and where <math>||\dots||</math> signifies the norm of a vector.

==== Accelerated Particle Swarm Optimization ====

Another simpler variant is the accelerated particle swarm optimization (APSO),<ref>X. S. Yang, S. Deb and S. Fong, [https://arxiv.org/abs/1203.6577 Accelerated particle swarm optimization and support vector machine for business optimization and applications], NDT 2011, Springer CCIS 136, pp. 53-66 (2011).</ref> which also does not need to use velocity and can speed up the convergence in many applications. A simple demo code of APSO is available.<ref>{{Cite web | url=http://www.mathworks.com/matlabcentral/fileexchange/?term=APSO | title=Search Results: APSO - File Exchange - MATLAB Central}}</ref>

In this variant of PSO one dispences with both the particle's velocity and the particle's best position. The particle position is updated according to the following rule,

:<math>

\vec x_i \leftarrow (1-\beta)\vec x_i + \beta \vec g + \alpha L \vec u \,,

</math>

where <math>\vec u</math> is a random uniformly distributed vector, <math>L</math> is the typical length of the problem at hand, and <math>\beta\sim 0.1-0.7</math> and <math>\alpha\sim 0.1-0.5</math> are the parameters of the method. As a refinement of the method one can decrease <math>\alpha</math> with each iteration, <math>\alpha_n=\alpha_0\gamma^n</math>, where <math>n</math> is the number of the iteration and <math>0 < \gamma < 1</math> is the decrease control parameter.

===Multi-objective optimization===

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===Binary, discrete, and combinatorial===

As the PSO equations given above work on real numbers, a commonly used method to solve discrete problems is to map the discrete search space to a continuous ___domain, to apply a classical PSO, and then to demap the result. Such a mapping can be very simple (for example by just using rounded values) or more sophisticated.<ref>Roy, R., Dehuri, S., & Cho, S. B. (2012). [http://sclab.yonsei.ac.kr/publications/Papers/IJ/A%20Novel%20Particle%20Swarm%20Optimization%20Algorithm%20for%20Multi-Objective%20Combinatorial%20Optimization%20Problem.pdf A Novel Particle Swarm Optimization Algorithm for Multi-Objective Combinatorial Optimization Problem] {{Webarchive|url=https://web.archive.org/web/20220120210030/http://sclab.yonsei.ac.kr/publications/Papers/IJ/A%20Novel%20Particle%20Swarm%20Optimization%20Algorithm%20for%20Multi-Objective%20Combinatorial%20Optimization%20Problem.pdf |date=2022-01-20 }}. 'International Journal of Applied Metaheuristic Computing (IJAMC)', 2(4), 41-57</ref>

However, it can be noted that the equations of movement make use of operators that perform four actions:

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* [[Fish School Search]]

* [[Dispersive flies optimisation]]

* [[Consensus based optimization]]

== References ==

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<ref name="xinchao10perturbed">{{cite journal | title=A perturbed particle swarm algorithm for numerical optimization | last=Xinchao | first=Z. | journal=Applied Soft Computing | year=2010 | volume=10 | issue=1 | pages=119–124 | doi=10.1016/j.asoc.2009.06.010}}</ref>

<ref name="zhan09adaptive">{{cite journal | first1=Z-H. | last1=Zhan | title=Adaptive Particle Swarm Optimization | last2=Zhang | first2=J. | last3=Li | first3=Y | last4=Chung | first4=H.S-H. | journal=IEEE Transactions on Systems, Man, and Cybernetics | year=2009 | volume=39 | issue=6 | pages=1362–1381 | doi=10.1109/TSMCB.2009.2015956 | pmid=19362911 | bibcode=2009ITSMB..39.1362Z | s2cid=11191625 | url=http://eprints.gla.ac.uk/7645/1/7645.pdf }}</ref>

<ref name="zhan10OLPSO">{{cite journal | first1=Z-H. | last1=Zhan | title=Orthogonal Learning Particle Swarm Optimization | last2=Zhang | first2=J. | last3=Li | first3=Y | last4=Shi | first4=Y-H. | journal=IEEE Transactions on Evolutionary Computation | year=2011 | volume=15 | issue=6 | pages=832–847| doi=10.1109/TEVC.2010.2052054 | bibcode=2011ITEC...15..832Z | url=http://eprints.gla.ac.uk/44801/1/44801.pdf }}</ref>

<ref name="Bonyadi2014">{{cite journal | first1=Mohammad reza. | last1=Bonyadi | title=A locally convergent rotationally invariant particle swarm optimization algorithm | last2=Michalewicz | first2=Z. | journal=Swarm Intelligence | year=2014 | volume=8 | issue=3 | pages=159–198 | doi=10.1007/s11721-014-0095-1| s2cid=2261683 | url=https://espace.library.uq.edu.au/view/UQ:396054/ERAUQ396054.pdf }}</ref>

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<ref name="trelea03particle">{{cite journal | title=The Particle Swarm Optimization Algorithm: convergence analysis and parameter selection | last=Trelea | first=I.C. | journal=Information Processing Letters | year=2003 | volume=85 | issue=6 | pages=317–325 | doi=10.1016/S0020-0190(02)00447-7}}</ref>

<ref name="clerc02explosion">{{cite journal | first1=M. | last1=Clerc | title=The particle swarm - explosion, stability, and convergence in a multidimensional complex space | last2=Kennedy | first2=J. | journal=IEEE Transactions on Evolutionary Computation | year=2002 | volume=6 | issue=1 | pages=58–73 | doi=10.1109/4235.985692| bibcode=2002ITEC....6...58C | citeseerx=10.1.1.460.6608 }}</ref>

<ref name="xzy02dpso">Xie, Xiao-Feng; Zhang, Wen-Jun; Yang, Zhi-Lian (2002). [http://www.wiomax.com/team/xie/paper/CEC02.pdf A dissipative particle swarm optimization]. ''Congress on Evolutionary Computation'' (CEC), Honolulu, HI, USA: 1456-1461.</ref>

<ref name="bratton08simplified">{{cite journal | first1=D. | last1=Bratton | url=http://downloads.hindawi.com/archive/2008/654184.pdf | title=A Simplified Recombinant PSO | last2=Blackwell | first2=T. | journal=Journal of Artificial Evolution and Applications | volume=2008 | pages=1–10 | year=2008| article-number=654184 | doi=10.1155/2008/654184 | doi-access=free }}</ref>

<ref name="kennedy01swarm">{{cite book | first1=J. | last1=Kennedy | title=Swarm Intelligence | publisher=Morgan Kaufmann | last2=Eberhart | first2=R.C. | year=2001 | isbn=978-1-55860-595-4}}</ref>

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<ref name="poli07analysis">{{cite journal | url=http://cswww.essex.ac.uk/technical-reports/2007/tr-csm469.pdf | title=An analysis of publications on particle swarm optimisation applications | last=Poli | first=R. | journal=Technical Report CSM-469 | year=2007 | access-date=2010-05-03 | archive-url=https://web.archive.org/web/20110716231935/http://cswww.essex.ac.uk/technical-reports/2007/tr-csm469.pdf | archive-date=2011-07-16 | url-status=dead }}</ref>

<ref name="poli08analysis">{{cite journal | url=http://downloads.hindawi.com/archive/2008/685175.pdf | title=Analysis of the publications on the applications of particle swarm optimisation | last=Poli | first=R. | journal=Journal of Artificial Evolution and Applications | year=2008 | volume=2008 | pages=1–10 | article-number=685175 | doi=10.1155/2008/685175| doi-access=free }}</ref>

<ref name="evers09thesis">{{cite book | url=http://www.georgeevers.org/publications.htm | title=An Automatic Regrouping Mechanism to Deal with Stagnation in Particle Swarm Optimization | publisher=The University of Texas - Pan American, Department of Electrical Engineering | last=Evers | first=G. | year=2009 | format=Master's thesis | access-date=2010-05-05 | archive-date=2011-05-18 | archive-url=https://web.archive.org/web/20110518164430/http://www.georgeevers.org/publications.htm | url-status=dead }}</ref>

<ref name="tu04robust">{{cite journal | first1=Z. | last1=Tu | title=A robust stochastic genetic algorithm (StGA) for global numerical optimization | last2=Lu | first2=Y. | journal=IEEE Transactions on Evolutionary Computation | year=2004 | volume=8 | issue=5 | pages=456–470 | doi=10.1109/TEVC.2004.831258| bibcode=2004ITEC....8..456T | s2cid=22382958 }}</ref>

<ref name="tu04corrections">{{cite journal | first1=Z. | last1=Tu | title=Corrections to "A Robust Stochastic Genetic Algorithm (StGA) for Global Numerical Optimization''" | last2=Lu | first2=Y. | journal=IEEE Transactions on Evolutionary Computation | year=2008 | volume=12 | issue=6 | pages=781 | doi=10.1109/TEVC.2008.926734| bibcode=2008ITEC...12..781T | s2cid=2864886 }}</ref>

<ref name="meissner06optimized">{{cite journal | first1=M. | last1=Meissner | title=Optimized Particle Swarm Optimization (OPSO) and its application to artificial neural network training | last2=Schmuker | first2=M. | last3=Schneider | first3=G. | journal=BMC Bioinformatics | pmc=1464136 | year=2006 | volume=7 | issue=1 | pages=125 | doi=10.1186/1471-2105-7-125 | pmid=16529661 | doi-access=free }}</ref>

<ref name="yang08nature">{{cite book | first1=X.S. | last1=Yang | title=Nature-Inspired Metaheuristic Algorithms | publisher=Luniver Press | year=2008 | isbn=978-1-905986-10-1}}</ref>

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<ref name="coellocoello02MOPSO">{{cite conference | first1=C. | last1=Coello Coello | url=http://portal.acm.org/citation.cfm?id=1252327 | title=MOPSO: A Proposal for Multiple Objective Particle Swarm Optimization | last2=Salazar Lechuga | first2=M. | book-title=Congress on Evolutionary Computation (CEC'2002) | year=2002 | pages=1051–1056}}</ref>

<ref name="Chen10SPSO">{{cite journal | first1=Wei-neng | last1=Chen | title=A novel set-based particle swarm optimization method for discrete optimization problem | last2=Zhang | first2=Jun | journal=IEEE Transactions on Evolutionary Computation | year=2010 | volume=14 | issue=2 | pages=278–300 | doi=10.1109/tevc.2009.2030331| bibcode=2010ITEC...14..278C | citeseerx=10.1.1.224.5378 | s2cid=17984726 }}</ref>

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*[http://www.particleswarm.info Particle Swarm Central] is a repository for information on PSO. Several source codes are freely available.

*[http://vimeo.com/17407010 A brief video] of particle swarms optimizing three benchmark functions.

*[http://www.mathworks.com/matlabcentral/fileexchange/11559-particle-swarm-optimization-simulation Simulation of PSO convergence in a two-dimensional space (Matlab).] {{Webarchive|url=https://web.archive.org/web/20240414152449/http://www.mathworks.com/matlabcentral/fileexchange/11559-particle-swarm-optimization-simulation |date=2024-04-14 }}

*[http://www.vocal.com/particle-swarm-optimization/ Applications] of PSO.

*{{cite journal|doi=10.1016/j.eswa.2008.10.086|title=Automatic calibration of a rainfall–runoff model using a fast and elitist multi-objective particle swarm algorithm|journal=Expert Systems with Applications|volume=36|issue=5|pages=9533–9538|year=2009|last1=Liu|first1=Yang}}

*[http://www.adaptivebox.net/research/bookmark/psocodes_link.html Links to PSO source code] {{Webarchive|url=https://web.archive.org/web/20210415022817/http://www.adaptivebox.net/research/bookmark/psocodes_link.html |date=2021-04-15 }}

{{Major subfields of optimization}}

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{{DEFAULTSORT:Particle swarm optimization}}

[[Category:MetaheuristicsNature-inspired metaheuristics]]

[[Category:EvolutionaryOptimization algorithms and methods]]

[[Category:Multi-agent systems]]