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| In [[computer science]], '''differential evolution''' (DE) is a method that [[optimization (mathematics)|optimizes]] a problem by [[iterative method|iteratively]] trying to improve a [[candidate solution]] with regard to a given measure of quality. Such methods are commonly known as [[metaheuristic]]s as they make few or no assumptions about the problem being optimized and can search very large spaces of candidate solutions. However, metaheuristics such as DE do not guarantee an optimal solution is ever found.
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| DE is used for multidimensional real-valued [[function (mathematics)|functions]] but does not use the [[gradient]] of the problem being optimized, which means DE does not require for the optimization problem to be [[differentiable]] as is required by classic optimization methods such as [[gradient descent]] and [[quasi-newton methods]]. DE can therefore also be used on optimization problems that are not even [[:wikt:continuous|continuous]], are noisy, change over time, etc.<ref name=elediadereview/>
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| DE optimizes a problem by maintaining a population of candidate solutions and creating new candidate solutions by combining existing ones according to its simple formulae, and then keeping whichever candidate solution has the best score or fitness on the optimization problem at hand. In this way the optimization problem is treated as a black box that merely provides a measure of quality given a candidate solution and the gradient is therefore not needed.
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| DE is originally due to Storn and Price.<ref name=storn97differential/><ref name=storn96usage/> Books have been published on theoretical and practical aspects of using DE in [[parallel computing]], [[multiobjective optimization]], [[constrained optimization]], and the books also contain surveys of application areas.<ref name=price05differential/><ref name=feoktistov06differential/><ref name=chakraborty08advances/>
| | '''MathML''' |
| | :<math forcemathmode="mathml">E=mc^2</math> |
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| == Algorithm ==
| | <!--'''PNG''' (currently default in production) |
| <!-- There is no need for extra pseudo-code when this algorithm description is made as detailed as it is --> | | :<math forcemathmode="png">E=mc^2</math> |
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| A basic variant of the DE algorithm works by having a population of [[candidate solutions]] (called agents). These agents are moved around in the search-space by using simple mathematical [[formula]]e to combine the positions of existing agents from the population. If the new position of an agent is an improvement it is accepted and forms part of the population, otherwise the new position is simply discarded. The process is repeated and by doing so it is hoped, but not guaranteed, that a satisfactory solution will eventually be discovered.
| | '''source''' |
| | :<math forcemathmode="source">E=mc^2</math> --> |
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| Formally, let <math>f: \Bbb{R}^n \to \Bbb{R}</math> be the cost function which must be minimized or fitness function which must be maximized. The function takes a candidate solution as argument in the form of a [[Row vector|vector]] of [[real number]]s and produces a real number as output which indicates the fitness of the given candidate solution. The [[gradient]] of <math>f</math> is not known. The goal is to find a solution <math>m</math> for which <math>f(m) \leq f(p)</math> for all <math>p</math> in the search-space, which would mean <math>m</math> is the global minimum. Maximization can be performed by considering the function <math>h := -f</math> instead.
| | <span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples]. |
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| Let <math>\mathbf{x} \in \Bbb{R}^n</math> designate a candidate solution (agent) in the population. The basic DE algorithm can then be described as follows:
| | ==Demos== |
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| * Initialize all agents <math>\mathbf{x}</math> with random positions in the search-space.
| | Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]: |
| * Until a termination criterion is met (e.g. number of iterations performed, or adequate fitness reached), repeat the following:
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| ** For each agent <math>\mathbf{x}</math> in the population do:
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| *** Pick three agents <math>\mathbf{a},\mathbf{b}</math>, and <math>\mathbf{c}</math> from the population at random, they must be distinct from each other as well as from agent <math>\mathbf{x}</math>
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| *** Pick a random index <math>R \in \{1, \ldots, n\}</math> (<math>n</math> being the dimensionality of the problem to be optimized).
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| *** Compute the agent's potentially new position <math>\mathbf{y} = [y_1, \ldots, y_n]</math> as follows:
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| **** For each <math>i</math>, pick a uniformly distributed number <math>r_i \equiv U(0,1)</math>
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| **** If <math>r_i < \text{CR}</math> or <math>i = R</math> then set <math>y_i = a_i + F \times (b_i-c_i)</math> otherwise set <math>y_i = x_i</math>
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| **** (In essence, the new position is outcome of binary crossover of agent <math>\mathbf{x}</math> with intermediate agent <math>\mathbf{z} = \mathbf{a} + F \times (\mathbf{b}-\mathbf{c})</math>.)
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| *** If <math>f(\mathbf{y}) < f(\mathbf{x})</math> then replace the agent in the population with the improved candidate solution, that is, replace <math>\mathbf{x}</math> with <math>\mathbf{y}</math> in the population.
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| * Pick the agent from the population that has the highest fitness or lowest cost and return it as the best found candidate solution.
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| Note that <math>F \in [0,2]</math> is called the ''differential weight'' and <math>\text{CR} \in [0,1]</math> is called the ''crossover probability'', both these parameters are selectable by the practitioner along with the population size <math>\text{NP} \geq 4</math> see below.
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| == Parameter selection ==
| | * accessibility: |
| | ** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]] |
| | ** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)] |
| | ** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)] |
| | ** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)] |
| | ** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]]. |
| | ** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]]. |
| | ** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode. |
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| [[Image:DE Meta-Fitness Landscape (Sphere and Rosenbrock).JPG|thumb|Performance landscape showing how the basic DE performs in aggregate on the Sphere and Rosenbrock benchmark problems when varying the two DE parameters <math>\text{NP}</math> and <math>\text{F}</math>, and keeping fixed <math>\text{CR}</math>=0.9.]]
| | ==Test pages == |
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| The choice of DE parameters <math>F, \text{CR}</math> and <math>\text{NP}</math> can have a large impact on optimization performance. Selecting the DE parameters that yield good performance has therefore been the subject of much research. [[Rules of thumb]] for parameter selection were devised by Storn et al.<ref name=storn96usage/><ref name=price05differential/> and Liu and Lampinen.<ref name=liu02setting/> Mathematical convergence analysis regarding parameter selection was done by Zaharie.<ref name=zaharie02critical/> [[Meta-optimization]] of the DE parameters was done by Pedersen <ref name=pedersen08thesis/><ref name=pedersen10good-de/> and Zhang et al.<ref name=zhang11fitting/>
| | To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages: |
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| == Variants ==
| | *[[Inputtypes|Inputtypes (private Wikis only)]] |
| | | *[[Url2Image|Url2Image (private Wikis only)]] |
| <!-- Please add proper reference to conference-paper / journal-paper / tech report / master's or phd thesis. Please only add representative works. There must be hundreds of DE variants and Wikipedia is not the proper place to list them all. -->
| | ==Bug reporting== |
| | | If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de . |
| Variants of the DE algorithm are continually being developed in an effort to improve optimization performance. Many different schemes for performing crossover and mutation of agents are possible in the basic algorithm given above, see e.g.<ref name=storn96usage/> More advanced DE variants are also being developed with a popular research trend being to perturb or adapt the DE parameters during optimization, see e.g. Price et al.,<ref name=price05differential/> Liu and Lampinen,<ref name=liu05fuzzy/> Qin and Suganthan,<ref name=qin05selfadaptive/> Civicioglu <ref name=civici/> and Brest et al.<ref name=brest06selfadapting/>
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| == See also ==
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| <!-- Please only add optimizers conceptually related to DE. -->
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| * [[CMA-ES]] | |
| * [[Artificial bee colony algorithm]] | |
| * [[Evolution strategy]]
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| * [[Genetic algorithm]]
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| * [[Differential search algorithm]] <ref name=civici/>
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| * [[Biogeography-based optimization]]
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| ==References== | |
| {{Reflist|refs=
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| <ref name=storn97differential>
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| {{cite journal
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| |last=Storn
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| |first=R.
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| |author2=Price, K.
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| |title=Differential evolution - a simple and efficient heuristic for global optimization over continuous spaces
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| |journal=Journal of Global Optimization
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| |year=1997
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| |volume=11
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| |pages=341–359
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| |doi=10.1023/A:1008202821328
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| }}
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| </ref>
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| | |
| <ref name=storn96usage>
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| {{cite conference
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| |last=Storn
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| |first=R.
| |
| |title=On the usage of differential evolution for function optimization
| |
| |booktitle=Biennial Conference of the North American Fuzzy Information Processing Society (NAFIPS)
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| |year=1996
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| |pages=519–523
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| }}
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| </ref>
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| | |
| <ref name=price05differential>
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| {{cite book
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| |title=Differential Evolution: A Practical Approach to Global Optimization
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| |url=http://www.springer.com/computer/theoretical+computer+science/foundations+of+computations/book/978-3-540-20950-8
| |
| |last1=Price
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| |first1=K.
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| |last2=Storn
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| |first2=R.M.
| |
| |last3=Lampinen
| |
| |first3=J.A.
| |
| |year=2005
| |
| |publisher=Springer
| |
| |isbn=978-3-540-20950-8
| |
| }}
| |
| </ref>
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| | |
| <ref name=feoktistov06differential>
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| {{cite book
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| |title=Differential Evolution: In Search of Solutions
| |
| |url=http://www.springer.com/mathematics/book/978-0-387-36895-5
| |
| |last=Feoktistov
| |
| |first=V.
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| |year=2006
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| |publisher=Springer
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| |isbn=978-0-387-36895-5
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| }}
| |
| </ref>
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| | |
| <ref name=chakraborty08advances>
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| {{citation
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| |title=Advances in Differential Evolution
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| |url=http://www.springer.com/engineering/book/978-3-540-68827-3
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| |editor-last=Chakraborty
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| |editor-first=U.K.
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| |year=2008
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| |publisher=Springer
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| |isbn=978-3-540-68827-3
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| }}
| |
| </ref>
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| | |
| <ref name=liu02setting>
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| {{cite conference
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| |title=On setting the control parameter of the differential evolution method
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| |booktitle=Proceedings of the 8th International Conference on Soft Computing (MENDEL)
| |
| |last=Liu
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| |first=J.
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| |last2=Lampinen
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| |first2=J.
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| |year=2002
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| |pages=11–18
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| |location=Brno, Czech Republic
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| }}
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| </ref>
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| | |
| <ref name=zaharie02critical>
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| {{cite conference
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| |title=Critical values for the control parameters of differential evolution algorithms
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| |booktitle=Proceedings of the 8th International Conference on Soft Computing (MENDEL)
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| |last=Zaharie
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| |first=D.
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| |year=2002
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| |pages=62–67
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| |location=Brno, Czech Republic
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| }}
| |
| </ref>
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| | |
| <ref name=brest06selfadapting>
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| {{cite journal
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| |last1=Brest
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| |first1=J.
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| |last2=Greiner
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| |first2=S.
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| |last3=Boskovic
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| |first3=B.
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| |last4=Mernik
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| |first4=M.
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| |last5=Zumer
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| |first5=V.
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| |title=Self-adapting control parameters in differential evolution: a comparative study on numerical benchmark functions
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| |journal=IEEE Transactions on Evolutionary Computation
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| |year=2006
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| |volume=10
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| |issue=6
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| |pages=646–657
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| |doi=10.1109/tevc.2006.872133
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| }}
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| </ref>
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| | |
| <ref name=qin05selfadaptive>
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| {{cite conference
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| |last1=Qin
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| |first1=A.K.
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| |last2=Suganthan
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| |first2=P.N.
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| |title=Self-adaptive differential evolution algorithm for numerical optimization
| |
| |booktitle=Proceedings of the IEEE congress on evolutionary computation (CEC)
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| |year=2005
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| |pages=1785–1791
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| }}
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| </ref>
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| | |
| <ref name=liu05fuzzy>
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| {{cite journal
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| |last1=Liu
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| |first1=J.
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| |last2=Lampinen
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| |first2=J.
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| |title=A fuzzy adaptive differential evolution algorithm
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| |journal=Soft Computing
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| |year=2005
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| |volume=9
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| |issue=6
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| |pages=448–462
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| |doi=10.1007/s00500-004-0363-x
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| }}
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| </ref>
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| | |
| <ref name=pedersen08thesis>
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| {{cite book
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| |type=PhD thesis
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| |title=Tuning & Simplifying Heuristical Optimization
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| |url=http://www.hvass-labs.org/people/magnus/thesis/pedersen08thesis.pdf
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| |last=Pedersen
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| |first=M.E.H.
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| |year=2010
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| |publisher=University of Southampton, School of Engineering Sciences, Computational Engineering and Design Group
| |
| }}
| |
| </ref>
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| | |
| <ref name=pedersen10good-de>
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| {{Cite journal
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| |last=Pedersen
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| |first=M.E.H.
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| |url=http://www.hvass-labs.org/people/magnus/publications/pedersen10good-de.pdf
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| |title=Good parameters for differential evolution
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| |journal=Technical Report HL1002
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| |publisher=Hvass Laboratories
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| |year=2010
| |
| }}
| |
| </ref>
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| | |
| <ref name=zhang11fitting>
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| {{cite conference
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| |last1=Zhang
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| |first1=X.
| |
| |last2=Jiang
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| |first2=X.
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| |last3=Scott
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| |first3=P.J.
| |
| |title=A Minimax Fitting Algorithm for Ultra-Precision Aspheric Surfaces
| |
| |booktitle=The 13th International Conference on Metrology and Properties of Engineering Surfaces
| |
| |year=2011
| |
| }}
| |
| </ref>
| |
| | |
| <ref name=civici>
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| {{cite journal
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| |last=Civicioglu
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| |first=P.
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| |title=Transforming geocentric cartesian coordinates to geodetic coordinates by using differential search algorithm
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| |journal=Computers & Geosciences
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| |year=2012
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| |volume=46
| |
| |pages=229–247
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| |doi=10.1016/j.cageo.2011.12.011
| |
| }}
| |
| </ref>
| |
| | |
| <ref name=elediadereview>
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| {{Cite journal
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| |last1=Rocca
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| |first1=P.
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| |last2=Oliveri
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| |first2=G.
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| |last3=Massa
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| |first3=A.
| |
| |title=Differential Evolution as Applied to Electromagnetics
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| |journal=IEEE Antennas and Propagation Magazine
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| |year=2011
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| |volume=53
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| |issue=1
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| |pages=38–49
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| |doi=10.1109/MAP.2011.5773566
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| }}
| |
| </ref>
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| }}
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| ==External links==
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| * [http://www.icsi.berkeley.edu/~storn/code.html Storn's Homepage on DE] featuring source-code for several programming languages.
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| * [http://www.sciencedirect.com/science/article/pii/S0957417410010493 Fast DE Algorithm] A Fast Differential Evolution Algorithm using k-Nearest Neighbour Predictor.
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| * [http://www.sciencedirect.com/science/article/pii/S0957417413000857 MODE Application] Parameter Estimation of a Pressure Swing Adsorption Model for Air Separation Using Multi-objective Optimisation and Support Vector Regression Model.
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| {{Major subfields of optimization}}
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| {{DEFAULTSORT:Differential Evolution}}
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| [[Category:Optimization algorithms and methods]]
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| [[Category:Evolutionary algorithms]]
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| [[Category:Mathematical optimization]]
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| [[Category:Operations research]]
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