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Citation bot (talk | contribs) Removed URL that duplicated identifier. Removed access-date with no URL. | Use this bot. Report bugs. | Suggested by Dominic3203 | Category:AfC submissions on science, mathematics and engineering | #UCB_Category 343/754 |
m section heads could be a bit more concise but otherwise I see no reason not to accept |
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where <math>f:\mathcal{X}\to\R</math> denotes the objective function acting on the state space <math>\cal{X}</math>, which is assumed to be a [[normed vector space]]. The function <math>f</math> can potentially be nonconvex and nonsmooth. The algorithm employs particles or agents to explore the state space, which communicate with each other to update their positions. Their dynamics follows the paradigm of [[Metaheuristic|metaheuristics]], which blend exporation with exploitation. In this sense, CBO is comparable to [[Ant colony optimization algorithms|ant colony optimization]], wind driven optimization<ref>{{Cite journal |title=The Wind Driven Optimization Technique and its Application in Electromagnetics |date=2013 |url=https://ieeexplore.ieee.org/document/6407788 |access-date=2024-02-03 |doi=10.1109/TAP.2013.2238654 |last1=Bayraktar |first1=Zikri |last2=Komurcu |first2=Muge |last3=Bossard |first3=Jeremy A. |last4=Werner |first4=Douglas H. |journal=IEEE Transactions on Antennas and Propagation |volume=61 |issue=5 |pages=2745–2757 |bibcode=2013ITAP...61.2745B |s2cid=38181295 }}</ref>, [[particle swarm optimization]] or [[Simulated annealing]].
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** In the limit <math>\alpha\to\infty</math> the consensus point corresponds to the particle with the best objective value, completely ignoring the position of other points in the ensemble.
== Implementation notes ==
In practice, the SDE is discretized via the [[Euler–Maruyama method]] such that the following explicit update formula for the ensemble <math>x = (x^1,\dots,x^N)</math> is obtained,<math display="block">x^i \gets x^i-\lambda\, (x^i-c_\alpha(x))\,dt + \sigma D(x^i-c_{\alpha}(x))\, B^i.</math>If one can employ an efficient implementation of the [[LogSumExp]] functions, this can be beneficial for numerical stability of the consensus point computation. We refer to existing implementation in [[Python (programming language)|Python]] [https://pdips.github.io/CBXpy/] and [[Julia (programming language)|Julia]] [https://github.com/PdIPS/CBX.jl].
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