Consensus based optimization: Difference between revisions

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'''Consensus-based optimization (CBO)'''<ref name=":0">{{Cite journal |last1=Pinnau |first1=René |last2=Totzeck |first2=Claudia |last3=Tse |first3=Oliver |last4=Martin |first4=Stephan |date=January 2017 |title=A consensus-based model for global optimization and its mean-field limit |url=https://www.worldscientific.com/doi/abs/10.1142/S0218202517400061 |journal=Mathematical Models and Methods in Applied Sciences |language=en |volume=27 |issue=1 |pages=183–204 |doi=10.1142/S0218202517400061 |arxiv=1604.05648 |s2cid=119296432 |issn=0218-2025}}</ref> is a multi-agent [[derivative-free optimization]] method, designed to obtain solutions for global optimization problems of the form <math display="block">\min_{x\in \cal{X}} f(x),</math>

[[File:CBORastrigin.gif|thumb|Behavior of CBO on the [[Rastrigin function]]. '''Blue:''' Particles, '''Pink:''' drift vectors and consensus point.]]

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 exporationexploration 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 |url-access=subscription }}</ref> [[particle swarm optimization]] or [[Simulated annealing]].

== Algorithm ==

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</math> determines the communication radius of particles. This choice corresponds to a local convex regularization of the objective function <math>f

</math>.

* '''[[Mean-shift algorithm]]''':<ref>{{Cite journal |last1=Fukunaga |first1=K. |last2=Hostetler |first2=L. |date=January 1975 |title=The estimation of the gradient of a density function, with applications in pattern recognition |url=https://ieeexplore.ieee.org/document/1055330 |journal=IEEE Transactions on Information Theory |language=en |volume=21 |issue=1 |pages=32–40 |doi=10.1109/TIT.1975.1055330 |issn=0018-9448|url-access=subscription }}</ref> Employing polarized CBO for a constant objective function <math>f

</math>, together with no noise (i.e. <math>\sigma = 0

</math>) and an Euler–Maruyama discretization with step size <math>dt=1