Content deleted Content added
Citation bot (talk | contribs) Altered template type. Add: class, eprint, s2cid, bibcode, pages, issue, volume, journal, date, authors 1-4. Removed parameters. Some additions/deletions were parameter name changes. | Use this bot. Report bugs. | Suggested by Headbomb | #UCB_toolbar |
m Bot: link syntax and minor changes |
||
Line 5:
'''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 we assume to be a [[normed vector space]] in the following. 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]].
== The algorithm ==
Line 21:
== Convergence ==
== Notes on Implementation ==
Line 34 ⟶ 32:
=== Polarization ===
If the function <math> f</math> is multi-modal, i.e., has more than one global minimum, the standard CBO algorithm can only find one of these points. However, one can “polarize”<ref>{{cite arXiv |last1=Bungert |first1=Leon |title=Polarized consensus-based dynamics for optimization and sampling |date=2023-10-09 |eprint=2211.05238 |last2=Roith |first2=Tim |last3=Wacker |first3=Philipp|class=math.OC }}</ref> the consensus computation by introducing a kernel <math>k: \cal{X}\times\cal{X}\to[0,\infty)</math> that includes local information into the weighting. In this case, every particle has its own version of the consensus point, which is computed as
<math display="block">c_\alpha^j(x) = \frac{1}{\sum_{i=1}^N \omega_\alpha^j(x^i)} \sum_{i=1}^N x^i\ \omega_\alpha^j(x^i), \quad\text{ with }\quad \omega_\alpha^j(\,\cdot\,) = \mathrm{exp}(-\alpha f(\,\cdot\,))\, k(\cdot,x^j).
Line 44 ⟶ 42:
</math> determines the communication radius of particles. This choice corresponds to a local convex regularization of the objective function <math>f
</math>.
*
</math>, together with no noise (i.e. <math>\sigma = 0
</math>) and an Euler–Maruyama discretization with step size <math>dt=1
| |||