Consensus based optimization: Difference between revisions

Revision as of 02:10, 15 February 2024

Revision as of 02:11, 15 February 2024

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=== 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>{{Citationcite arxiv |last1=Bungert |first1=Leon |title=Polarized consensus-based dynamics for optimization and sampling |date=2023-10-09 |url=http://arxiv.org/abs/2211.05238 |access-date=2024-02-05 |arxiv=2211.05238 |last2=Roith |first2=Tim |last3=Wacker |first3=Philipp}}</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).