A Python package for consensus-based particle dynamics, focusing on optimization and sampling.
Minimizing a function using CBXPy can be done as follows:
from cbx.dynamics import CBO # import the CBO class
f = lambda x: x[0]**2 + x[1]**2 # define the function to minimize
x = CBO(f, d=2).optimize() # run the optimizationA documentation together with more examples and usage instructions is available at https://pdips.github.io/CBXpy.
Currently CBXPy can only be installed from PyPI with pip.
pip install cbxOriginally designed for optimization problems of the form
the scheme was introduced as CBO (Consensus-Based Optimization). Given an ensemble of points
where
with a parameter
Among others, CBXPy currently implements
- CBO (Consensus-Based Optimization) [1]
- CBS (Consensus-Based Sampling) [2]
- CBO with memory [3]
- Batching schemes [4]
- Polarized CBO [5]
- Mirror CBO [6]
- Adamized CBO [7]
- Constrained CBO methods, including
[1] A consensus-based model for global optimization and its mean-field limit, Pinnau, R., Totzeck, C., Tse, O. and Martin, S., Mathematical Models and Methods in Applied Sciences 2017
[2] Consensus-based sampling, Carrillo, J.A., Hoffmann, F., Stuart, A.M., and Vaes, U., Studies in Applied Mathematics 2022
[3] Leveraging Memory Effects and Gradient Information in Consensus-Based Optimization: On Global Convergence in Mean-Field Law, Riedl, K., 2022
[4] A consensus-based global optimization method for high dimensional machine learning problems, Carrillo, J.A., Jin, S., Li, L. and Zhu, Y., ESAIM: Control, Optimisation and Calculus of Variations 2021
[5] Bungert, L., Roith, T., & Wacker, P. (2024). Polarized consensus-based dynamics for optimization and sampling. Mathematical Programming, 1-31.
[6] Bungert, L., Hoffmann, F., Kim, D. Y., & Roith, T. (2025). MirrorCBO: A consensus-based optimization method in the spirit of mirror descent. arXiv preprint arXiv:2501.12189.
[7] Chen, J., Jin, S., & Lyu, L. (2020). A consensus-based global optimization method with adaptive momentum estimation. arXiv preprint arXiv:2012.04827.
[8] Carrillo, J. A., Jin, S., Zhang, H., & Zhu, Y. (2024). An interacting particle consensus method for constrained global optimization. arXiv preprint arXiv:2405.00891.
[9] Borghi, G., Herty, M., & Pareschi, L. (2023). Constrained consensus-based optimization. SIAM Journal on Optimization, 33(1), 211-236.
[10] Fornasier, M., Huang, H., Pareschi, L., & Sünnen, P. (2020). Consensus-based optimization on hypersurfaces: Well-posedness and mean-field limit. Mathematical Models and Methods in Applied Sciences, 30(14), 2725-2751.