For d>1, smooth solutions of the CBO Fokker-Planck equation are positive away from the consensus point, so the usual initial-support condition for global convergence can be dropped.
CBX: Python and Julia packages for consensus-based interacting particle methods
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We introduce CBXPy and ConsensusBasedX.jl, Python and Julia implementations of consensus-based interacting particle systems (CBX), which generalise consensus-based optimization methods (CBO) for global, derivative-free optimisation. The raison d'\^etre of our libraries is twofold: on the one hand, to offer high-performance implementations of CBX methods that the community can use directly, while on the other, providing a general interface that can accommodate and be extended to further variations of the CBX family. Python and Julia were selected as the leading high-level languages in terms of usage and performance, as well as their popularity among the scientific computing community. Both libraries have been developed with a common ethos, ensuring a similar API and core functionality, while leveraging the strengths of each language and writing idiomatic code.
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Regularity and positivity of solutions of the Consensus-Based Optimization equation: unconditional global convergence
For d>1, smooth solutions of the CBO Fokker-Planck equation are positive away from the consensus point, so the usual initial-support condition for global convergence can be dropped.