Consensus-Based Optimization on Hypersurfaces: Well-Posedness and Mean-Field Limit
This work addresses optimization problems on curved surfaces, which is incremental as it extends existing consensus-based methods to hypersurfaces.
The paper tackles global optimization of nonconvex functions on compact hypersurfaces by introducing a stochastic differential model based on Consensus-Based Optimization, and it establishes the well-posedness of the model and rigorously derives its mean-field limit for large particle numbers.
We introduce a new stochastic differential model for global optimization of nonconvex functions on compact hypersurfaces. The model is inspired by the stochastic Kuramoto-Vicsek system and belongs to the class of Consensus-Based Optimization methods. In fact, particles move on the hypersurface driven by a drift towards an instantaneous consensus point, computed as a convex combination of the particle locations weighted by the cost function according to Laplace's principle. The consensus point represents an approximation to a global minimizer. The dynamics is further perturbed by a random vector field to favor exploration, whose variance is a function of the distance of the particles to the consensus point. In particular, as soon as the consensus is reached, then the stochastic component vanishes. In this paper, we study the well-posedness of the model and we derive rigorously its mean-field approximation for large particle limit.