  In this paper we are concerned with the global minimization of a possibly
non-smooth and non-convex objective function constrained on the unit
hypersphere by means of a multi-agent derivative-free method. The proposed
algorithm falls into the class of the recently introduced Consensus-Based
Optimization. In fact, agents move on the sphere driven by a drift towards an
instantaneous consensus point, which is computed as a convex combination of
agent locations, weighted by the cost function according to Laplaces principle,
and it represents an approximation to a global minimizer. The dynamics is
further perturbed by an anisotropic random vector field to favor exploration.
The main results of this paper are about the proof of convergence of the
numerical scheme to global minimizers provided conditions of well-preparation
of the initial datum. The proof of convergence combines a mean-field limit
result with a novel asymptotic analysis, and classical convergence results of
numerical methods for SDE. The main innovation with respect to previous work is
the introduction of an anisotropic stochastic term, which allows us to ensure
the independence of the parameters of the algorithm from the dimension and to
scale the method to work in very high dimension. We present several numerical
experiments, which show that the algorithm proposed in the present paper is
extremely versatile and outperforms previous formulations with isotropic
stochastic noise.
