  In this paper we study consensus-based optimization (CBO), a versatile,
flexible and customizable optimization method suitable for performing nonconvex
and nonsmooth global optimizations in high dimensions. CBO is a multi-particle
metaheuristic, which is effective in various applications and at the same time
amenable to theoretical analysis thanks to its minimalistic design. The
underlying dynamics, however, is flexible enough to incorporate different
mechanisms widely used in evolutionary computation and machine learning, as we
show by analyzing a variant of CBO which makes use of memory effects and
gradient information. We rigorously prove that this dynamics converges to a
global minimizer of the objective function in mean-field law for a vast class
of functions under minimal assumptions on the initialization of the method. The
proof in particular reveals how to leverage further, in some applications
advantageous, forces in the dynamics without loosing provable global
convergence. To demonstrate the benefit of the herein investigated memory
effects and gradient information in certain applications, we present numerical
evidence for the superiority of this CBO variant in applications such as
machine learning and compressed sensing, which en passant widen the scope of
applications of CBO.
