  Using jointly geometric and stochastic reformulations of nonconvex problems
and exploiting a Monge-Kantorovich gradient system formulation with vanishing
forces, we formally extend the simulated annealing method to a wide class of
global optimization methods. Due to an inbuilt combination of a gradient-like
strategy and particles interactions, we call them swarm gradient dynamics. As
in the original paper of Holley-Kusuoka-Stroock, the key to the existence of a
schedule ensuring convergence to a global minimizer is a functional inequality.
One of our central theoretical contributions is the proof of such an inequality
for one-dimensional compact manifolds. We conjecture the inequality to be true
in a much wider setting. We also describe a general method allowing for global
optimization and evidencing the crucial role of functional inequalities {\`a}
la {\L}ojasiewicz.
