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

Swarm gradient dynamics for global optimization: the density case