Sara Grassi, Hui Huang, Lorenzo Pareschi, Jinniao Qiu (3/8/2021 to appear in

*Modeling and Simulation for Collective Dynamics*, IMS Lecture Note Series, World Scientific, preprint arXiv:2108.00393)

In this work we survey some recent results on the global minimization of a non-convex and possibly non-smooth high dimensional objective function by means of particle based gradient-free methods. Such problems arise in many situations of contemporary interest in machine learning and signal processing. After a brief overview of metaheuristic methods based on particle swarm optimization (PSO), we introduce a continuous formulation via second-order systems of stochastic differential equations that generalize PSO methods and provide the basis for their theoretical analysis. Subsequently, we will show how through the use of mean-field techniques it is possible to derive in the limit of large particles number the corresponding mean-field PSO description based on Vlasov-Fokker-Planck type equations.