We address a large-scale and non-convex optimization problem, involving an aggregative term. This term can be interpreted as the sum of the contributions of N agents to some common good, with N large. We investigate a relaxation of this problem, obtained by randomization. The relaxation gap is proved to have an order O(1/N). Introducing the stochastic Frank-Wolfe (SFW) algorithm, we establish its sublinear convergence rate towards the primal problem, both in expectation and probability contexts.
In the subsequent segment, we extend this relaxation concept to encompass scenarios with an infinite number of agents, resulting in the formulation of the mean-field optimization problem (MFO). We ascertain the stability of MFO, enabling the application of the SFW algorithm to obtain solutions for the Lagrangian discretization of MFO problems.
