  We propose a zero-order optimization method for sequential min-max problems
based on two populations of interacting particles. The systems are coupled so
that one population aims to solve the inner maximization problem, while the
other aims to solve the outer minimization problem. The dynamics are
characterized by a consensus-type interaction with additional stochasticity to
promote exploration of the objective landscape. Without relying on convexity or
concavity assumptions, we establish theoretical convergence guarantees of the
algorithm via a suitable mean-field approximation of the particle systems.
Numerical experiments illustrate the validity of the proposed approach. In
particular, the algorithm is able to identify a global min-max solution, in
contrast to gradient-based methods, which typically converge to possibly
suboptimal stationary points.
