  This paper considers a variational inequality (VI) problem arising from a
game among multiple agents, where each agent aims to minimize its own cost
function subject to its constrained set represented as the intersection of a
(possibly infinite) number of convex functional level sets. A direct
projection-based approach or Lagrangian-based techniques for such a problem can
be computationally expensive if not impossible to implement. To deal with the
problem, we consider randomized methods that avoid the projection step on the
whole constraint set by employing random feasibility updates. In particular, we
propose and analyze such random methods for solving VIs based on the projection
method, Korpelevich method, and Popov method. We establish the almost sure
convergence of the methods and, also, provide their convergence rate
guarantees. We illustrate the performance of the methods in simulations for
two-agent games.
