  We consider a class of hierarchical multi-agent optimization problems over
networks where agents seek to compute an approximate solution to a single-stage
stochastic mathematical program with equilibrium constraints (MPEC). MPECs
subsume several important problem classes including Stackelberg games, bilevel
programs, and traffic equilibrium problems, to name a few. Our goal in this
work is to provably resolve stochastic MPECs in distributed regimes where the
agents only have access to their local objectives and an inexact best-response
to the lower-level equilibrium problem. To this end, we devise a new method
called randomized smoothed distributed zeroth-order gradient tracking
(rs-DZGT). This is a novel gradient tracking scheme where agents employ a
zeroth-order implicit scheme to approximate their (unavailable) local
gradients. Leveraging the properties of a randomized smoothing technique, we
establish the convergence of the method and derive complexity guarantees for
computing a stationary point of an optimization problem with a smoothed
implicit global objective. We also provide preliminary numerical experiments
where we compare the performance of rs-DZGT on networks under different
settings with that of its centralized counterpart.
