  Motivated by multi-user optimization problems and non-cooperative Nash games
in stochastic regimes, we consider stochastic variational inequality (SVI)
problems on matrix spaces where the variables are positive semidefinite
matrices and the mapping is merely monotone. Much of the interest in the theory
of variational inequality (VI) has focused on addressing VIs on vector
spaces.Yet, most existing methods either rely on strong assumptions, or require
a two-loop framework where at each iteration, a projection problem, i.e., a
semidefinite optimization problem needs to be solved. Motivated by this gap, we
develop a stochastic mirror descent method where we choose the distance
generating function to be defined as the quantum entropy. This method is a
single-loop first-order method in the sense that it only requires a
gradient-type of update at each iteration. The novelty of this work lies in the
convergence analysis that is carried out through employing an auxiliary
sequence of stochastic matrices. Our contribution is three-fold: (i) under this
setting and employing averaging techniques, we show that the iterate generated
by the algorithm converges to a weak solution of the SVI; (ii) moreover, we
derive a convergence rate in terms of the expected value of a suitably defined
gap function; (iii) we implement the developed method for solving a
multiple-input multiple-output multi-cell cellular wireless network composed of
seven hexagonal cells and present the numerical experiments supporting the
convergence of the proposed method.
