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

A First-order Method for Monotone Stochastic Variational Inequalities on Semidefinite Matrix Spaces