Stochastic approximation approaches for CVaR-based variational inequalities

This paper considers variational inequalities (VI) defined by the conditional value-at-risk (CVaR) of uncertain functions and provides three stochastic approximation schemes to solve them. All methods use an empirical estimate of the CVaR at each iteration. The first algorithm constrains the iterates to the feasible set using projection. To overcome the computational burden of projections, the second one handles inequality and equality constraints defining the feasible set differently. Particularly, projection onto to the affine subspace defined by the equality constraints is achieved by matrix multiplication and inequalities are handled by using penalty functions. Finally, the third algorithm discards projections altogether by introducing multiplier updates. We establish asymptotic convergence of all our schemes to any arbitrary neighborhood of the solution of the VI. A simulation example concerning a network routing game illustrates our theoretical findings.