  We consider minimizing a sum of agent-specific nondifferentiable merely
convex functions over the solution set of a variational inequality (VI) problem
in that each agent is associated with a local monotone mapping. This problem
finds an application in computation of the best equilibrium in nonlinear
complementarity problems arising in transportation networks. We develop an
iteratively regularized incremental gradient method where at each iteration,
agents communicate over a cycle graph to update their solution iterates using
their local information about the objective and the mapping. The proposed
method is single-timescale in the sense that it does not involve any excessive
hard-to-project computation per iteration. We derive non-asymptotic agent-wise
convergence rates for the suboptimality of the global objective function and
infeasibility of the VI constraints measured by a suitably defined dual gap
function. The proposed method appears to be the first fully iterative scheme
equipped with iteration complexity that can address distributed optimization
problems with VI constraints over cycle graphs. Preliminary numerical
experiments for a transportation network problem and a support vector machine
model are presented.
