An Incremental Gradient Method for Optimization Problems with Variational Inequality Constraints

TL;DR

This paper introduces an incremental gradient method for solving distributed optimization problems with variational inequality constraints, applicable to transportation networks and machine learning, with proven convergence rates and preliminary numerical validation.

Contribution

It presents the first fully iterative, single-timescale scheme with complexity guarantees for distributed VI-constrained optimization over cycle graphs.

Findings

Convergence rates established for the proposed method.

Applicable to transportation networks and SVM models.

Numerical experiments demonstrate effectiveness.

Abstract

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.