Regularized Projection Algorithms for Monotone Inverse Variational Inequalities

Abstract

Stochastic inverse variational inequalities (SIVIs) arise in applications in which equilibrium responses are observed under uncertainty, such as inverse road pricing and network equilibrium control. Existing methods typically rely on co-coercivity or strong monotonicity, while general monotone SIVIs remain less understood. We propose a regularized projection algorithm that combines Tikhonov regularization with increasing batch sizes. Under monotonicity and Lipschitz continuity, we prove almost sure boundedness of the iterates and almost sure convergence of their distance to the SIVI solution set. We further establish, to the best of our knowledge, the first explicit nonasymptotic rate of $O(T^{-1/2})$ for the expected squared residual under general monotonicity. This yields $O(\epsilon^{-2})$ iterations and $O(\epsilon^{-4-2\delta})$ stochastic oracle calls, for any $\delta>0$, to obtain an $\epsilon$-solution in expected squared residual. A deterministic variant attains the same iteration complexity using $O(\epsilon^{-2})$ exact operator evaluations. Numerical experiments illustrate the proposed methods on monotone SIVI problems.