Solving polynomial variational inequality problems via Lagrange multiplier expressions and Moment-SOS relaxations

This paper focuses on the development of numerical methods for solving variational inequality problems (VIPs) with involved mappings and feasible sets characterized by polynomial functions. We propose a numerical algorithm for computing solutions to polynomial VIPs based on Lagrange multiplier expressions and the Moment-SOS hierarchy of semidefinite relaxations. Building upon this algorithm, we also extend to finding more or even all solutios to polynomial VIPs. This algorithm can find solutions to polynomial VIPs or determine their nonexistence within a finite number of steps, under some general assumptions. Moreover, it is demonstrated that if the VIP is represented by generic polynomial functions, a finite number of Karush–Kuhn–Tucker (KKT) points exist, and all solutions to the polynomial VIP are KKT points. The paper establishes that in such cases, the method is guaranteed to terminate within a finite number of iterations, with an upper bound for the number of KKT points determined using intersection theory. Finally, even when algorithms lack finite convergence, the paper demonstrates asymptotic convergence under specific continuity assumptions. Numerical experiments are conducted to illustrate the efficiency of the proposed methods.