Improved guarantees for optimal Nash equilibrium seeking and bilevel variational inequalities

Abstract

We consider a class of hierarchical variational inequality (VI) problems that subsumes VI-constrained optimization and several other important problem classes including the optimal solution selection problem, the optimal Nash equilibrium (NE) seeking problem, and the generalized NE seeking problem. Our main contributions are threefold. (i) We consider bilevel VIs with merely monotone and Lipschitz continuous mappings and devise a single-timescale iteratively regularized extragradient method (IR-EG). We improve the existing iteration complexity results for addressing both bilevel VI and VI-constrained convex optimization problems. (ii) Under the strong monotonicity of the outer level mapping, we develop a variant of IR-EG, called R-EG, and derive significantly faster guarantees than those in (i). These results appear to be new for both bilevel VIs and VI-constrained optimization. (iii) To our knowledge, complexity guarantees for computing the optimal NE in nonconvex settings do not exist. Motivated by this lacuna, we consider VI-constrained nonconvex optimization problems and devise an inexactly-projected gradient method, called IPR-EG, where the projection onto the unknown set of equilibria is performed using R-EG with prescribed adaptive termination criterion and regularization parameters. We obtain new complexity guarantees in terms of a residual map and an infeasibility metric for computing a stationary point. We validate the theoretical findings using preliminary numerical experiments for computing the best and the worst Nash equilibria