Improved Guarantees for Optimal Nash Equilibrium Seeking and Bilevel Variational Inequalities

Sepideh Samadi, Farzad Yousefian

Abstract:

We consider a class of hierarchical variational inequality (VI) problems that subsumes VI-constrained optimization and several other problem classes, including the optimal solution
selection problem and the optimal Nash equilibrium (NE) seeking problem. Our main contribution
is threefold. (i) We consider bilevel VIs with monotone and Lipschitz continuous mappings and devise a single-timescale iteratively regularized extragradient method, named IR-EGm,m. 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
method named IR-EGs,m and derive faster guarantees than those in (i). We also study the iteration
complexity of this method under a constant regularization parameter. 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, named IPR-EG, where the projection onto the unknown set of equilibria is performed using
IR-EGs,m with a prescribed termination criterion and an adaptive regularization parameter. 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 NEs.