  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 and the optimal Nash
equilibrium (NE) seeking problem. Our main contributions are threefold. (i) We
consider bilevel VIs with monotone and Lipschitz continuous mappings and devise
a single-timescale iteratively regularized extragradient method, named
IR-EG$_{{\texttt{m,m}}}$. 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-EG$_{{\texttt{s,m}}}$ 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-EG$_{{\texttt{s,m}}}$ 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 Nash equilibria.
