An explicit Tikhonov algorithm for nested variational inequalities

Abstract

We consider nested variational inequalities consisting in a (upper-level) variational inequality whose feasible set is given by the solution set of another (lower-level) variational inequality. Purely hierarchical convex bilevel optimization problems and certain multi-follower games are particular instances of nested variational inequalities. We present an explicit and ready-to-implement Tikhonov-type solution method for such problems. We give conditions that guarantee the convergence of the proposed method. Moreover, inspired by recent works in the literature, we provide a convergence rate analysis. In particular, for the simple bilevel instance, we are able to obtain enhanced convergence results.