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We aim at building a bridge between bilevel programming and generalized Nash equilibrium problems. First, we present two Nash games that turn out to be linked to the (approximated) optimistic version of the bilevel problem. Specifically, on the one hand we establish relations between the equilibrium set of a Nash game and global optima of the (approximated) optimistic bilevel problem. On the other hand, correspondences between equilibria of another Nash game and stationary points of the (approximated) optimistic bilevel problem are obtained. Then, building on these ideas, we also propose different Nash-like models that are related to the (approximated) pessimistic version of the bilevel problem. This analysis, being of independent theoretical interest, leads also to algorithmic developments. Finally, we discuss the intrinsic complexity characterizing both the optimistic bilevel and the Nash game models.

We consider a class of optimistic bilevel problems. Specifically, we address bilevel problems in which at the lower level the objective function is fully convex and the feasible set does not depend on the upper level variables. We show that this nontrivial class of mathematical programs is sufficiently broad to encompass significant real-world applications and proves to be numerically tractable. From this respect, we establish that the stationary points for a relaxation of the original problem can be obtained addressing a suitable generalized Nash equilibrium problem. The latter game is proven to be convex and with a nonempty solution set. Leveraging this correspondence, we provide a provably convergent, easily implementable scheme to calculate stationary points of the relaxed bilevel program. As witnessed by some numerical experiments on an application in economics, this algorithm turns out to be numerically viable also for big dimensional problems.

The optimality system of a quasi-variational inequality can be reformulated as a non-smooth equation or a constrained equation with a smooth function. Both reformulations can be exploited by algorithms, and their convergence to solutions usually relies on the nonsingularity of the Jacobian, or the fact that the merit function has no nonoptimal stationary points. We prove new sufficient conditions for the absence of nonoptimal constrained or unconstrained stationary points that are weaker than some known ones. All these conditions exploit some properties of a certain matrix, but do not require the nonsingularity of the Jacobian. Further, we present new necessary and sufficient conditions for the nonsingularity of the Jacobian that are based on the signs of certain determinants. Additionally, we consider generalized Nash equilibrium problems that are a special class of quasi-variational inequalities. Exploiting their structure, we also prove some new sufficient conditions for stationarity and nonsingularity results.