A Bridge between Bilevel Programs and Nash Games

TL;DR

This paper explores the relationship between bilevel programming problems and Generalized Nash Equilibrium Problems, proposing a new model and a globally convergent algorithm to solve complex bilevel problems effectively.

Contribution

It introduces a novel GNEP model linked to bilevel problems without requiring unique lower-level solutions and provides a convergent algorithm for solving these problems.

Findings

Effective solution method for large nonlinear bilevel problems

Complete analysis of the relationship between bilevel and GNEP models

Algorithm demonstrates global convergence and practical efficiency

Abstract

We study connections between optimistic bilevel programming problems and Generalized Nash Equilibrium Problems (GNEP)s. We remark that, when addressing bilevel problems, we consider the general case in which the lower level program is not assumed to have a unique solution. Inspired by the optimal value approach, we propose a new GNEP model that is closely related to the bilevel program. We provide a complete analysis of the relationship between the "vertical" bilevel problem and the corresponding "horizontal" (one-level) GNEP model. We define classes of problems for which solutions of the bilevel program can be computed by finding equilibria of the GNEP. We develop a simple algorithm, which turns out to be globally convergent, for the solution of classes of our GNEP; we study how it is then possible to recover a solution of the bilevel problem from the computed equilibrium. Numerical experience shows the effectiveness of our approach, even when addressing big nonlinear bilevel problems.