Exact Penalization of Generalized Nash Equilibrium Problems

Abstract

This paper presents an exact penalization theory of the generalized Nash equilibrium problem (GNEP) that has its origin from the renowned Arrow–Debreu general economic equilibrium model. Whereas the latter model is the foundation of much of mathematical economics, the GNEP provides a mathematical model of multiagent noncooperative competition that has found many contemporary applications in diverse engineering domains. The most salient feature of the GNEP that distinguishes it from a standard noncooperative (Nash) game is that each player’s optimization problem contains constraints that couple all players’ decision variables. Extending results for stand-alone optimization problems, the penalization theory aims to convert the GNEP into a game of the standard kind without the coupled constraints, which is known to be more readily amenable to solution methods and analysis. Starting with an illustrative example to motivate the development, this paper focuses on two kinds of coupled constraints, shared (i.e., common) and finitely representable. Constraint residual functions and the associated error bound theory play an important role throughout the development.

Rational Generalized Nash Equilibrium Problems

“…), and the feasible extension p 1 of (0, 1) at u (1) is (x 1,1 , 1 − x 2,1 − x 2,2 − x 1,1 ). At u (2) , the minimizer of F 1 (u…”

“…A semidefinite relaxation method for convex GNEPs of polynomials is given in [40]. The penalty functions are used to solve GNEPs in [2,12]. An Augmented-Lagrangian method is given in [21].…”

First-Order Algorithms for Nonlinear Generalized Nash Equilibrium Problems

“…The first attempts to design such algorithms are due to Robinson [1993a,b]. Further progress has been made over the ensuing three decades by considering generic algorithms based on either the Nikaido-Isoda function [Uryas' ev and Rubinstein, 1994, Krawczyk and Uryasev, 2000, Von Heusinger and Kanzow, 2009, Dreves et al, 2011, von Heusinger et al, 2012, Dreves et al, 2013, Izmailov and Solodov, 2014, Fischer et al, 2016 or penalty functions [Pang and Fukushima, 2005, Facchinei and Lampariello, 2011, Fukushima, 2011, Facchinei and Kanzow, 2010b, Kanzow, 2016, Kanzow and Steck, 2016, Ba and Pang, 2020]. Global convergence rates and local convergence rates have been established for some of these algorithms under suitable assumptions.…”

“…Global convergence rates and local convergence rates have been established for some of these algorithms under suitable assumptions. In particular, for an overview of recent progress on penalty-type algorithms, we refer to Ba and Pang [2020].…”

Convex generalized Nash equilibrium problems and polynomial optimization

“…The Lemke's method is used to solve affine GNEPs [54]. For general nonconvex GNEPs, we refer to [5,12,16,28,47]. It is generally quite difficult to solve GNEPs, even if they are convex.…”