Augmented Lagrangian Methods for the Solution of Generalized Nash Equilibrium Problems

Abstract

We propose an augmented Lagrangian-type algorithm for the solution of generalized Nash equilibrium problems (GNEPs). Specifically, we discuss the convergence properties with regard to both feasibility and optimality of limit points. This is done by introducing a secondary GNEP as a new optimality concept. In this context, special consideration is given to the role of suitable constraint qualifications that take into account the particular structure of GNEPs. Furthermore, we consider the behaviour of the method for jointly-convex GNEPs and describe a modification which is tailored towards the computation of variational equilibria. Numerical results are included to illustrate the practical performance of the overall method.Note that we do not include equality constraints in our GNEP simply for the sake of notational convenience; our subsequent approach can easily be extended to equality and inequality constraints. Apart from this, the above setting is very general since, so far, we do not assume any convexity assumptions on the mappings θ ν and c ν as is done in many other GNEP papers where only the player-convex or jointly-convex case is considered, cf. [2,8,7,10,12,17,28] for more details. It follows that our framework can, in principle, be applied to very general classes of GNEPs.In the meantime, there exist a variety of methods for the solution of GNEPs, though most of them are designed for player-or jointly-convex GNEPs and therefore do not cover the GNEP in its full generality. We refer the interested reader once again to the two survey papers [12,17] and the references therein for a quite complete overview of the existing approaches. One of the main problems when solving a GNEP is an inherent singularity property that arises when some players share the same constraints, see [11] for more details. Hence, second-order methods with fast local convergence are difficult to design. This also motivates us to consider methods which may not be locally superlinearly or quadratically convergent, but have nice global convergence properties.Penalty-type schemes belong to this class of methods. The first penalty method for GNEPs that we are aware of is due to Fukushima [18]. A related penalty algorithm was also proposed in [13], and a modification of this algorithm is described in [14] where only some of the constraints are penalized. While all these approaches prove exactness results under suitable assumptions, they suffer from the drawback that the resulting penalized subproblems are nonsmooth Nash equilibrium problems and therefore difficult to solve numerically.Taking this into account, it is natural to apply an augmented Lagrangian-type approach in order to solve GNEPs because the resulting subproblems then have a higher degree of smoothness and should therefore be easier to solve. This idea is not completely new since Pang and Fukushima [22] applied this idea to quasi-variational inequalities (QVIs). An improved version of that method can be found in [20], also for QVIs. Since the GNEP is a special instance of a QVI...

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