Generalized Nash Equilibrium Problems - Recent Advances and Challenges

Fischer, Andreas; Herrich, Markus; Schönefeld, K.

doi:10.1590/0101-7438.2014.034.03.0521

Cited by 107 publications

(74 citation statements)

References 62 publications

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“…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.…”

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confidence: 99%

“…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.…”

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confidence: 99%

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Augmented Lagrangian Methods for the Solution of Generalized Nash Equilibrium Problems

Kanzow¹,

Steck²

2016

SIAM J. Optim.

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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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confidence: 99%

Augmented Lagrangian Methods for the Solution of Generalized Nash Equilibrium Problems

Kanzow¹,

Steck²

2016

SIAM J. Optim.

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“…The notation (x ν , x −ν ) for a strategy vector x ∈ R n is typically used in the context of GNEPs to emphasize the variables x ν of the ν-th player and the variables x −ν of all the rival players. For more information on GNEPs see [9,13] and references therein. The KKT system associated with the ν-th player's optimization problem (7.1) is given by…”

Section: Individual Error Bounds For Active Pieces In Case Of Complemmentioning

confidence: 99%

Convergence conditions for Newton-type methods applied to complementarity systems with nonisolated solutions

et al. 2015

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Abstract. We consider a class of Newton-type methods for constrained systems of equations that involve complementarity conditions. In particular, at issue are the constrained Levenberg-Marquardt method and the recently introduced Linear-Programming-Newton method, designed for the difficult case when solutions need not be isolated, and the equation mapping need not be differentiable at the solutions. We show that the only structural assumption needed for rapid local convergence of those algorithms is the piecewise error bound, i.e., a local error bound holding for the branches of the solution set resulting from partitions of the bi-active complementarity indices. The latter error bound is implied by various piecewise constraint qualifications, including some relatively weak ones. As further applications of our results, we consider Karush-Kuhn-Tucker systems arising from optimization or variational problems, and from generalized Nash equilibrium problems. In the first case, we show convergence under the assumption that the dual part of the solution is a noncritical Lagrange multiplier, and in the second case convergence follows under a certain relaxed constant rank condition. In both cases, this improves on results previously available for the corresponding problems.

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“…Hence, the governing concept is no longer a Nash equilibrium (cf. Nash, , ) but a generalized Nash equilibrium (GNE; see, e.g., von Heusinger, ; Fischer et al., ). Recall that, in classical Nash equilibrium problems, the strategies of the players, that is, the decision makers in the noncooperative game, affect the utility functions of other players, but the feasible set of each player depends only on their strategies.…”

Section: Introductionmentioning

confidence: 99%

Cybersecurity investments with nonlinear budget constraints and conservation laws: variational equilibrium, marginal expected utilities, and Lagrange multipliers

Colajanni

Daniele

Giuffrè

et al. 2018

Int Trans Operational Res

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In this paper, we propose a new cybersecurity investment supply chain game theory model, assuming that the demands for the product are known and fixed and, hence, the conservation law of each demand market is fulfilled. The model is a generalized Nash equilibrium model with nonlinear budget constraints for which we define the variational equilibrium, which provides us with a variational inequality formulation. We construct an equivalent formulation, enabling the analysis of the influence of the conservation laws and the importance of the associated Lagrange multipliers. We find that the marginal expected transaction utility of each retailer depends on this Lagrange multiplier and its sign. Finally, numerical examples with reported equilibrium product flows, cybersecurity investment levels, and Lagrange multipliers, along with individual firm vulnerability and network vulnerability, illustrate the obtained results.

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Generalized Nash Equilibrium Problems - Recent Advances and Challenges

Cited by 107 publications

References 62 publications

Augmented Lagrangian Methods for the Solution of Generalized Nash Equilibrium Problems

Augmented Lagrangian Methods for the Solution of Generalized Nash Equilibrium Problems

Convergence conditions for Newton-type methods applied to complementarity systems with nonisolated solutions

Cybersecurity investments with nonlinear budget constraints and conservation laws: variational equilibrium, marginal expected utilities, and Lagrange multipliers

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