Nonsmooth optimization reformulations of player convex generalized Nash equilibrium problems

Dreves, Axel; Kanzow, Christian; Stein, Oliver

doi:10.1007/s10898-011-9727-9

Cited by 35 publications

(24 citation statements)

References 23 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%

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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“…We computed 0-approximate equilibria of the game by solving problem (9) with different ordinal potential functions (see section 3). In particular, we denote with P the solution obtained with the exact potential function P (x) = N ν=1 θ ν (x ν ), with Pmin the solution obtained by solving problem (13) with α = 1e4, and with Pmax the solution obtained by solving problem (14) with α = 1e4 and θ 1 = θ 2 = θ 3 = 0. We observe that in all the cases we solved a mixed-integer linear problem, and we used CPLEX.…”

Section: Experiments On the Market Described In Examplementioning

confidence: 99%

Algorithms for generalized potential games with mixed-integer variables

Sagratella

2017

Comput Optim Appl

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We consider generalized potential games, that constitute a fundamental subclass of generalized Nash equilibrium problems. We propose different methods to compute solutions of generalized potential games with mixedinteger variables, i.e., games in which some variable are continuous while the others are discrete. We investigate which types of equilibria of the game can be computed by minimizing a potential function over the common feasible set. In particular, for a wide class of generalized potential games, we characterize those equilibria that can be computed by minimizing potential functions as Pareto solutions of a particular multi-objective problem, and we show how different potential functions can be used to select equilibria. We propose a new Gauss-Southwell algorithm to compute approximate equilibria of any generalized potential game with mixed-integer variables. We show that this method converges in a finite number of steps and we also give an upper bound on this number of steps. Moreover, we make a thorough analysis on the behaviour of approximate equilibria with respect to exact ones. Finally, we make many numerical experiments to show the viability of the proposed approaches. Keywords Generalized Nash equilibrium problem • Generalized potential game • Mixed-integer nonlinear problem • Parametric optimization The work of the author has been partially supported by Avvio alla Ricerca 2015 Sapienza University of Rome, under grant 488.

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“…Borrowing the terminology from the related case of (quasi) variational inequalities ( [11,12]) in the following, we shall refer to V as a gap function of LGNEP. According to [16], the function V is nonnegative on the unfolded common strategy set…”

Section: Global Extension Of the Gap Functionmentioning

confidence: 99%

The Cone Condition and Nonsmoothness in Linear Generalized Nash Games

Stein

Sudermann‐Merx

2015

J Optim Theory Appl

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We consider linear generalized Nash games and introduce the so-called cone condition, which characterizes the smoothness of a gap function that arises from a reformulation of the generalized Nash equilibrium problem as a piecewise linear optimization problem based on the Nikaido-Isoda function. Other regularity conditions such as the linear independence constraint qualification or the strict MangasarianFromovitz condition are only sufficient for smoothness, but have the advantage that they can be verified more easily than the cone condition. Therefore, we present special cases, where these conditions are not only sufficient, but also necessary for smoothness of the gap function. Our main tool in the analysis is a global extension of the gap function that allows us to overcome the common difficulty that its domain may not cover the whole space.

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Nonsmooth optimization reformulations of player convex generalized Nash equilibrium problems

Cited by 35 publications

References 23 publications

Augmented Lagrangian Methods for the Solution of Generalized Nash Equilibrium Problems

Augmented Lagrangian Methods for the Solution of Generalized Nash Equilibrium Problems

Algorithms for generalized potential games with mixed-integer variables

The Cone Condition and Nonsmoothness in Linear Generalized Nash Games

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