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Optimization reformulations of the generalized Nash equilibrium problem using the NikaidoIsoda type functions
 COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 43(3)
, 2006
"... We consider the generalized Nash equilibrium problem which, in contrast to the standard Nash equilibrium problem, allows joint constraints of all players involved in the game. Using a regularized NikaidoIsodafunction, we then present three optimization problems related to the generalized Nash eq ..."
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We consider the generalized Nash equilibrium problem which, in contrast to the standard Nash equilibrium problem, allows joint constraints of all players involved in the game. Using a regularized NikaidoIsodafunction, we then present three optimization problems related to the generalized Nash equilibrium problem. The first optimization problem is a complete reformulation of the generalized Nash game in the sense that the global minima are precisely the solutions of the game. However, this reformulation is nonsmooth. We then modify this approach and obtain a smooth constrained optimization problem whose global minima correspond to socalled normalized Nash equilibria. The third approach uses the difference of two regularized NikaidoIsodafunctions in order to get a smooth unconstrained optimization problem whose global minima are, once again, precisely the normalized Nash equilibria. Conditions for stationary points to be global minima of the two smooth optimization problems are also given. Some numerical results illustrate the behaviour of our approaches.
Restricted generalized Nash equilibria and controlled penalty algorithm
, 2008
"... The generalized Nash equilibrium problem (GNEP) is a generalization of the standard Nash equilibrium problem (NEP), in which each player’s strategy set may depend on the rival players ’ strategies. The GNEP has recently drawn much attention because of its capability of modeling a number of interesti ..."
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The generalized Nash equilibrium problem (GNEP) is a generalization of the standard Nash equilibrium problem (NEP), in which each player’s strategy set may depend on the rival players ’ strategies. The GNEP has recently drawn much attention because of its capability of modeling a number of interesting conflict situations in, for example, an electricity market and an international pollution control. However, a GNEP usually has multiple or even infinitely many solutions, and it is not a trivial matter to choose a meaningful solution from those equilibria. The purpose of this paper is twofold. First we present an incremental penalty method for the broad class of GNEPs and show that it can find a GNE under suitable conditions. Next, we formally define the restricted GNE for the GNEPs with shared constraints and propose a controlled penalty method, which includes the incremental penalty method as a subprocedure, to compute a restricted GNE. Numerical examples are provided to illustrate the proposed approach. Key words. Generalized Nash equilibrium, shared constraints, shadow price, penalty method, restricted GNE. 1
Gap Function Approach to the Generalized Nash Equilibrium Problem
, 2009
"... Abstract. We consider an optimization reformulation approach for the generalized Nash equilibrium problem (GNEP) that uses the regularized gap function of a quasivariational inequality (QVI). The regularized gap function for QVI is in general not differentiable, but only directionally differentiabl ..."
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Abstract. We consider an optimization reformulation approach for the generalized Nash equilibrium problem (GNEP) that uses the regularized gap function of a quasivariational inequality (QVI). The regularized gap function for QVI is in general not differentiable, but only directionally differentiable. Moreover, a simple condition has yet to be established, under which any stationary point of the regularized gap function solves the QVI. We tackle these issues for the GNEP in which the shared constraints are given by linear equalities, while the individual constraints are given by convex inequalities. First, we formulate the minimization problem involving the regularized gap function, and show the equivalence to GNEP. Next, we establish the differentiability of the regularized gap function and show that any stationary point of the minimization problem solves the original GNEP under some suitable assumptions. Then, by using a barrier technique, we propose an algorithm that sequentially solves minimization problems obtained from GNEPs with the shared equality constraints only. Further, we discuss the case of shared inequality constraints and present an algorithm that utilizes the transformation of the inequality constraints to equality constraints by means of slack variables. We present some results of numerical experiments to illustrate the proposed approach.
SC 1 optimization reformulations of the generalized Nash equilibrium problem
 Optimization Methods Software
, 2008
"... Abstract. The generalized Nash equilibrium problem is a Nash game which, in contrast to the standard Nash equilibrium problem, allows the strategy sets of each player to depend on the decision variables of all other players. It was recently shown by the authors that this generalized Nash equilibrium ..."
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Abstract. The generalized Nash equilibrium problem is a Nash game which, in contrast to the standard Nash equilibrium problem, allows the strategy sets of each player to depend on the decision variables of all other players. It was recently shown by the authors that this generalized Nash equilibrium problem can be reformulated as both an unconstrained and a constrained optimization problem with continuously differentiable objective functions. This paper further investigates these approaches and shows, in particular, that the objective functions are SC1functions. Moreover, conditions for the local superlinear convergence of a semismooth Newton method being applied to the unconstrained optimization reformulation are also given. Some numerical results indicate that this method works quite well on a number of problems coming from different application areas.
How to use Rosen’s normalised equilibrium to enforce a socially desirable Pareto efficient solution
, 2009
"... We consider a situation, in which a regulator believes that constraining a complex good created jointly by competitive agents, is socially desirable. Individual levels of outputs that generate the constrained amount of the externality can be computed as a Pareto efficient solution of the agents ’ jo ..."
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We consider a situation, in which a regulator believes that constraining a complex good created jointly by competitive agents, is socially desirable. Individual levels of outputs that generate the constrained amount of the externality can be computed as a Pareto efficient solution of the agents ’ joint utility maximisation problem. However, generically, a Pareto efficient solution is not an equilibrium. We suggest the regulator calculates a NashRosen coupledconstraint equilibrium (or a “generalised ” Nash equilibrium) and uses the coupledconstraint Lagrange multiplier to formulate a threat, under which the agents will play a decoupled Nash game. An equilibrium of this game will possibly coincide with the Pareto efficient solution. We focus on situations when the constraints are saturated and examine, under which conditions a match between an equilibrium and a Pareto solution is possible. We illustrate our findings using a model for a coordination problem, in which firms ’ outputs depend on each other and where the output levels are important for the regulator.
comscifacpub/4 Stochastic Stability in Internet Router Congestion
"... Abstract. Congestion control at bottleneck routers on the internet is a long standing problem. Many policies have been proposed for effective ways to drop packets from the queues of these routers so that network endpoints will be inclined to share router capacity fairly and minimize the overflow o ..."
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Abstract. Congestion control at bottleneck routers on the internet is a long standing problem. Many policies have been proposed for effective ways to drop packets from the queues of these routers so that network endpoints will be inclined to share router capacity fairly and minimize the overflow of packets trying to enter the queues. We study just how effective some of these queuing policies are when each network endpoint is a selfinterested player with no information about the other players ’ actions or preferences. By employing the adaptive learning model of evolutionary game theory, we study policies such as Droptail, RED, and the greedyflowpunishing policy proposed by Gao et al. [10] to find the stochastically stable states: the states of the system that will be reached in the long run.
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