Generalized Nash Equilibrium Problem Solving Algorithm Under Penalty Framework

The generalized Nash equilibrium problem, GNEP for short, is a noncooperative game, where each player’s strategy set depends on the rival players’ strategies in contrast to the standard Nash equilibrium problem (NEP). The GNEP has recently drawn much attention because of its fruitful use in many different fields of economics, mathematics and engineering, etc, ranging from telecommunications to structural engineering, and from computer science to the analysis of pollution scenarios, so that its potential importance should not be underestimated. However, sparse solution algorithms are available for its study.When it comes to the solution of a general GNEP, most of provably convergent solution algorithms are available for some specific problems, see [32] for a review. Recently, an increasing effort based on the penalty technique has been made to solve the GNEP, which opens a new road for studying the GNEP. In [10,11], the penalty scheme is simple and it gives a rule to choose the penalty parameters, thus guaranteeing convergence to a solution of the GNEP. But the convergence analysis is based on a constraint qualification, and the nondifferentiation of the subproblem is somewhat problematic from the numerical point of view; another important issue that certainly deserves considering is the method used to solve the penalized Nash problem. A natural idea is how to avoid the occurrence of the bad situation and how to put the penalty approach into practise.In this paper, we propose a new penalty algorithm for solving a general GNEP, where we use a interior point penalty scheme to define the updating rule of penalty parameters, i.e., apply a penalty technique in nonlinear programming to the dif-ficulty constraints, and thereby formulating the GNEP into a constrained NEP. Depending on the related variational inequality formulation of the NEP, we use the Alternating Direction Method (ADM) to solve it. The global convergence of the new method is established under certain appropriate assumptions. Preliminary numerical results demonstrate the proposed method is efficient.