Abstract
The generalized Nash equilibrium problem is a generalization of the standard Nash equilibrium problem, in which both the utility function and the strategy space of each player may depend on the strategies chosen by all other players. This problem has been used to model various problems in applications but convergent solution algorithms are extremely scare in the literature. In this article, we show that a generalized Nash equilibrium can be calculated by solving a variational inequality (VI). Moreover, conditions for the local superlinear convergence of a semismooth Newton method being applied to the VI are also given. Some numerical results are presented to illustrate the performance of the method.
Highlights
In this article, We consider the generalized Nash equilibrium problem (GNEP)
We first recall the definition of the Nash equilibrium problem (NEP)
X∗ = x∗,1, ..., x∗,N T ∈ n is called a Nash equilibrium, or a solution of the NEP, if each block component x*,ν is a solution of the optimization problem min θ ν xν, x∗,−ν xν s.t. xν ∈ Xν
Summary
We consider the generalized Nash equilibrium problem (GNEP). To this end, we first recall the definition of the Nash equilibrium problem (NEP). Given a closed convex set K ⊆ n and a continuous function G : K → n, solving the VI defined by K and G (which is denoted by VI(G, K)) means finding a vector x Î K such that Lemma 2.1 Suppose that the GNEP satisfies Assumption 1.1 and assume further that the sets Xν(x-ν) are defined by (1.1) with X closed and convex.
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