Abstract
The generalized Nash equilibrium problem (GNEP) is an extension of the standard Nash equilibrium problem (NEP), in which each player's strategy set may depend on the rival player's strategies. In this paper, we present two descent type methods. The algorithms are based on a reformulation of the generalized Nash equilibrium using Nikaido-Isoda function as unconstrained optimization. We prove that our algorithms are globally convergent and the convergence analysis is not based on conditions guaranteeing that every stationary point of the optimization problem is a solution of the GNEP.
Highlights
The generalized Nash equilibrium problem (GNEP for short) is an extension of the standard Nash equilibrium problem (NEP for short), in which the strategy set of each player depends on the strategies of all the other players as well as on his own strategy
Let us first recall the definition of the GNEP
If X is defined as the Cartesian product of certain sets Xv ∈ Rnv, that is, X = X1 × X2 × ⋅ ⋅ ⋅ × XN, the GNEP reduces to the standard Nash equilibrium problem
Summary
The generalized Nash equilibrium problem (GNEP for short) is an extension of the standard Nash equilibrium problem (NEP for short), in which the strategy set of each player depends on the strategies of all the other players as well as on his own strategy. X∗,N)T is called a solution of the GNEP or a generalized Nash equilibrium, if for each player v = 1, . If X is defined as the Cartesian product of certain sets Xv ∈ Rnv , that is, X = X1 × X2 × ⋅ ⋅ ⋅ × XN, the GNEP reduces to the standard Nash equilibrium problem. A basic tool for both the theoretical and the numerical solution of (generalized) Nash equilibrium problems is the Nikaido-Isoda function defined as. X∗ is a normalized Nash equilibrium of the GNEP, if maxyΨ(x∗, y) = 0 holds, where Ψ(x, y) denotes the Nikaido-Isoda function defined as (2). We will show that our algorithms are globally convergent to a normalized Nash equilibrium under appropriate assumption on the cost function, which is not stronger than the one considered in [8].
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