Methods for Solving Generalized Nash Equilibrium

The generalizedNash equilibriumproblem (GNEP) is an extension of the standardNash equilibriumproblem (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.


Introduction
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.The GNEP has recently attracted much attention due to its applications in various fields like mathematics, computer science, economics, and engineering [1][2][3][4][5][6][7][8][9][10][11].For more details, we refer the reader to a recent survey paper by Facchinei and Kanzow [3] and the references therein.
Throughout this paper, we can make the following assumption.

Assumption 1. (a)
The set  is nonempty, closed, and convex.(b) The utility function   is continuously differentiable and, as a function of   alone, convex.
A basic tool for both the theoretical and the numerical solution of (generalized) Nash equilibrium problems is the Nikaido-Isoda function defined as Sometimes also the name Ky-Fan function can be found in the literature, see [12,13].In the following, we state a definition which we have taken from [9].Definition 1.  * is a normalized Nash equilibrium of the GNEP, if max  Ψ( * , ) = 0 holds, where Ψ(, ) denotes the Nikaido-Isoda function defined as (2).
In order to overcome the nondifferentiable property of the mapping Ψ(, ), von Heusinger and Kanzow [8] used a simple regularization of the Nikaido-Isoda function.For a parameter  > 0, the following regularized Nikaido-Isoda function was considered: (3) Since under the given Assumption 1, Ψ  (, ) is strongly concave in , the maximization problem has a unique solution for each , denoted by   ().
The corresponding value function is then defined by Let  >  > 0 be a given parameter.The corresponding value function is then defined by Define In [8], the following important properties of the function   () have been proved.(c)   () is continuously differentiable on   and that From Theorem 2, we know that the normalized Nash equilibrium of the GNEP is precisely the global minima of the smooth unconstrained optimization problem (see [5]) as with zero optimal value.
In this paper, we develop two new descent methods for finding a normalized Nash equilibrium of the GNEP by solving the optimization problem (9).The key to our methods is a strategy for adjusting  and  when a stationary point of V  () is not a solution of the GNEP.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].
The organization of the paper is as follows.In Section 2, we state the main assumption underlying our algorithms and present some examples of the GNEP satisfying it.In Section 3, we derive some useful properties of the function   ().In Section 4, we formally state our algorithms and prove that they are both globally convergent to a normalized Nash equilibrium.

Main Assumption
In order to construct algorithms and guarantee the convergence of them, we give the following assumption.
We next consider three examples which satisfy Assumption 2.
Example 3. Let us consider the case in which all the cost functions are separable, that is, where   :    →  is convex and   :  −  → .A simple calculation shows that, for any  ∈   , we have Hence Assumption 2 holds.
Example 4. Consider the case where the cost function   () is quadratic, that is, for  = 1, . . ., .We have Therefore, if the matrix is positive semidefinite, Assumption 2 is satisfied.
In the following example, we show the relationship between our assumption and the one considered in [8] as follows.
For any  >  > 0, a given  ∈   with   () ̸ =   (), the inequality Example 5. Consider the GNEP with  = 2 as and the cost function The point  * = (1, 9)  is the unique normalized Nash equilibrium.For any  ∈  2 , we have Therefore Assumption 2 holds, but (16) does not hold for any  >  > 0.

Properties of 𝑉 𝛼𝛽 (𝑥)
Lemma 6.For any  >  > 0 and  ∈   , we have (20) Proof.Since   () satisfies the optimality condition, then In a similar way, it follows that   () satisfies Since   () as a function of   alone is convex, we have respectively.Thus, using the definition of   () and ( 23 (25) Similarly, using the definition of   () and (24), we have The proof is complete.which is the second inequality in (33).This completes the proof.

Two Methods for Solving the GNEP
In this section, we introduce two methods for solving the GNEP, motivated by the D-gap function scheme for solving monotone variational inequalities [14,15].We first formally describe our methods below and then analyze their convergence using Lemma 8.
Apply a descent method to the unconstrained minimization of the function      , with   as the starting point and using    −    as a safeguard descent direction at , until the method generates an  ∈   satisfying      () ≤   .The resulting  is denoted by   .Theorem 10.Assume X is bounded.Let {  ,   ,   ,   ,   ,   } =0,1,2,... be generated by Algorithm 9. Then {  } is bounded;   → ∞;   → 0; and every cluster point of {  } is a normalized Nash equilibrium of the GNEP.
The proof is completed.