Methods for Solving Generalized Nash Equilibrium

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.

1. 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 [1–11]. For more details, we refer the reader to a recent survey paper by Facchinei and Kanzow [3] and the references therein.

Let us first recall the definition of the GNEP. There are 𝑁 players labelled by an integer 𝑣=1,…,𝑁. Each player 𝑣 controls the variables 𝑥^𝑣∈𝑅^𝑛𝑣. Let 𝑥=(𝑥¹⋯𝑥^𝑛)^𝑇 be the vector formed by all these decision variables, where 𝑛∶=𝑛₁+𝑛₂+⋯+𝑛_𝑣. To emphasize the 𝑣th player variable within the vector 𝑥, we sometimes write 𝑥=(𝑥^𝑣,𝑥^−𝑣)^𝑇∈𝑅^𝑛, where 𝑥^−𝑣 denotes all the other player's variables. In the games, each player controls the variables 𝑥^𝑣 and tries to minimize a cost function 𝜃_𝑣(𝑥^𝑣,𝑥^−𝑣) subjects to the constraint (𝑥^𝑣,𝑥^−𝑣)^𝑇∈𝑋 with 𝑥^−𝑣 given as exogenous, where 𝑋 is a common strategy set. A vector 𝑥^∗∶=(𝑥^∗,1,…,𝑥^∗,𝑁)^𝑇 is called a solution of the GNEP or a generalized Nash equilibrium, if for each player 𝑣=1,…,𝑁, 𝑥^∗,𝑣 solves the following optimization problem with 𝑥^∗,−𝑣 being fixed: min_𝑥𝑣𝜃_𝑣𝑥^𝑣,𝑥^∗,−𝑣,s.t.𝑥^𝑣,𝑥^∗,−𝑣∈𝑋. (1)

If 𝑋 is defined as the Cartesian product of certain sets 𝑋_𝑣∈𝑅^𝑛𝑣, that is, 𝑋=𝑋₁×𝑋₂×⋯×𝑋_𝑁, then the GNEP reduces to the standard Nash equilibrium problem.

Throughout this paper, we can make the following assumption.

Assumption 1. (a) The set 𝑋 is nonempty, closed, and convex.
(b) The utility function 𝜃_𝑣 is continuously differentiable and, as a function of 𝑥^𝑣 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 Ψ𝑥,𝑦=^𝑁_𝑣=1𝜃_𝑣𝑥^𝑣,𝑥^−𝑣−𝜃_𝑣𝑦^𝑣,𝑥^−𝑣. (2)

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. 𝑥^∗ is a normalized Nash equilibrium of the GNEP, if max_𝑦Ψ(𝑥^∗,𝑦)=0 holds, where Ψ(𝑥,𝑦) denotes the Nikaido-Isoda function defined as (2).

In order to overcome the nondifferentiable property of the mapping Ψ(𝑥,𝑦), von Heusinger and Kanzow [8] used a simple regularization of the Nikaido-Isoda function. For a parameter 𝛼>0, the following regularized Nikaido-Isoda function was considered: Ψ_𝛼𝑥,𝑦=^𝑁_𝑣=1𝜃_𝑣𝑥^𝑣,𝑥^−𝑣−𝜃_𝑣𝑦^𝑣,𝑥^−𝑣−𝛼2𝑥−𝑦². (3) Since under the given Assumption 1, Ψ_𝛼(𝑥,𝑦) is strongly concave in 𝑦, the maximization problemmax_𝑦Ψ_𝛼𝑥,𝑦,s.t.𝑦∈𝑋 (4) has a unique solution for each 𝑥, denoted by 𝑦_𝛼(𝑥).

The corresponding value function is then defined by 𝑉_𝛼(𝑥)=max_𝑦Ψ_𝛼𝑥,𝑦=Ψ_𝛼𝑥,𝑦_𝛼(𝑥). (5)

Let 𝛽>𝛼>0 be a given parameter. The corresponding value function is then defined by 𝑉_𝛽(𝑥)=max_𝑦Ψ_𝛽𝑥,𝑦=Ψ_𝛽𝑥,𝑦_𝛽(𝑥). (6)

Define 𝑉_𝛼𝛽(𝑥)=𝑉_𝛼(𝑥)−𝑉_𝛽(𝑥). (7)

In [8], the following important properties of the function 𝑉_𝛼𝛽(𝑥) have been proved.

Theorem 2. The following statements hold: (a)𝑉_𝛼𝛽(𝑥)≥0 for any 𝑥∈𝑅^𝑛; (b)𝑥^∗ is a normalized Nash equilibrium of the GNEP if and only if 𝑉_𝛼𝛽(𝑥^∗)=0; (c)𝑉_𝛼𝛽(𝑥) is continuously differentiable on 𝑅^𝑛 and that ∇𝑉_𝛼𝛽(𝑥)=∇𝑉_𝛼(𝑥)−∇𝑉_𝛽(𝑥)=^𝑁_𝑣=1∇𝜃_𝑣𝑦_𝛽(𝑥)^𝑣,𝑥^−𝑣−∇𝜃_𝑣𝑦_𝛼(𝑥)^𝑣,𝑥^−𝑣+∇_𝑥1𝜃₁𝑦_𝛼(𝑥)¹,𝑥⁻¹−∇_𝑥1𝜃₁𝑦_𝛽(𝑥)¹,𝑥⁻¹⋮∇_𝑥𝑁𝜃_𝑁𝑦_𝛼(𝑥)^𝑁,𝑥^−𝑁−∇_𝑥𝑁𝜃_𝑁𝑦_𝛽(𝑥)^𝑁,𝑥^−𝑁−𝛼𝑥−𝑦_𝛼(𝑥)+𝛽𝑥−𝑦_𝛽(𝑥). (8)

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 min_{𝑥∈𝑅𝑛}𝑉_𝛼𝛽(𝑥) (9) 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 𝛼 and 𝛽 when a stationary point of V_𝛼𝛽(𝑥) 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 𝑉_𝛼𝛽(𝑥). In Section 4, we formally state our algorithms and prove that they are both globally convergent to a normalized Nash equilibrium.

2. Main Assumption

In order to construct algorithms and guarantee the convergence of them, we give the following assumption.

Assumption 2. For any 𝛽>𝛼>0 and 𝑥∈𝑅^𝑛, if 𝑦_𝛼(𝑥)≠𝑦_𝛽(𝑥), we have ^𝑁_𝑣=1∇𝜃_𝑣𝑦_𝛽(𝑥)^𝑣,𝑥^−𝑣−∇𝜃_𝑣𝑦_𝛼(𝑥)^𝑣,𝑥^−𝑣𝑇⋅𝑦_𝛽(𝑥)−𝑦_𝛼(𝑥≥_𝑁𝑣=1)∇_𝑥𝑣𝜃_𝑣𝑦_𝛽(𝑥)^𝑣,𝑥^−𝑣−∇_𝑥𝑣𝜃_𝑣𝑦_𝛼(𝑥)^𝑣,𝑥^−𝑣𝑇⋅𝑦_𝛽(𝑥)^𝑣−𝑦_𝛼(𝑥)^𝑣. (10)
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, 𝜃_𝑣(𝑥)=𝑓_𝑣𝑥^𝑣+𝑔_𝑣𝑥^−𝑣, (11) where 𝑓_𝑣∶𝑅^𝑛𝑣→𝑅 is convex and 𝑔_𝑣∶𝑅^{𝑛−𝑛𝑣}→𝑅. A simple calculation shows that, for any 𝑦∈𝑅^𝑛, we have ^𝑁_𝑣=1∇_𝑥𝑣𝜃_𝑣𝑦_𝛽(𝑥)^𝑣,𝑥^−𝑣−∇_𝑥𝑣𝜃_𝑣𝑦_𝛼(𝑥)^𝑣,𝑥^−𝑣𝑇⋅𝑦_𝛽(𝑥)^𝑣−𝑦_𝛼(𝑥)^𝑣=^𝑁_𝑣=1∇𝑓_𝑣𝑦_𝛽(𝑥)^𝑣−∇𝑓_𝑣𝑦_𝛼(𝑥)^𝑣𝑇𝑦_𝛽(𝑥)^𝑣−𝑦_𝛼(𝑥)^𝑣,_𝑁𝑣=1∇𝜃_𝑣𝑦_𝛽(𝑥)^𝑣,𝑥^−𝑣−∇𝜃_𝑣𝑦_𝛼(𝑥)^𝑣,𝑥^−𝑣𝑇⋅𝑦_𝛽(𝑥)^𝑣−𝑦_𝛼(𝑥)^𝑣=^𝑁_𝑣=1∇𝑓_𝑣𝑦_𝛽(𝑥)^𝑣−∇𝑓_𝑣𝑦_𝛼(𝑥)^𝑣𝑇𝑦_𝛽(𝑥)^𝑣−𝑦_𝛼(𝑥)^𝑣. (12)
Hence Assumption 2 holds.

Example 4. Consider the case where the cost function 𝜃_𝑣(𝑥) is quadratic, that is, 𝜃_𝑣(𝑥)=12𝑥^𝑣𝑇𝐴_𝑣𝑣𝑥^𝑣+^𝑁_{𝜇=1,𝜇≠𝑣}𝑥^𝑣𝑇𝐴_𝑣𝜇𝑥^𝜇 (13) For 𝑣=1,…,𝑁. We have ^𝑁_𝑣=1∇_𝑥𝑣𝜃_𝑣𝑦_𝛽(𝑥)^𝑣,𝑥^−𝑣−∇_𝑥𝑣𝜃_𝑣𝑦_𝛼(𝑥)^𝑣,𝑥^−𝑣𝑇⋅𝑦_𝛽(𝑥)^𝑣−𝑦_𝛼(𝑥)^𝑣=^𝑁_𝑣=1𝑦_𝛽(𝑥)^𝑣−𝑦_𝛼(𝑥)^𝑣,𝐴_𝑣𝑣𝑦_𝛽(𝑥)^𝑣−𝑦_𝛼(𝑥)^𝑣,_𝑁𝑣=1∇𝜃_𝑣𝑦_𝛽(𝑥)^𝑣,𝑥^−𝑣−∇𝜃_𝑣𝑦_𝛼(𝑥)^𝑣,𝑥^−𝑣𝑇⋅𝑦_𝛽(𝑥)−𝑦_𝛼(𝑥)=𝑦_𝛽(𝑥)−𝑦_𝛼(𝑥),𝐴₁₁𝐴₂₁⋯𝐴_1𝑁𝐴₂₁𝐴₂₂⋯𝐴_2𝑁𝐴⋮⋮⋱⋮_1𝑁𝐴_2𝑁⋯𝐴_𝑁𝑁𝑦_𝛽(𝑥)−𝑦_𝛼(𝑥). (14)
Therefore, if the matrix 0𝐴₁₂⋯𝐴_1𝑁𝐴₂₁0⋯𝐴_2𝑁𝐴⋮⋮⋱⋮_𝑁1𝐴_𝑁2⋯0 (15) 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 𝛽>𝛼>0, a given 𝑥∈𝑅^𝑛 with 𝑦_𝛼(𝑥)≠𝑦_𝛽(𝑥), the inequality ^𝑁_𝑣=1∇𝜃_𝑣𝑦_𝛽(𝑥)^𝑣,𝑥^−𝑣−∇𝜃_𝑣𝑦_𝛼(𝑥)^𝑣,𝑥^−𝑣𝑇×𝑦_𝛽(𝑥)−𝑦_𝛼(𝑥)>0 (16) holds.

Example 5. Consider the GNEP with 𝑁=2 as 𝑋=𝑥∈𝑅^𝑛∶𝑥¹≥1,𝑥²≥1,𝑥¹+𝑥²≤10 (17) and the cost function 𝜃₁(𝑥)=𝑥¹𝑥² and 𝜃₂(𝑥)=−𝑥¹𝑥². The point 𝑥^∗=(1,9)^𝑇 is the unique normalized Nash equilibrium. For any 𝑦∈𝑅², we have ^𝑁_𝑣=1∇𝜃_𝑣𝑦_𝛽(𝑥)^𝑣,𝑥^−𝑣−∇𝜃_𝑣𝑦_𝛼(𝑥)^𝑣,𝑥^−𝑣𝑇⋅𝑦_𝛽(𝑥)−𝑦_𝛼(𝑥)=0,_𝑁𝑣=1∇_𝑥𝑣𝜃_𝑣𝑦_𝛽(𝑥)^𝑣,𝑥^−𝑣−∇_𝑥𝑣𝜃_𝑣𝑦_𝛼(𝑥)^𝑣,𝑥^−𝑣𝑇⋅𝑦_𝛽(𝑥)^𝑣−𝑦_𝛼(𝑥)^𝑣=0. (18)
Therefore Assumption 2 holds, but (16) does not hold for any 𝛽>𝛼>0.

3. Properties of 𝑉_𝛼𝛽(𝑥)

Lemma 6. For any 𝛽>𝛼>0 and 𝑥∈𝑅^𝑛, we have 𝑉_𝛼𝛽(𝑥)≥𝛽−𝛼2𝑥−𝑦_𝛽(𝑥)²+𝛼2𝑦_𝛼(𝑥)−𝑦_𝛽(𝑥)²,𝑉_𝛼𝛽(𝑥)≤𝛽−𝛼2𝑥−𝑦_𝛼(𝑥)²−𝛽2𝑦_𝛼(𝑥)−𝑦_𝛽(𝑥)². (19)(20)

Proof. Since 𝑦_𝛼(𝑥) satisfies the optimality condition, then ^𝑁_𝑣=1∇_𝑥𝑣𝜃_𝑣𝑦_𝛼(𝑥)^𝑣,𝑥^−𝑣−𝛼𝑥^𝑣−𝑦_𝛼(𝑥)^𝑣𝑇⋅𝑦_𝛽(𝑥)^𝑣−𝑦_𝛼(𝑥)^𝑣≥0. (21)
In a similar way, it follows that 𝑦_𝛽(𝑥) satisfies ^𝑁_𝑣=1∇_𝑥𝑣𝜃_𝑣𝑦_𝛽(𝑥)^𝑣,𝑥^−𝑣−𝛽𝑥^𝑣−𝑦_𝛽(𝑥)^𝑣𝑇⋅𝑦_𝛼(𝑥)^𝑣−𝑦_𝛽(𝑥)^𝑣≥0. (22)
Since 𝜃_𝑣(𝑥) as a function of 𝑥^𝑣 alone is convex, we have ^𝑁_𝑣=1𝜃_𝑣𝑦_𝛽(𝑥)^𝑣,𝑥^−𝑣−𝜃_𝑣𝑦_𝛼(𝑥)^𝑣,𝑥^−𝑣−𝛼𝑥−𝑦_𝛼(𝑥)^𝑇𝑦_𝛽(𝑥)−𝑦_𝛼(𝑥)≥0,_𝑁𝑣=1𝜃_𝑣𝑦_𝛼(𝑥)^𝑣,𝑥^−𝑣−𝜃_𝑣𝑦_𝛽(𝑥)^𝑣,𝑥^−𝑣−𝛽𝑥−𝑦_𝛽(𝑥)^𝑇𝑦_𝛼(𝑥)−𝑦_𝛽(𝑥)≥0 (23)(24) respectively. Thus, using the definition of 𝑉_𝛼𝛽(𝑥) and (23), we have 𝑉_𝛼𝛽(𝑥)=^𝑁_𝑣=1𝜃_𝑣𝑦_𝛼(𝑥)^𝑣,𝑥^−𝑣−𝜃_𝑣𝑦_𝛽(𝑥)^𝑣,𝑥^−𝑣+𝛽2𝑥−𝑦_𝛽(𝑥)²−𝛼2𝑥−𝑦_𝛼(𝑥)²≥𝛼𝑥−𝑦_𝛼(𝑥)^𝑇𝑦_𝛽(𝑥)−𝑦_𝛼(𝑥)+𝛽2𝑥−𝑦_𝛽(𝑥)²−𝛼2𝑥−𝑦_𝛼(𝑥)²=𝛼𝑥−𝑦_𝛼(𝑥)^𝑇𝑦_𝛽(𝑥)−𝑦_𝛼(𝑥)+𝛽2𝑥−𝑦_𝛽(𝑥)²−𝛼2𝑥−𝑦_𝛽(𝑥)+𝑦_𝛽(𝑥)−𝑦_𝛼(𝑥)²=𝛽−𝛼2𝑥−𝑦_𝛽(𝑥)²+𝛼2𝑦_𝛼(𝑥)−𝑦_𝛽(𝑥)². (25)
Similarly, using the definition of 𝑉_𝛼𝛽(𝑥) and (24), we have 𝑉_𝛼𝛽(𝑥)≤𝛽−𝛼2𝑥−𝑦_𝛼(𝑥)²−𝛽2𝑦_𝛼(𝑥)−𝑦_𝛽(𝑥)². (26)
The proof is complete.

Lemma 7. Assume 𝑋 is bounded. For any 𝛽>𝛼>0 and 𝑥∈𝑅^𝑛, we have limsup_{𝛽′→∞,𝛼′→0}𝑉_{𝛼′𝛽′}(𝑥)𝛽^′−𝛼^′≤𝑉_𝛼𝛽(𝑥).𝛽−𝛼 (27)

Proof. We have from (19) that 2𝑉_𝛼𝛽(𝑥)≥𝛽−𝛼𝑥−𝑦_𝛽(𝑥)²+𝛼𝑦𝛽−𝛼^𝛼(𝑥)−𝑦_𝛽(𝑥)²≥𝑥−𝑦_𝛽(𝑥)². (28)
By the definition of 𝑉_𝛼𝛽(𝑥), we have 2𝑉_{𝛼′𝛽′}𝛽^′−𝛼^′=2∑^𝑁_𝑣=1𝜃_𝑣𝑦_𝛽′(𝑥)^𝑣,𝑥^−𝑣−𝜃_𝑣𝑦_𝛼′(𝑥)^𝑣,𝑥^−𝑣𝛽^′−𝛼^′−𝛼^′𝑥−𝑦_𝛼′(𝑥)²+𝛽^′𝑥−𝑦_𝛽′(𝑥)²𝛽^′−𝛼^′. (29)
Since 𝑦_𝛼(𝑥)∈𝑋,𝑦_𝛽(𝑥)∈𝑋 and 𝑋 is bounded, we get that limsup_{𝛽′→∞,𝛼′→0}𝑉_{𝛼′𝛽′}(𝑥)𝛽^′−𝛼^′≤12𝑥−𝑦_𝛽(𝑥)²≤𝑉_𝛼𝛽.𝛽−𝛼 (30)
This completes the proof.

Equation (8) and Assumption 2 yield 𝑉_𝛼𝛽(𝑥)^𝑇𝑦_𝛽(𝑥)−𝑦_𝛼(𝑥=_𝑁𝑣=1)∇𝜃_𝑣𝑦_𝛽(𝑥)^𝑣,𝑥^−𝑣−∇𝜃_𝑣𝑦_𝛼(𝑥)^𝑣,𝑥^−𝑣𝑇⋅𝑦_𝛽(𝑥)−𝑦_𝛼(𝑥)+∇_𝑥1𝜃₁𝑦_𝛼(𝑥)¹,𝑥⁻¹−∇_𝑥1𝜃₁𝑦_𝛽(𝑥)¹,𝑥⁻¹⋮∇_𝑥𝑁𝜃_𝑁𝑦_𝛼(𝑥)^𝑁,𝑥^−𝑁−∇_𝑥𝑁𝜃_𝑁𝑦_𝛽(𝑥)^𝑁,𝑥^−𝑁𝑇⋅𝑦_𝛽(𝑥)−𝑦_𝛼(𝑥)−𝛼𝑥−𝑦_𝛼(𝑥)−𝛽𝑥−𝑦_𝛽(𝑥)^𝑇⋅𝑦_𝛽(𝑥)−𝑦_𝛼(𝑥)≥−𝛼𝑥−𝑦_𝛼(𝑥)−𝛽𝑥−𝑦_𝛽(𝑥)^𝑇⋅𝑦_𝛽(𝑥)−𝑦_𝛼(𝑥)=∶𝑒_𝛼𝛽(𝑥)≥0, (31) where nonnegativity of 𝑒_𝛼𝛽(𝑥) follows from the inequalities (23) and (24). In particular, either 𝑒_𝛼𝛽(𝑥) is above a tolerance 𝜀>0, in which case 𝑦_𝛼(𝑥)−𝑦_𝛽(𝑥) is a direction of sufficient descent for 𝑉_𝛼𝛽(𝑥) at 𝑥 or else, as we show in the lemma below, and 𝑥 is an approximate solution of the GNEP with accuracy depending on 𝜀, 𝛼, 𝛽. This result will lead to our methods.

Lemma 8. For any 𝛽>𝛼>0 and 𝑥∈𝑅^𝑛, we have 𝑥−𝑦_𝛽(𝑥)≤2𝑉_𝛼𝛽(𝑥),𝛾𝛽−𝛼_𝛼𝛽(𝑥)≤𝑉_𝛼(𝑥)≤𝛾_𝛼𝛽(𝑥)+𝑒_𝛼𝛽(𝑥)+𝛼2𝑦_𝛼(𝑥)−𝑦_𝛽(𝑥)², (32)(33) where 𝛾_𝛼𝛽(𝑥)=∑^𝑁_𝑣=1[𝜃_𝑣(𝑥^𝑣,𝑥^−𝑣)−𝜃_𝑣(𝑦_𝛽(𝑥)^𝑣,𝑥^−𝑣)]−(𝛼/2)‖𝑥−𝑦_𝛽(𝑥)‖².

Proof. Inequality (32) follows immediately from (19) in Lemma 6.
The definition of 𝑉_𝛼(𝑥) implies that 𝑉_𝛼(𝑥)≥^𝑁_𝑣=1𝜃_𝑣𝑥^𝑣,𝑥^−𝑣−𝜃_𝑣𝑦_𝛽(𝑥)^𝑣,𝑥^−𝑣−𝛼2𝑥−𝑦_𝛽(𝑥)², (34) which proves the first inequality in (33).
Since 𝑒_𝛼𝛽(𝑥) is the sum of the nonnegative quantity (∑^𝑁_𝑣=1[𝜃_𝑣(𝑦_𝛽(𝑥)^𝑣,𝑥^−𝑣)−𝜃_𝑣(𝑦_𝛼(𝑥)^𝑣,𝑥^−𝑣)]−𝛼(𝑥−𝑦_𝛼(𝑥))^𝑇⋅(𝑦_𝛽(𝑥)−𝑦_𝛼(𝑥))) with another nonnegative quantity (see (23) and (24)), we have 𝑒_𝛼𝛽(𝑥)≥^𝑁_𝑣=1𝜃_𝑣𝑦_𝛽(𝑥)^𝑣,𝑥^−𝑣−𝜃_𝑣𝑦_𝛼(𝑥)^𝑣,𝑥^−𝑣−𝛼𝑥−𝑦_𝛼(𝑥)^𝑇𝑦_𝛽(𝑥)−𝑦_𝛼(𝑥). (35)
Thus, 𝑉_𝛼(𝑥)≤𝛾_𝛼𝛽(𝑥)+𝑒_𝛼𝛽(𝑥)+𝛼2𝑦_𝛼(𝑥)−𝑦_𝛽(𝑥)², (36) which is the second inequality in (33). This completes the proof.

4. 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.

Algorithm 9. Choose an arbitrary initial point 𝑥⁰∈𝑅^𝑛, and any 𝛽₀>𝛼₀>0. Choose any sequences of numbers 𝜀^𝑘>0, 𝜂_𝑘≥0, 𝜆_𝑘∈[0,1), 𝑘=1,2,…, such that lim_𝑘→∞𝜀^𝑘=lim_𝑘→∞𝜂_𝑘1−𝜆_𝑘=0,^∞_𝑘=11−𝜆_𝑘=∞. (37)
For 𝑘=1,2,…, we iterate the following.
Iteration k. Choose any 0<𝛼_𝑘≤(1/2)𝛼_𝑘−1. Choose any 𝛽_𝑘≥2𝛽_𝑘−1 and 𝑥^𝑘∈𝑅^𝑛 satisfying 𝑉_{𝛼𝑘𝛽𝑘}𝑥^𝑘𝛽_𝑘−𝛼_𝑘≤𝜂_𝑘+𝜆_𝑘𝑉_{𝛼𝑘−1𝛽𝑘−1}𝑥^𝑘−1𝛽_𝑘−1−𝛼_𝑘−1. (38)
Apply a descent method to the unconstrained minimization of the function 𝑉_{𝛼𝑘𝛽𝑘}, with 𝑥^𝑘 as the starting point and using 𝑦_𝛼𝑘−𝑦_𝛽𝑘 as a safeguard descent direction at 𝑥, until the method generates an 𝑥∈𝑅^𝑛 satisfying 𝑒_{𝛼𝑘𝛽𝑘}(𝑥)≤𝜀^𝑘. The resulting 𝑥 is denoted by 𝑥^𝑘.

Theorem 10. Assume 𝑋 is bounded. Let {𝑥^𝑘,𝛼_𝑘,𝛽_𝑘,𝜀_𝑘,𝜂_𝑘,𝜆_𝑘}_{𝑘=0,1,2,…} be generated by Algorithm 9. Then {𝑥^𝑘} is bounded; 𝛽_𝑘→∞;𝛼_𝑘→0; and every cluster point of {𝑥^𝑘} is a normalized Nash equilibrium of the GNEP.

Proof. Denote 𝑎^𝑘=𝑉_{𝛼𝑘𝛽𝑘}(𝑥^𝑘)/(𝛽_𝑘−𝛼_𝑘). By (38), we have 𝑎^𝑘≤𝜂_𝑘+𝜆_𝑘𝑎^𝑘−1 for 𝑘=1,2,… and it follows from (37) that 𝑎^𝑘→0 ([16], Lemma 3). For each 𝑘∈{1,2,…}, we have from Lemma 8 that (32) and (33) hold with 𝛼=𝛼_𝑘, 𝛽=𝛽_𝑘, 𝑥=𝑥^𝑘. This together with 𝑒_{𝛼𝑘,𝛽𝑘}(𝑥^𝑘)≤𝜀^𝑘 and ‖𝑦_𝛼𝑘(𝑥^𝑘)−𝑦_𝛽𝑘(𝑥^𝑘)‖≤diam(𝑋) yields 𝑥^𝑘−𝑦_𝛽𝑘𝑥^𝑘≤2𝑎^𝑘,𝛾^𝑘≤𝑉_𝛼𝑘𝑥^𝑘≥𝛾^𝑘+𝜀^𝑘+𝛼_𝑘diam(𝑋)²2, (39) where 𝛾^𝑘=∑^𝑁_𝑣=1[𝜃_𝑣(𝑥^𝑘,𝑣,𝑥^𝑘,−𝑣)−𝜃_𝑣(𝑦^𝑣_𝛽𝑘(𝑥^𝑘),𝑥^𝑘,−𝑣)]−(𝛼/2)‖𝑥^𝑘−𝑦_𝛽𝑘(𝑥^𝑘)‖² and diam(𝑋)=max_{𝑥,𝑦∈𝑋}‖𝑥−𝑦‖. Since 𝑎^𝑘→0, the first inequality in (39) implies {𝑥^𝑘} is bounded. Moreover, this also implies 𝑟^𝑘→0.
Since 𝛼_𝑘→0, the last two inequalities in (39) yield 𝑉_𝛼𝑘(𝑥^𝑘)→0. Since for each 𝑦∈𝑋, we have 𝑉_𝛼𝑘(𝑥^𝑘)≥Ψ(𝑥,𝑦)−(𝛼_𝑘/2)‖𝑥^𝑘−𝑦‖², and this yields 0≥Ψ(𝑥^∞,𝑦) for each cluster point 𝑥^∞ of {𝑥^𝑘}. Thus, each cluster point 𝑥^∞ is a normalized Nash equilibrium of the GNEP. This completes the proof.

Algorithm 11. Choose any 𝑥⁰∈𝑅^𝑛, any 𝛽₀>𝛼₀, and two sequences of nonnegative numbers 𝜌_𝑘,𝜂_𝑘,𝑘=1,2… such that 𝜂_𝑘+𝜌_𝑘>0∀𝑘,^∞_𝑘=1𝜌_𝑘<∞,^∞_𝑘=1𝜂_𝑘<∞. (40)
Choose any continuous function 𝜙∶𝑅₊→𝑅₊ with 𝜙(𝑡)=0⇔𝑡=0. For 𝑘=1,2,…, we iterate the following.
Iteration k. Choose any 0<𝛼_𝑘≤(1/2)𝛼_𝑘−1 and then choose 𝛽_𝑘≥2𝛽_𝑘−1 satisfying 𝑉_{𝛼𝑘𝛽𝑘}𝑥^𝑘−1𝛽_𝑘−𝛼_𝑘≤1+𝜌_𝑘𝑉_{𝛼𝑘−1𝛽𝑘−1}𝑥^𝑘−1𝛽_𝑘−1−𝛼_𝑘−1+𝜂_𝑘. (41)
Apply a descent method to the unconstrained minimization of the function 𝑉_{𝛼𝑘𝛽𝑘}(𝑥^𝑘) with 𝑥^𝑘−1 as the starting point. We assume the descent method has the property that the amount of descent achieved at 𝑥 per step is bounded away from zero whenever 𝑥 is bounded and ‖∇𝑉_{𝛼𝑘𝛽𝑘}(𝑥)‖ is bounded away from zero. Then, either the method in a finite number of steps generates an x satisfying ∇𝑉_{𝛼𝑘𝛽𝑘}(𝑥𝑉)≤𝜙^{𝛼𝑘𝛽𝑘}(𝑥)𝛽_𝑘−𝛼_𝑘, (42) which we denote by 𝑥^𝑘, or else 𝑉_{𝛼𝑘𝛽𝑘}(𝑥) must decrease towards zero, in which case any cluster point of 𝑥 solves the GNEP.

Theorem 12. Assume 𝑋 is bounded. Let {𝑥^𝑘,𝛼_𝑘,𝛽_𝑘,𝜌_𝑘,𝜂_𝑘}_{𝑘=0,1,2,…} be generated by Algorithm 11. (a)Suppose 𝑥^𝑘 is obtained for all 𝑘. Then, {𝑥^𝑘}is bounded; 𝛽_𝑘→∞; 𝛼_𝑘→0, and every cluster point of {𝑥^𝑘} is a normalized Nash equilibrium of the GNEP. (b)Suppose 𝑥^𝑘 is not obtained for some 𝑘. Then, the descent method generates a bounded sequence of 𝑥 with 𝑉_{𝛼𝑘𝛽𝑘}(𝑥)→0 so every cluster point of 𝑥 solves the GNEP.

Proof. (a) Since we use a descent method at iteration 𝑘 to obtain 𝑥^𝑘 from 𝑥^𝑘−1, then 𝑉_{𝛼𝑘𝛽𝑘}(𝑥^𝑘)≤𝑉_{𝛼𝑘𝛽𝑘}(𝑥^𝑘−1), so (41) yields 𝑉_{𝛼𝑘𝛽𝑘}𝑥^𝑘𝛽_𝑘−𝛼_𝑘≤1+𝜌_𝑘𝑉_{𝛼𝑘−1𝛽𝑘−1}𝑥^𝑘−1𝛽_𝑘−1−𝛼_𝑘−1+𝜂_𝑘. (43)
Denote 𝑎^𝑘=𝑉_{𝛼𝑘𝛽𝑘}(𝑥^𝑘)/(𝛽_𝑘−𝛼_𝑘). This can then be written as 𝑎^𝑘≤(1+𝜌_𝑘)𝑎^𝑘−1+𝜂_𝑘 for 𝑘=1,2,…. Using 𝑎^𝑘≥0 and (41), it follows that the sequence {𝑎^𝑘} converges to some 𝑎≥0 ([16], Lemma 2). Since (32) implies 𝑥^𝑘−𝑦_𝛽𝑘𝑥^𝑘≤2𝑎^𝑘,∀𝑘 (44) the sequence {𝑥^𝑘} is bounded.
We claim that 𝑎=0. Suppose the contrary. Then for all 𝑘 sufficiently large, it holds that 𝑎^𝑘≥𝑎/2. Then, 𝑎2≤∑^𝑁_𝑣=1𝜃_𝑣𝑦^𝑣_𝛽𝑘𝑥^𝑘,𝑥^𝑘,−𝑣−𝜃_𝑣𝑦^𝑣_𝛼𝑘𝑥^𝑘,𝑥^𝑘,−𝑣𝛽_𝑘−𝛼_𝑘−𝛼_𝑘/2𝑥^𝑘−𝑦_𝛼𝑘𝑥^𝑘2+𝛽_𝑘/2𝑥^𝑘−𝑦_𝛽𝑘𝑥^𝑘2𝛽_𝑘−𝛼_𝑘 (45)
Since, by the construction of the algorithm, 𝛽_𝑘→∞ and 𝛼_𝑘→0, and {𝑥^𝑘} is bounded (as are 𝑦_𝛼𝑘(𝑥^𝑘) and 𝑦_𝛽𝑘(𝑥^𝑘)), we get 0<𝑎2≤liminf_𝑘→∞𝑥^𝑘−𝑦_𝛽𝑘𝑥^𝑘2. (46)
Then lim_𝑘→∞‖𝑥^𝑘−𝑦_𝛽𝑘(𝑥^𝑘)‖𝛽_𝑘=∞, so lim_𝑘→∞∇𝑉_{𝛼𝑘,𝛽𝑘}𝑥^𝑘=∞. (47)
Since 𝑥^𝑘 satisfies (42), ‖∇𝑉_{𝛼𝑘,𝛽𝑘}(𝑥^𝑘)‖≤𝜙(𝑎^𝑘)for all 𝑘, this contradicts convergence of {𝜙(𝑎^𝑘)} (recall that 𝜙 is a continuous function). Hence, 𝑎=0. For each 𝑘∈{1,2,…}, we have from Lemma 8 that (33) holds with 𝛼=𝛼_𝑘, 𝛽=𝛽_𝑘, 𝑥=𝑥^𝑘 and from the fact 𝑒_𝛼𝛽(𝑥)≥0 that 𝑒_{𝛼𝑘𝛽𝑘}𝑥^𝑘≤𝜀^𝑘∶=∇𝑉_{𝛼𝑘𝛽𝑘}𝑥^𝑘𝑇𝑦_𝛽𝑘𝑥^𝑘−𝑦_𝛼𝑘𝑥^𝑘. (48)
This, together with ‖𝑦_𝛼𝑘(𝑥^𝑘)−𝑦_𝛽𝑘(𝑥^𝑘)‖≤diam(𝑋), yields 𝛾^𝑘≤𝑉_𝛼(𝑥)≤𝛾^𝑘+𝜀^𝑘+𝛼_𝑘diam(𝑋)²2, (49) where 𝛾^𝑘=∑^𝑁_𝑣=1[𝜃_𝑣(𝑥^𝑘,𝑣,𝑥^𝑘,−𝑣)−𝜃_𝑣(𝑦_𝛽𝑘(𝑥^𝑘)^𝑣,𝑥^𝑘,−𝑣)]−(𝛼_𝑘/2)‖𝑥^𝑘−𝑦_𝛽𝑘(𝑥^𝑘)‖² and diam(𝑋)=max_{𝑥,𝑦∈𝑋}‖𝑥−𝑦‖. Since 𝑎^𝑘→0, (44) implies {𝑥^𝑘}is bounded. Moreover, (44) implies 𝛾^𝑘→0. Also, we have ‖∇𝑉_{𝛼𝑘𝛽𝑘}(𝑥^𝑘)‖≤𝜙(𝑎^𝑘)→0, so𝜀^𝑘→0.
From the facts that 𝛼_𝑘→0, (49), 𝛾^𝑘→0 and 𝜀^𝑘→0, we get 𝑉_𝛼𝑘(𝑥^𝑘)→0. Since for each 𝑦∈𝑋, we have from the definition of 𝑉_𝛼(𝑥) that 𝑉_𝛼𝑘𝑥^𝑘−𝛼≥Ψ𝑥,𝑦^𝑘2𝑥^𝑘−𝑦², (50) which yields 0≥Ψ(𝑥^∞,𝑦) for each cluster point 𝑥^∞ of {𝑥^𝑘}. Thus, each cluster point 𝑥^∞ is a normalized Nash equilibrium of the GNEP.
(b) It is easy to proof that 𝑉_{𝛼𝑘𝛽𝑘}(𝑥)→0. Hence 𝑥^∞ is a normalized Nash equilibrium of the GNEP.
The proof is completed.

Acknowledgments

This research was partly supported by the National Natural Science Foundation of China (11271226, 10971118) and the Promotive Research Fund for Excellent Young and Middle-Aged Scientists of Shandong Province (BS2010SF010).