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A smoothing method for a class of generalized Nash equilibrium problems
Journal of Inequalities and Applications volumeÂ 2015, ArticleÂ number:Â 90 (2015)
Abstract
The generalized Nash equilibrium problem is an extension of the standard Nash equilibrium problem where both the utility function and the strategy space of each player depend on the strategies chosen by all other players. Recently, the generalized Nash equilibrium problem has emerged as an effective and powerful tool for modeling a wide class of problems arising in many fields and yet solution algorithms are extremely scarce. In this paper, using a regularized NikaidoIsoda function, we reformulate the generalized Nash equilibrium problem as a mathematical program with complementarity constraints (MPCC). We then propose a suitable method for this MPCC and under some conditions, we establish the convergence of the proposed method by showing that any accumulation point of the generated sequence is a Mstationary point of the MPCC. Numerical results on some generalized Nash equilibrium problems are reported to illustrate the behavior of our approach.
1 Introduction
This paper considers the generalized Nash equilibrium problem with jointly convex constraints (GNEP). To be more specific, let us now give formal definitions of the standard Nash equilibrium problem (NEP) and the GNEP. We assume there are N players, each player \(\nu\in\{1,\ldots,N\}\) controls the variables \(x^{\nu}\in\Re^{n_{\nu}}\) and \(x=(x^{1},\ldots,x^{N})^{T}\in \Re^{n}\) with \(n=n_{1}+\cdots+n_{N}\) denotes the vector comprised of all these decision variables. To emphasize the Î½th playerâ€™s variables within the vector x, we sometimes write \(x=(x^{\nu},x^{\nu})^{T}\), where \(x^{\nu}\) subsumes all the other playersâ€™ variables. We will also write \(n_{\nu}=nn_{\nu}\). Moreover, for both NEPs and GNEPs, let \(\theta ^{\nu}:\Re^{n}\rightarrow\Re\) be the Î½th playerâ€™s payoff (or loss or utility) function.
For a NEP, there is a separate strategy set \(X^{\nu}\subseteq\Re^{n_{\nu}} \) for each player Î½. Let
be the Cartesian product of the strategy sets of all players, then a vector \(x^{*}\in X\) is called a Nash equilibrium, or a solution of the NEP, if each block component \(x^{*,\nu}\) is a solution of the optimization problem
i.e., \(x^{*}\) is a Nash equilibrium if no player can improve his situation by unilaterally changing his strategy.
On the other hand, in a GNEP, there is a common strategy set \(X\subseteq\Re^{n}\) for all players, and a vector \(x^{*}=(x^{*,1},\ldots,x^{*,N})^{T}\in\Re^{n}\) is called a generalized Nash equilibrium or a solution of the GNEP if each block component \(x^{*,\nu}\) is a solution of the optimization problem
If X has the Cartesian product structure as (1.1), then a GNEP reduces to a NEP. Throughout this paper, we assume that X can be represented as
for some functions \(g:\Re^{n} \rightarrow\Re^{m}\). Note that usually, a player Î½ might have some additional constraints of the form \(h^{\nu}(x^{\nu})\leq0\) depending on his decision variables only. However, these additional constraints can be viewed as part of the joint constraints \(g(x)\leq0\), with some of the component functions \(g_{i}\) of g depending on the block component \(x^{\nu}\) of x only. So, we include these latter constraints in the former ones.
The GNEP was formally introduced by Debreu [1] as early as 1952, but it is only from the mid1990s that the GNEP attracted much attention because of its capability of modeling a number of interesting problems in economy, computer science, telecommunications and deregulated markets (for example, see [2â€“4]). Motivated by the fact that a NEP can be reformulated as a variational inequality problem (VI); see, for example, [5, 6], Harker [7] characterized the GNEP as a quasivariational inequality (QVI). But unlike VI, there are few efficient methods for solving QVI, and therefore such a reformulation is not used widely in designing implementable algorithms. The idea of using an exact penalty approach to the GNEP was proposed by Facchinei and Pang [8] and Facchinei and Kanzow [9], but the disadvantage of this method is that a nondifferentiable NEP has to be solved to obtain a generalized Nash equilibrium.
Another approach for solving the GNEP is based on the NikaidoIsoda function. Relaxation methods and proximallike methods using the NikaidoIsoda function are investigated in [10â€“12]. A regularized version of the NikaidoIsoda function was first introduced in [13] for NEPs then further investigated by Heusinger and Kanzow [14], they reformulated the GNEP as a constrained optimization problem with continuously differentiable objective function.
Motivated by [14], in this paper, we further reformulate the GNEP as a MPCC. Moreover, we propose a smoothing method to this problem and give some suitable conditions for the convergence of the proposed method. The organization of the paper is as follows. In the next section, we recall some preliminaries and basic facts and definitions. In SectionÂ 3, we give details of our optimization reformulation of the GNEP and discuss the convergence properties of our method. Finally, in SectionÂ 4, we present some numerical results.
We use the following notations throughout the paper. For a differentiable function \(g:\Re^{n}\rightarrow\Re^{m}\), the Jacobian of g at \(x\in\Re^{n}\) is denoted by \(\mathcal{J}g(x)\), and it is transposed by \(\nabla g(x)\). Given a differentiable function \(\Psi:\Re^{n} \rightarrow\Re\), the symbol \(\nabla_{x^{\nu}} \Psi (x)\) denotes the partial gradient with respect to \(x^{\nu}\)part only. For a function \(f:\Re^{n}\times\Re^{n}\rightarrow \Re\), \(f(x,\cdot):\Re^{n}\rightarrow\Re\) denotes the function with x being fixed. For vectors \(x,y\in\Re^{n}\), \(\langle x,y \rangle\) denotes the inner product defined by \(\langle x,y \rangle:=x^{T}y\) and \(x\perp y\) means \(\langle x,y \rangle=0\). Finally, throughout the paper, \(\\cdot\\) denotes the Euclidean vector norm.
2 Preliminaries
Throughout this paper, we make the following blanket assumptions.
Assumption 2.1

(i)
The utility functions \(\theta^{\nu}\) are twice continuously differentiable and as a function of \(x^{\nu}\) along, \(\theta^{\nu}\) are convex.

(ii)
The function g is twice continuously differentiable, its components \(g_{i}\) are convex (inÂ x), and the corresponding strategy space X defined by (1.2) is nonempty.
Note that the convexity assumptions are absolutely standard setting under which the GNEP is usually investigated in the literature, and AssumptionÂ 2.1(ii) implies that the strategy set \(X\subseteq\Re^{n}\) is nonempty, closed, and convex. An important tool for both NEPs and GNEPs is the NikaidoIsoda function (NI function for short) \(\Psi:\Re^{n} \times\Re^{n} \rightarrow\Re\),
In particular, the NI function provides an important subset of all the solutions of a GNEP.
Definition 2.1
A vector \(x^{*}\in X\) is called a normalized Nash equilibrium of the GNEP if
However, the supremum in (2.1) may not be attained, or it may be attained at more than one point. In order to overcome these disadvantages, in [14] authors provided a regularized version of the NI function. Let \(\alpha>0\) be a given parameter that is assumed to be fixed throughout this paper. The regularized NI function is given by
We now define the corresponding value function by
where \(y_{\alpha}(x)\) denotes the unique solution of the uniformly concave maximization problem
As noted in [14], the function \(V_{\alpha}\) is continuously differentiable with gradient given by
and \(x^{*}\) is a normalized Nash equilibrium of the GNEP if and only if it solves the constrained optimization problem
with optimal function value \(V_{\alpha}(x^{*})= 0\).
3 Problem reformulation and a smoothing method
We now use the regularized NI function in order to obtain a MPCC reformulation of the GNEP.
Based on (2.3), \(x^{*}\) is a normalized Nash equilibrium of the GNEP if and only if \(x^{*}\) solves the following optimization problem:
We consider Problem (2.2). For every \(x\in X\), let the linear independence constraint qualification (LICQ) hold at \(y_{\alpha}(x)\), then by AssumptionÂ 2.1, \(y_{\alpha}(x)\) is a solution of (2.2) if and only if \(y_{\alpha}(x)\) satisfies
where \(\lambda_{\alpha}(x)\) is a Lagrangian multiplier. Thus, Problem (3.1) is equivalent to
This problem is a MPCC.
Now, we can easily get the following result as regards the normalized Nash equilibrium of the GNEP and the solution of the MPCC.
Proposition 3.1
For every \(x\in X\), let the LICQ hold at \(y_{\alpha}(x)\), then \(x^{*}\) is a normalized Nash equilibrium if and only if there exists a vector \((y^{*},\lambda^{*})\) such that \((x^{*},y^{*},\lambda^{*})\) is a solution of Problem (3.2).
Let \(z=(x,y,\lambda)\), \(f(z)=\Psi_{\alpha}(x,y)\), \(h(z)=\nabla_{y} \Psi_{\alpha}(x,y) \nabla g(y)\lambda\), \(\bar{g}(z)=g(x)\), \(G(z)=\lambda\), and \(H(z)=g(y)\), we rewrite (3.2) more compactly as
Define the Lagrangian of (3.3) as
and the index sets of active constraints as
the MPCCLICQ for (3.3) at a feasible point \(\bar{z}\) says that the following vectors:
are linearly independent.
We next consider two simple GNEPs which show that the MPCCLICQ for (3.3) holds at a solution \(z^{*}\).
Example 3.1
Consider the GNEP with \(N=2\), \(X=\{x\in\Re^{2}\mid x^{1}\geq1, x^{2}\geq1\}\), and payoff functions \(\theta^{1}(x)=x^{1}x^{2}\) and \(\theta^{2}(x)=x^{2}\). Now let us consider \(y_{\alpha}(x)\), the unique solution of
An elementary calculation shows that
Furthermore, we get
We see that \(x^{*}=(1,1)\) is the normalized Nash equilibrium and \(z^{*}=(1,1,1,1,1,1)\) is a solution of (3.3). It is easy to compute that
Moreover, we can see \(\nabla h_{i}(z^{*})\), \(i=1,2\), \(\nabla \bar{g}_{i}(z^{*})\), \(i=1,2\), \(\nabla H_{i}(z^{*})\), \(i=1,2\), are linearly independent, hence the MPCCLICQ holds at \(z^{*}\).
Example 3.2
Consider the GNEP with two players:
The regularized NI function is
and \(X=\{x\mid 1x^{1}\leq0, 1x^{2}\leq0, x^{1}+x^{2}10\leq0\}\). It can be seen that \(x^{*}=(1,9)\) is the unique normalized Nash equilibrium and
is the solution of (3.3). Moreover, we have
Obviously, \(\nabla h_{i}(z^{*})\), \(i=1,2\), \(\nabla\bar{g}_{i}(z^{*})\), \(i=1,3\), \(\nabla H_{i}(z^{*})\), \(i=1,3\), \(\nabla G_{i}(z^{*})\), \(i=2\), are linearly independent, hence the MPCCLICQ for (3.3) holds at \(z^{*}\).
In the study of MPCCs, there are several kinds of stationarity defined for Problem (3.3).
Definition 3.1

(1)
A feasible point \(\bar{z}\) of (3.3) is called a critical point if there exist multipliers \(\bar{\mu}\), \(\bar{\eta}\), \(\bar{\xi}^{G}\), and \(\bar{\xi}^{H}\) such that
$$ \begin{aligned} &\nabla_{z}L\bigl(\bar{z}, \bar{\mu},\bar{\eta},\bar {\xi}^{G},\bar{\xi}^{H}\bigr)=0, \\ &\bar{\eta}\geq0,\qquad \bar{\eta}^{T}\bar{g}({\bar {z}})=0, \\ &\bar{\xi}_{i}^{G}=0, \quad \mbox{if }i\notin I_{G}(\bar{z}), \\ &\bar{\xi}_{i}^{H}=0,\quad \mbox{if }i\notin I_{H}(\bar{z}). \end{aligned} $$(3.4) 
(2)
Clarke (C)stationarity: \(\bar{\eta}_{i}\geq0\) and \(\bar{\xi}_{k}^{G}\bar{\xi}_{k}^{H}\geq0\) for all \(k\in I_{G}(\bar{z})\cap I_{H}(\bar{z})\).

(3)
Mordukhovich (M)stationarity: \(\bar{\eta}_{i}\geq0\) and either \(\bar{\xi}_{k}^{G}, \bar{\xi}_{k}^{H}>0\) or \(\bar{\xi}_{k}^{G}\bar{\xi}_{k}^{H}= 0\) for all \(k\in I_{G}(\bar{z})\cap I_{H}(\bar{z})\).
We now propose our smoothing method for (3.3). This method is similar to one given in [15] which, however, uses a different reformulation of the complementarity constraints. Let
and \(\epsilon>0\) is the smoothing parameter. We have
and
By the definition of the function Ï• and the calculation formulas for its first and secondorder partial derivatives, we can easily obtain the following properties of Ï•.
Lemma 3.1
Let \((a,b,\epsilon)\) satisfy \(\phi(a,b,\epsilon)=0\) and \(\epsilon >0\).

(i)
We have
$$\begin{aligned} \begin{aligned} &\frac{ \partial}{ \partial a}\phi(a,b,0)= 0, \quad \textit{if } a>0=b, \\ &\frac{ \partial}{ \partial b}\phi(a,b,0)= 0, \quad \textit{if } a=0< b. \end{aligned} \end{aligned}$$ 
(ii)
Let \((a^{k},b^{k})\rightarrow(0,0)\) as \(\epsilon ^{k}\rightarrow 0^{+}\) with \(\phi(a^{k},b^{k},\epsilon^{k})=0\). If
$$\lim_{k\rightarrow\infty}\frac{\frac{ \partial}{ \partial a}\phi(a^{k},b^{k},\epsilon ^{k})}{\frac{ \partial}{ \partial b}\phi (a^{k},b^{k},\epsilon^{k})}\rightarrow r>0, $$we have
$$\bigl(V^{k}\bigr)H^{k}\bigl(V^{k} \bigr)^{T}\rightarrow\infty, \quad \textit{as }k\rightarrow \infty, $$where \(V^{k}=(\frac{ \partial\phi}{ \partial b}, \frac{ \partial\phi}{ \partial a})\) and \(H^{k}\) is the Hessian of Ï• with respect to a and b evaluated at \((a^{k},b^{k}, \epsilon^{k})\).
Now, we consider the following problem with \(\epsilon>0\):
where \(\Phi^{\epsilon} (z)= (\phi(G_{1}(z),H_{1}(z),\epsilon),\ldots,\phi (G_{m}(z),H_{m}(z),\epsilon))^{T}\). We recall that \(z^{\epsilon}\) is stationary for (3.5) if it is feasible and there exist Lagrangian multiplier vectors \(\mu^{\epsilon}\in\Re^{n}\), \(\eta^{\epsilon}\in\Re^{m}\), and \(\xi^{\epsilon}\in\Re^{m}\) satisfying
where the Lagrangian function is
A stationary point \(z^{\epsilon}\) with Lagrangian multipliers \(\mu ^{\epsilon}\), \(\eta^{\epsilon}\), \(\xi^{\epsilon}\) of (3.5) is said to satisfy a secondorder necessary condition (SONC) if
for any d in the critical cone,
We need a slightly weaker condition that we call the weak secondorder necessary condition (WSONC), which requires the positive semidefiniteness of \(\nabla^{2}_{zz} L(z^{\epsilon},\mu^{\epsilon},\eta^{\epsilon},\xi^{\epsilon})\) on the critical subspace
Now, we state a convergence result for the smoothing method (3.5).
Theorem 3.1
Let \(\{z^{k},\mu^{k},\eta^{k},\xi^{k}\}\) be a KarushKuhnTucher (KKT) point of (3.5) for each \(\epsilon=\epsilon^{k}\), where \(\epsilon^{k}\rightarrow0^{+}\). Suppose that \(\bar{z}\) is a limit point of \(\{z^{k}\}\) and the MPCCLICQ holds at \(\bar{z}\) for (3.3). Then

(i)
\(\bar{z}\) is a Cstationary point of (3.3);

(ii)
if WSONC holds for (3.5) at each \(z^{k}\), then \(\bar{z}\) is a Mstationary point of (3.3).
Proof
By taking a subsequence if necessary, we assume that \(z^{k}\rightarrow \bar{z}\), and it is easy to see that \(\bar{z}\) is feasible for (3.3). To simplify notation, in the following, we denote
First, we show that \(\bar{z}\) is a critical point of (3.3). The gradient equation of the KKT system for (3.5) at \(z^{k}\) is
Equation (3.6) can be equivalently expressed as
Let \(r_{i}^{k}=\xi_{i}^{k}\frac{ \partial \phi_{i}}{ \partial a}\) and \(v_{i}^{k}=\xi_{i}^{k}\frac{ \partial \phi_{i}}{ \partial b}\), we show that \(\lim_{k\rightarrow\infty}r_{i}^{k}\) exists and \(r_{i}^{k}\rightarrow0\) if \(i\notin I_{G}(\bar{z})\). Let \(i\notin I_{G}(\bar{z})\), then \(i\in I_{H}(\bar{z})\) by the feasibility of \(\bar{z}\). We assume that there exist a positive number \(\bar{\alpha} >0\) and a subsequence (we denote the subsequence by the sequence itself for the sake of notational simplicity) such that \( r_{i}^{k} \geq\bar{\alpha}\) for sufficiently large k. Since
then \(\lim_{k\rightarrow\infty}  \xi_{i}^{k} =+\infty\). Let \(\beta^{k}:=\(\mu^{k},\eta^{k},\xi^{k})\\), then \(\beta^{k} \rightarrow+\infty\). It is not difficult to obtain for sufficiently large k, \(I_{\bar{g}}(z^{k}) \subseteq I_{\bar {g}}(\bar{z})\). Dividing (3.7) by \(\beta^{k}\), and taking any limit point \((\tilde{\mu},\tilde{\eta},\tilde{r},\tilde{v})\) of \((\mu^{k},\eta^{k},r^{k},v^{k})/\beta^{k} \) yields \((\tilde{\mu},\tilde{\eta},\tilde{r},\tilde {v})\neq 0\) and
Equation (3.8) contradicts the MPCCLICQ at \(\bar{z}\). Therefore, \(\lim_{k\rightarrow\infty}r_{i}^{k}=0\), for \(i\notin I_{G}(\bar{z})\). In the same way, we can also prove that \(\lim_{k\rightarrow\infty}v_{i}^{k}=0\), for \(i\notin I_{H}(\bar{z})\). Furthermore, \(\{\mu_{i}^{k}\}_{i=1}^{n}\), \(\{\eta_{i}^{k}\}_{i\in I_{\bar{g}}(\bar{z})}\), \(\{r_{i}^{k}\}_{i\in I_{G}(\bar{z})}\), and \(\{v_{i}^{k}\}_{i\in I_{H}(\bar{z})}\) are bounded. Otherwise, dividing (3.7) by \(\beta^{k}\) and taking the limit will lead to a contradiction to the MPCCLICQ at \(\bar{z}\) as done above. Due to the MPCCLICQ at \(\bar{z}\), Let \((\bar{\mu},\bar{\eta},\bar{r},\bar{v})\) denote the unique limit of \((\mu^{k},\eta^{k},r^{k},v^{k})\), with \(r^{k}=(r_{1}^{k},\ldots,r_{m}^{k})^{T}\) and \(v^{k}=(v_{1}^{k},\ldots,v_{m}^{k})^{T}\), we can see, \((\bar{z},\bar{\mu},\bar{\eta},\bar {r},\bar{v})\) satisfies (3.4), so, \(\bar{z}\) is a critical point of (3.3).
Note that, for any \(i\in I_{G}(\bar{z})\cap I_{H}(\bar{z})\),
the Cstationarity of \(\bar{z}\) follows.
For the Mstationarity of \(\bar{z}\). If \(\bar{z}\) is not a Mstationary point of (3.3) which means that there exists at least one index, denoted by \(l\in I_{G}(\bar{z})\cap I_{H}(\bar{z})\) such that \(\bar{r}_{l}>0\) and \(\bar{v}_{l}>0\), so we have \(\xi_{l}^{k}>0\) and away from zero for sufficiently large k. First, it is easy to see that
Second, by the MPCCLICQ at \(\bar{z}\), we have
has full row rank. Then for sufficiently large k,
is full row rank also. Therefore there exists \(d^{k}\) which is bounded and satisfies
It is easy to see that \(d^{k}\) is in the critical subspace \(\operatorname{lin} C(z^{k})\) of Problem (3.5) at \(z^{k}\), and
We know \((d^{k})^{T}\{\nabla^{2}f(z^{k})+ \sum_{i=1}^{n}\mu_{i}^{k}\nabla^{2}h_{i}(z^{k}) + \sum_{i=1}^{m}\eta_{i}^{k}\nabla^{2}\bar{g}_{i}(z^{k}) \}d^{k}\), \((d^{k})^{T}( \sum_{i=1}^{m}\xi_{i}^{k}\frac{ \partial\phi _{i}}{ \partial a}\times \nabla^{2}G_{i}(z^{k}))d^{k} \) and \((d^{k})^{T}( \sum_{i=1}^{m}\xi_{i}^{k}\frac{ \partial\phi_{i}}{ \partial b} \nabla^{2}H_{i}(z^{k}))d^{k}\) are bounded, and
For sufficiently large k, we have \(\xi_{l}^{k}>0\) and (3.9) is equal to
which tends to âˆ’âˆž by LemmaÂ 3.1. This contradicts that \(z^{k}\) satisfies WSONC. The Mstationarity of \(\bar{z}\) follows.â€ƒâ–¡
4 Numerical experiments
We have tested the method on various examples of the GNEP. We applied MATLAB 7.0 builtin solver function fmincon to solve the nonlinear programs for positive Ïµvalues. The computational results are summarized in TablesÂ 1, 2, 46, 8, which indicate that the proposed method produces good approximate solutions.
Example 4.1
This problem is taken from [16]. There are two players, each player Î½ has a onedimensional decision variable \(x^{\nu} \in \Re\). The optimization problems of the two players are given by
This problem has infinitely many solutions \(\{(\alpha,1\alpha)\mid \alpha\in[0.5,1]\}\), but has only one normalized Nash equilibrium at \(\bar{x}=( \frac{3}{4}, \frac{1}{4})^{T}\). TableÂ 1 is for the corresponding numerical results.
Example 4.2
This is a duopoly model with two players taken from [10]. Each player Î½ controls one variable \(x^{\nu} \in\Re\). The payoff functions are given by
and the constraints are given by
where \(d=20\), \(\lambda=4\), \(\bar{\rho} =1\).
Example 4.3
This example is a river basin pollution game also taken from [10]. There are three players, each player controls a single variable \(x^{\nu}\in{\Re}\). The objective functions are given by
for \(\nu=1,2,3\), and the constraints are
The economic constants \(d_{1}\) and \(d_{2}\) determine the inverse demand law and set to 3.0 and 0.01, respectively. Values for constants \(c_{1,\nu}\), \(c_{2,\nu}\), \(e_{\nu}\), \(\mu_{\nu,1}\), and \(\mu_{\nu,2}\) are given in TableÂ 3, and \(K_{1}=K_{2}=100\).
Example 4.4
This test problem is an Internet switching model introduced by Kesselman et al. [17]. There are N players, the cost function of each player is given by
with constraints \(x^{\nu}\geq0.01\), \(\nu=1,\ldots,N\), and \(\sum_{\nu=1}^{N}x^{\nu}\leq B\). We set \(N=10\), \(B=1\). The exact solution of this problem is \(x^{*}=(0.09,0.09,\ldots,0.09)^{T}\). We only state the first three components of x in TableÂ 5.
Example 4.5
Let us consider the following GNEP. There are two players in the game, where player 1 controls a twodimensional variable \(x^{1}=(x_{1},x_{2})^{T}\in\Re^{2}\) and player 2 controls a onedimensional variable \(x^{2}=x_{3}\in\Re\). The problem is
The problem has infinitely many solutions given by
but it has only one normalized Nash equilibrium at \(\alpha=0\).
Example 4.6
This GNEP from [18] is the electricity market problem. This model has three players, player 1 controls a single variable \(x^{1}\in\Re\), player 2 controls a twodimensional vector \(x^{2}=(x_{1}^{2},x_{2}^{2})\), and player 3 controls a threedimensional decision variable \(x^{3}=(x_{1}^{3},x_{2}^{3},x_{3}^{3})\). Let
The utility functions are given by
where \(\psi(x)=2(x_{1}+\cdots+x_{6})378.4\) and the constants \(c_{i}\), \(d_{i}\), \(e_{i}\) are given in TableÂ 7.
The constraints are
TableÂ 8 is for the corresponding numerical results.
The numerical experiments show that the method proposed in this paper is implementable for solving GNEPs with jointly convex constraints.
5 Remarks
The main idea of this paper is to try use a smoothing method to solve the GNEP. Based on the regularized NikaidoIsoda function, we reformulate the set of normalized Nash equilibria, which is a subset of the generalized Nash equilibria, as solutions of a MPCC and we solve the MPCC by a smoothing method. There are some problems as regards the smoothing method worth further investigating:

(i)
In this paper, some conditions are given to establish the convergence of the smoothing method by showing that any accumulation point of the generated sequence is a Mstationary point of the MPCC. For the next step, less strict assumptions than TheoremÂ 3.1 to obtain the results of TheoremÂ 3.1 are worth considering.

(ii)
Based on the special structure of the MPCC defined in (3.3), can we derive convergence results tailored to the GNEP, which may possibly be stronger than those known for the MPCC? This problem is worth studying.
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Acknowledgements
The research was supported by the Technology Research plan of Inner Mongolia under project Nos. 20130603 and 20120812.
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JH and JFL carried out the design of the study and performed the analysis. ZCW participated in its design and coordination. All authors read and approved the final manuscript.
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Lai, JF., Hou, J. & Wen, ZC. A smoothing method for a class of generalized Nash equilibrium problems. J Inequal Appl 2015, 90 (2015). https://doi.org/10.1186/s1366001506076
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DOI: https://doi.org/10.1186/s1366001506076