A variational inequality method for computing a normalized equilibrium in the generalized Nash game

In this article, We consider the generalized Nash equilibrium problem (GNEP). To this end, we first recall the definition of the Nash equilibrium problem (NEP). There are N players, each player ν ∈ {1,...,N} controls the variables $x^{\nu }\in {\Re }^{n_{\nu }}$. All players' strategies are collectively denoted by a vector $x={\left(x^1,...,x^N\right)}^T\in {\Re }^n$, where n = n₁ + ⋯ + n _N . To emphasize the ν th player's variables within the vector x, we sometimes write x = (x ^ν , x^-ν) ^T , where $x^{-\nu }\in {\Re }^{n_{-\nu }}$ subsumes all the other players' variables.

Such a vector x* is called a generalized Nash equilibrium or, more simply a solution of the GNEP.

for some function $g:{\Re }^n\to {\Re }^m$. Additional equality constraints are also allowed, but for notational simplicity, we prefer not to include them explicitly. In many cases, a player ν might have some additional constraints depending on his decision variables only. However, these additional constraints can be viewed as part of the joint constraints g(x) ≤ 0, so, we include these latter constraints in the former ones.

Throughout this article, we make the following blanket assumptions.

Assumption 1.1 (i) The utility functions θ ^ν are twice continuously differentiable and as a function of x ^ν along, convex.

(ii) The function g is twice continuously differentiable, its components g _i are convex (in x), and the corresponding strategy space X defined by (1.2) is nonempty.

The convexity assumptions are standard in the context of GNEPs. The smoothness assumptions are also very natural since our aim is to develop locally fast convergent methods for the solution of GNEPs.

The GNEP was formally introduced by Debreu [1] as early as 1952, but it is only from the mid-1990s 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 (e.g., see [2–4]). Another approach for solving the GNEP is based on the Nikaido-Isoda function. Relaxation methods and proximal-like methods using the Nikaido-Isoda function are investigated in [5–7]. A regularized version of the Nikaido-Isoda function was first introduced in [8] for standard NEPs then further investigated by Heusinger and Kanzow [9], they reformulated the GNEP as a constrained optimization problem with continuously differentiable objective function.

Motivated by the fact that a standard NEP can be reformulated as a variational inequality problem (VI for short), see, for example, [10, 11], Harker [12] characterized the GNEP as a quasi-variational 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. On the other hand, it was noted in [13], for example, that certain solutions of the GNEP (the normalized Nash equilibria, to be defined later) can be found by solving a suitable standard VI associated to the GNEP.

Here, we further investigate the properties of the normalized Nash equilibria. The rest of the article is organized as follows. Section 2 gives some preliminaries. In Section 3, we use the fact that the normalized Nash equilibria can be found by solving a suitable VI, we reformulate the VI associated to the GNEP as a semismooth system of equations and the nonsingularity of the B-subdifferential for the system is explored. Finally, in Section 4, we implement a semismooth Newton method to some examples of the GNEP.

We use the following notations throughout the article. A function $G:{\Re }^n\to {\Re }^t$ is called a C ^k -function if it is k times continuously differentiable. For a differentiable function $g:{\Re }^n\to {\Re }^m$, the Jacobian of g at $x\in {\Re }^n$ is denoted by $\mathcal{J}g\left(x\right)$, and its transposed by ∇g(x). Given a differentiable function $\Psi :{\Re }^n\to \Re$, the symbol ${\nabla }_{x^{\nu }}\Psi \left(x\right)$ denotes the partial gradient with respect to x ^ν -part only, and ${\nabla }_{x^{\nu }{x}^{\mu }}^2\Psi \left(x\right)$ denotes the second-order partial derivative with respect to x ^ν -part and x ^μ -part. For a function $f:{\Re }^n\times {\Re }^n\to \Re ,f\left(x,\cdot \right):{\Re }^n\to \Re$ denotes the function with x being fixed. For vectors $x,y\in {\Re }^n,⟨x,y⟩$ denotes the inner product defined by ⟨x,y⟩ := x ^T y and x ⊥ y means ⟨x,y⟩ = 0.

is Clarke's generalized Jacobian of F at x (see [15]).

Based on this notation, we next recall the definition of a semismooth function. This concept was firstly introduced by Mifflin [16] for real-valued mappings and extended by Qi and Sun [17] to vector-valued mappings.

In the study of algorithms for locally Lipschitzian systems of equations, the following regularity condition plays a role similar to that of the nonsingularity of the Jacobian in the study of algorithms for smooth systems of equations.

Definition 2.2 Let$G:{\Re }^n\to {\Re }^n$be Lipschitzian around x, G is said to be BD-regular at x if all the elements in ∂ _B G(x) are nonsingular. If$\bar{x}$is a solution of the system G(x) = 0 and G is BD-regular at$\bar{x}$, then$\bar{x}$is called a BD-regular solution of this system.

we state a result due to [13] which will be used later.

Lemma 2.1 Suppose that the GNEP satisfies Assumption 1.1 and assume further that the sets X _ν (x^-ν) are defined by (1.1) with X closed and convex. Then, every solution of the VI(F, X) is a solution of the GNEP.

Consider the GNEP from Section 1 with utility functions θ ^ν and a strategy set X satisfying the requirements from Assumption 1.1. In this section, our aim is to show that the GNEP can be reformulated as a nonsmooth equation and then we present several conditions guaranteeing the BD-regularity condition of the equation.

are satisfied.

The next lemma from [13] relates the normalized Nash equilibria to the KKT conditions (3.3).

Lemma 3.1 (i) Let x be a solution of VI(F,X) at which the KKT conditions (3.3) hold. Then x is a solution of the GNEP (normalized Nash equilibria) at which the KKT conditions (3.1) hold with λ¹ = λ² = ⋯ = λ ^N = λ.

(ii) Viceversa, let x be a solution of the GNEP at which KKT conditions (3.1) hold with λ¹ = λ² = ⋯ = λ ^N . Then x is a solution of VI(F, X).

From Assumption 1.1, we know that Φ is semismooth.

In the following, our aim is to present several conditions guaranteeing that all elements in the generalized Jacobian ∂Φ(ω) (and hence in the B-subdifferential ∂ _B Φ(ω)) are nonsingular. Our first result gives a description of the structure of the matrices in the generalized Jacobian ∂Φ(ω).

for any μ _i ∈ [0,1].

This gives the representation of H ∈ ∂Φ(ω) ^T .

Our next aim is to establish conditions guaranteeing that all elements in the generalized Jacobian ∂Φ(ω) at a point ω = (x,λ) satisfying Φ(ω) = 0 are nonsingular.

It holds that (a) ⇒ (b).

This together with Δx₁ = 0 shows that the matrix (3.6) is nonsingular, and then, H is nonsingular.

Now, we are able to apply Theorem 3.1 to some classes of GNEPs.

where$f^{\nu }:{\Re }^{n_{\nu }}\to \Re$is stongly convex and$h^{\nu }:{\Re }^{n-n_{\nu }}\to \Re$. Assume that LICQ holds at x*. Then all elements H ∈ ∂Φ(ω*) are nonsingular.

By the strong convexity of f ^ν , we can conclude that ∇F(x*) is positive definite.

is positive definite. Thus, the strong second-order sufficient condition for the VI(F, X) holds at x*. From Theorem 3.1, we obtain any element in ∂Φ(ω*) is nonsingular.

is positive definite. Then all the elements in the generalized Jacobian ∂Φ(ω*) are nonsingular.

is positive semidefinite. Hence, we obtain that ∇ _x L(ω*) is positive definite. The statement therefore follows from Theorem 3.1.

A simple Armijo-type line search is used in the algorithm and we switch to the steepest direction whenever the generalized Newton direction is not computable or does not satisfy a sufficient decrease condition.

Algorithm 4.1

Step 0 Choose ${\omega }^0=\left(x^0,{\lambda }^0\right)\in {\Re }^{n+m},\rho >0,\kappa >2,\sigma \in \left(0,\frac{1}{2}\right),\beta \in \left(0,1\right),\epsilon \ge 0$, and set k = 0.

Step 1 If ∥∇Ψ(ω ^k )∥ ≤ ε, stop.

Step 4 Set ω^k+1= ω ^k + t ^k d ^k , k = k + 1 and go to step 1.

The following result about the convergence property of Algorithm 4.1 comes from [18] directly.

Theorem 4.1 Assume that Algorithm 4.1 does not terminate within a finite number of iterations, let {ω ^k } be generated by Algorithm 4.1 having an accumulation point ω*, then ω* is a stationary point of Ψ. Moreover, if ω* is a BD-regular solution of the system Φ(ω) = 0, then {ω ^k } convergence to ω* Q-superlinearly.

with constraints x ^v ≥ 0.01, ν = 1,..., N and${\sum }_{\nu =1}^N{x}^{\nu }\le B$. According to[20], we also set N = 10, B = 1 and use the starting point$x^0={\left(0.1,0.1,0.1,...,\right)}^T\in {\Re }^{10}$. The exact solution of this problem is x* = (0.09,0.09,..., 0.09) ^T . We only state the first three components of the iteration vectors in Table1.

be positive definite, The strategy space X is defined by some linear constraints. For convenience, we set the elements of x⁰ all 1. The elements of λ⁰ all 0. We set other parameters in the algorithm as ρ = 10^-8, κ = 2.1, σ = 10^-4, β = 0.55. Our numerical results are reported in Table 3, where Iter., Func, Res0. and Res*. stand for, respectively, the number of iterations, the number of function evaluations, the residual ∥∇Ψ(·)∥ at the starting point and the residual ∥∇Ψ(·)∥ at the final iterate of implementation.