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 BDregularity condition of the equation.
Suppose that x is a solution of the GNEP. Then if for player ν, a suitable constraint qualification (like the slater condition) holds, it follows that there exists a Lagrange multiplier {\lambda}^{\nu}\in {\Re}^{m} such that the KarushKuhnTucker (KKT) conditions
\begin{array}{l}{\nabla}_{{x}^{\nu}}{\theta}^{\nu}\left({x}^{\nu},{x}^{\nu}\right)+{\nabla}_{{x}^{\nu}}g\left({x}^{\nu},{x}^{\nu}\right){\lambda}^{\nu}=0,\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}0\le {\lambda}^{\nu}\perp g\left({x}^{\nu},{x}^{\nu}\right)\ge 0\phantom{\rule{2em}{0ex}}\end{array}
(3.1)
are satisfied.
Let us consider the KKT conditions for the VI(F,X). Assuming that a suitable constraint qualification holds at a solution x, the KKT conditions can be expressed as
\begin{array}{l}F\left(x\right)+\nabla g\left(x\right)\lambda =0,\phantom{\rule{2em}{0ex}}\\ 0\le \lambda \perp g\left(x\right)\ge 0,\phantom{\rule{2em}{0ex}}\end{array}
(3.2)
which is equivalent to
\begin{array}{c}\left(\begin{array}{c}\hfill {\nabla}_{{x}^{1}}{\theta}^{1}\left(x\right)\hfill \\ \hfill \vdots \hfill \\ \hfill {\nabla}_{{x}^{N}}{\theta}^{N}\left(x\right)\hfill \end{array}\right)+\left(\begin{array}{c}\hfill {\nabla}_{{x}^{1}}g\left(x\right)\hfill \\ \hfill \vdots \hfill \\ \hfill {\nabla}_{{x}^{N}}g\left(x\right)\hfill \end{array}\right)\lambda =0,\\ 0\le \lambda \perp g\left(x\right)\ge 0.\end{array}
(3.3)
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 λ^{1} = λ^{2} = ⋯ = λ^{N}= λ.
(ii) Viceversa, let x be a solution of the GNEP at which KKT conditions (3.1) hold with λ^{1} = λ^{2} = ⋯ = λ^{N}. Then x is a solution of VI(F, X).
Using the minimum function \phi :\Re \times \Re \to \Re ,\phantom{\rule{2.77695pt}{0ex}}\phi \left(a,b\right):=\text{min}\left\{a,b\right\}, the KKT conditions (3.2) can equivalently be written as the nonlinear system of equations
\Phi \left(\omega \right):=\Phi \left(x,\lambda \right)=0,
(3.4)
where \Phi :{\Re}^{n+m}\to {\Re}^{n+m} is defined by
\Phi \left(\omega \right)=\Phi \left(x,\lambda \right):=\left(\begin{array}{c}\hfill L\left(x,\lambda \right)\hfill \\ \hfill \varphi \left(g\left(x\right),\lambda \right)\hfill \end{array}\right),
and
\begin{array}{l}L\left(x,\lambda \right):=F\left(x\right)+\nabla g\left(x\right)\lambda ,\phantom{\rule{2em}{0ex}}\\ \varphi \left(g\left(x\right),\lambda \right):={\left(\phi \left({g}_{1}\left(x\right),{\lambda}_{1}\right),...,\phi \left({g}_{m}\left(x\right),{\lambda}_{m}\right)\right)}^{T}\in {\Re}^{m}.\phantom{\rule{2em}{0ex}}\end{array}
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 Bsubdifferential ∂_{
B
}Φ(ω)) are nonsingular. Our first result gives a description of the structure of the matrices in the generalized Jacobian ∂Φ(ω).
Lemma 3.2 Let\omega =\left(x,\lambda \right)\in {\Re}^{n+m}. Then, each element H ∈ ∂Φ(ω)^{T}can be represented as follows:
H=\left[\begin{array}{cc}\hfill {\nabla}_{x}L\left(\omega \right)\hfill & \hfill \nabla g\left(x\right){D}_{a}\left(\omega \right)\hfill \\ \hfill \nabla g{\left(x\right)}^{T}\hfill & \hfill {D}_{b}\left(\omega \right)\hfill \end{array}\right],
where
{D}_{a}\left(\omega \right):=\text{diag}\left({a}_{1}\left(\omega \right),...,{a}_{m}\left(\omega \right)\right),\phantom{\rule{2.77695pt}{0ex}}{D}_{b}\left(\omega \right):=\text{diag}\left({b}_{1}\left(\omega \right),...,{b}_{m}\left(\omega \right)\right)\in {\Re}^{m\times m}
are diagonal matrices whose ith diagonal elements are given by
{a}_{i}\left(\omega \right)=\left\{\begin{array}{ccc}\hfill 1,\hfill & \hfill if\hfill & \hfill {g}_{i}\left(x\right)<{\lambda}_{i},\hfill \\ \hfill 0,\hfill & \hfill if\hfill & \hfill {g}_{i}\left(x\right)>{\lambda}_{i},\hfill \\ \hfill {\mu}_{i},\hfill & \hfill if\hfill & \hfill {g}_{i}\left(x\right)={\lambda}_{i},\hfill \end{array}\right.\phantom{\rule{2.77695pt}{0ex}}and\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}{b}_{i}\left(\omega \right)=\left\{\begin{array}{ccc}\hfill 0,\hfill & \hfill if\hfill & \hfill {g}_{i}\left(x\right)<{\lambda}_{i},\hfill \\ \hfill 1,\hfill & \hfill if\hfill & \hfill {g}_{i}\left(x\right)>{\lambda}_{i},\hfill \\ \hfill 1{\mu}_{i},\hfill & \hfill if\hfill & \hfill {g}_{i}\left(x\right)={\lambda}_{i},\hfill \end{array}\right.
for any μ_{
i
}∈ [0,1].
Proof. The first n components of the vector function Φ are continuously differentiable, so the expression for the first n columns of H readily follows. Then, consider the last m columns. Use the fact that
\partial \varphi {\left(g\left(x\right),\lambda \right)}^{T}\subset \partial \phi {\left({g}_{1}\left(x\right),{\lambda}_{1}\right)}^{T}\times \cdots \times \partial \phi {\left({g}_{m}\left(x\right),{\lambda}_{m}\right)}^{T},
if i is such that g_{
i
}(x) ≠ λ_{
i
}, then φ is continuously differentiable at (g_{
i
}(x), λ_{
i
}) and the expression for the (n + i)th column of H follows. If instead g_{
i
}(x) = λ_{
i
}, then, using the definition of the Bsubdifferential, it follows that
{\partial}_{B}\phi {\left({g}_{i}\left(x\right),{\lambda}_{i}\right)}^{T}=\left\{\left(\nabla {g}_{i}{\left(x\right)}^{T},0\right),\left(0,{e}_{i}^{T}\right)\right\}.
Taking the convex hull, we get
\partial \phi {\left({g}_{i}\left(x\right),{\lambda}_{i}\right)}^{T}=\left\{\left({\mu}_{i}\nabla {g}_{i}{\left(x\right)}^{T},\left(1{\mu}_{i}\right){e}_{i}^{T}\right){\mu}_{i}\in \left[0,1\right]\right\}.
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.
Theorem 3.1 Let{\omega}^{*}=\left({x}^{*},{\lambda}^{*}\right)\in {\Re}^{n+m}be a solution of the system Φ(ω) = 0. Consider the following two statements:

(a)
The strong secondorder sufficient condition and the linear independence constraint qualification (LICQ) for VI(F,X) holds at x*.

(b)
Any element in ∂Φ(ω*) is nonsingular.
It holds that (a) ⇒ (b).
Proof. For the sake of notational simplicity, let us define the following subsets of the index set I := {1,...,m},
{I}_{0}:=\left\{i{g}_{i}\left({x}^{*}\right)=0,{\lambda}_{i}^{*}\ge 0\right\},\phantom{\rule{1em}{0ex}}{I}_{<}:=\left\{i{g}_{i}\left({x}^{*}\right)<0,{\lambda}_{i}^{*}=0\right\}.
Moreover, we need
\begin{array}{ll}\hfill {I}_{00}& :=\left\{i{g}_{i}\left({x}^{*}\right)=0,{\lambda}_{i}^{*}=0\right\},\phantom{\rule{1em}{0ex}}{I}_{+}:=\left\{i{g}_{i}\left({x}^{*}\right)=0,{\lambda}_{i}^{*}>0\right\},\phantom{\rule{2em}{0ex}}\\ \hfill {I}_{01}& :=\left\{i\in {I}_{00}{\mu}_{i}=1\right\},\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{0.3em}{0ex}}{I}_{02}:=\left\{i\in {I}_{00}{\mu}_{i}\in \left(0,1\right)\right\},\phantom{\rule{2em}{0ex}}\\ \hfill {I}_{03}& :=\left\{i\in {I}_{00}{\mu}_{i}=0\right\}.\phantom{\rule{2em}{0ex}}\end{array}
The following relationships between these index sets can easily be seen to hold:
I={I}_{0}\cup {I}_{<},\phantom{\rule{1em}{0ex}}{I}_{0}={I}_{00}\cup {I}_{+},\phantom{\rule{1em}{0ex}}{I}_{00}={I}_{01}\cup {I}_{02}\cup {I}_{03}.
Using a suitable reordering of the constraints, every element H ∈ ∂Φ(ω*)^{T}has the following structure:
H=\left[\begin{array}{cccccc}\hfill {\nabla}_{x}L\left({\omega}^{*}\right)\hfill & \hfill \nabla {g}_{+}\left({x}^{*}\right)\hfill & \hfill \nabla {g}_{01}\left({x}^{*}\right)\hfill & \hfill \nabla {g}_{02}\left({x}^{*}\right){D}_{a}{\left({\omega}^{*}\right)}_{02}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \nabla {g}_{+}{\left({x}^{*}\right)}^{T}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \nabla {g}_{01}{\left({x}^{*}\right)}^{T}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \nabla {g}_{02}{\left({x}^{*}\right)}^{T}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {D}_{b}{\left({\omega}^{*}\right)}_{02}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \nabla {g}_{03}{\left({x}^{*}\right)}^{T}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill I\hfill & \hfill 0\hfill \\ \hfill \nabla {g}_{<}{\left({x}^{*}\right)}^{T}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill I\hfill \end{array}\right],
(3.5)
where D_{
a
}(ω*)_{02} and D_{
b
}(ω*)_{02} are positive definite diagonal matrices. Note that we abbreviated {g}_{{I}_{+}} etc. by g_{+} etc. in (3.5). It is obvious that H is nonsingular if and only if the following matrix is nonsingular,
\left[\begin{array}{cccccc}\hfill {\nabla}_{x}L\left({\omega}^{*}\right)\hfill & \hfill \nabla {g}_{+}\left({x}^{*}\right)\hfill & \hfill \nabla {g}_{01}\left({x}^{*}\right)\hfill & \hfill \nabla {g}_{02}\left({x}^{*}\right)\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \nabla {g}_{+}{\left({x}^{*}\right)}^{T}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \nabla {g}_{01}{\left({x}^{*}\right)}^{T}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \nabla {g}_{02}{\left({x}^{*}\right)}^{T}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {D}_{b}{\left({\omega}^{*}\right)}_{02}{D}_{a}{\left({\omega}^{*}\right)}_{02}^{1}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \nabla {g}_{03}{\left({x}^{*}\right)}^{T}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill I\hfill & \hfill 0\hfill \\ \hfill \nabla {g}_{<}{\left({x}^{*}\right)}^{T}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill I\hfill \end{array}\right].
In turn, this matrix is nonsingular if and only if the following matrix is nonsingular:
\left[\begin{array}{cccc}\hfill {\nabla}_{x}L\left({\omega}^{*}\right)\hfill & \hfill \nabla {g}_{+}\left({x}^{*}\right)\hfill & \hfill \nabla {g}_{01}\left({x}^{*}\right)\hfill & \hfill \nabla {g}_{02}\left({x}^{*}\right)\hfill \\ \hfill \nabla {g}_{+}{\left({x}^{*}\right)}^{T}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \nabla {g}_{01}{\left({x}^{*}\right)}^{T}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \nabla {g}_{02}{\left({x}^{*}\right)}^{T}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {D}_{b}{\left({\omega}^{*}\right)}_{02}{D}_{a}{\left({\omega}^{*}\right)}_{02}^{1}\hfill \end{array}\right].
(3.6)
Let \left(\Delta {x}_{1},\Delta {x}_{2},\Delta {x}_{3},\Delta {x}_{4}\right)\in {\Re}^{n}\times {\Re}^{\left{I}_{+}\right}\times {\Re}^{\left{I}_{01}\right}\times {\Re}^{\left{I}_{02}\right} be such that
\left[\begin{array}{cccc}\hfill {\nabla}_{x}L\left({\omega}^{*}\right)\hfill & \hfill \nabla {g}_{+}\left({x}^{*}\right)\hfill & \hfill \nabla {g}_{01}\left({x}^{*}\right)\hfill & \hfill \nabla {g}_{02}\left({x}^{*}\right)\hfill \\ \hfill \nabla {g}_{+}{\left({x}^{*}\right)}^{T}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \nabla {g}_{01}{\left({x}^{*}\right)}^{T}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \nabla {g}_{02}{\left({x}^{*}\right)}^{T}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {D}_{b}{\left({\omega}^{*}\right)}_{02}{D}_{a}{\left({\omega}^{*}\right)}_{02}^{1}\hfill \end{array}\right]\left[\begin{array}{c}\hfill \Delta {x}_{1}\hfill \\ \hfill \Delta {x}_{2}\hfill \\ \hfill \Delta {x}_{3}\hfill \\ \hfill \Delta {x}_{4}\hfill \end{array}\right]=0,
(3.7)
we know that
\begin{array}{c}{\nabla}_{x}L\left({\omega}^{*}\right)\Delta {x}_{1}\nabla {g}_{+}\left({x}^{*}\right)\Delta {x}_{2}\nabla {g}_{01}\left({x}^{*}\right)\Delta {x}_{3}\nabla {g}_{02}\left({x}^{*}\right)\Delta {x}_{4}=0,\\ \nabla {g}_{+}{\left({x}^{*}\right)}^{T}\Delta {x}_{1}=0,\\ \nabla {g}_{01}{\left({x}^{*}\right)}^{T}\Delta {x}_{1}=0,\\ \nabla {g}_{02}{\left({x}^{*}\right)}^{T}\Delta {x}_{1}+\left[{D}_{b}{\left({\omega}^{*}\right)}_{02}{D}_{a}{\left({\omega}^{*}\right)}_{02}^{1}\right]\Delta {x}_{4}=0.\end{array}
(3.8)
By the first, second and third equations of (3.8), we obtain that
\begin{array}{ll}\hfill 0& =\u27e8\Delta {x}_{1},{\nabla}_{x}L\left({\omega}^{*}\right)\Delta {x}_{1}\nabla {g}_{+}\left({x}^{*}\right)\Delta {x}_{2}\nabla {g}_{01}\left({x}^{*}\right)\Delta {x}_{3}\nabla {g}_{02}\left({x}^{*}\right)\Delta {x}_{4}\u27e9\phantom{\rule{2em}{0ex}}\\ =\u27e8\Delta {x}_{1},{\nabla}_{x}L\left({\omega}^{*}\right)\Delta {x}_{1}\u27e9\u27e8\Delta {x}_{1},\nabla {g}_{+}\left({x}^{*}\right)\Delta {x}_{2}\u27e9\u27e8\Delta {x}_{1},\nabla {g}_{01}\left({x}^{*}\right)\Delta {x}_{3}\u27e9\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}\u27e8\Delta {x}_{1},\nabla {g}_{02}\left({x}^{*}\right)\Delta {x}_{4}\u27e9\phantom{\rule{2em}{0ex}}\\ =\u27e8\Delta {x}_{1},{\nabla}_{x}L\left({\omega}^{*}\right)\Delta {x}_{1}\u27e9\u27e8\Delta {x}_{1},\nabla {g}_{02}\left({x}^{*}\right)\Delta {x}_{4}\u27e9,\phantom{\rule{2em}{0ex}}\end{array}
which, together with the last equation of (3.8), implies that
\u27e8\Delta {x}_{1},{\nabla}_{x}L\left({\omega}^{*}\right)\Delta {x}_{1}\u27e9=\Delta {x}_{4}^{T}\left[{D}_{b}{\left({\omega}^{*}\right)}_{02}{D}_{a}{\left({\omega}^{*}\right)}_{02}^{1}\right]\Delta {x}_{4}\le 0.
(3.9)
From the second equation of (3.8), we know that
\Delta {x}_{1}\in \text{a}\text{f}\text{f}\left(C\left({x}^{*}\right)\right),
where C(x*) denotes the critical cone of VI(F,X). Then, by (3.9) and the strong secondorder sufficient condition that
Thus, the first equation of (3.8) reduces to
\nabla {g}_{+}\left({x}^{*}\right)\Delta {x}_{2}+\nabla {g}_{01}\left({x}^{*}\right)\Delta {x}_{3}+\nabla {g}_{02}\left({x}^{*}\right)\Delta {x}_{4}=0.
(3.10)
By the LICQ for VI(F,X), we have
\Delta {x}_{2}=0,\phantom{\rule{2.77695pt}{0ex}}\Delta {x}_{3}=0,\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\text{and}\phantom{\rule{2.77695pt}{0ex}}\Delta {x}_{4}=0.
This together with Δx_{1} = 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.
Proposition 3.1 Let{\omega}^{*}=\left({x}^{*},{\lambda}^{*}\right)\in {\Re}^{n+m}satisfying Φ(ω*) = 0, for all ν = 1,..., N the payoff functions θ^{ν}are separable, that is
{\theta}^{\nu}\left(x\right)={f}^{\nu}\left({x}^{\nu}\right)+{h}^{\nu}\left({x}^{\nu}\right),
where{f}^{\nu}:{\Re}^{{n}_{\nu}}\to \Reis 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.
Proof. We know that
F\left({x}^{*}\right)=\left(\begin{array}{c}\hfill {\nabla}_{{x}^{1}}{\theta}^{1}\left({x}^{*}\right)\hfill \\ \hfill \vdots \hfill \\ \hfill {\nabla}_{{x}^{N}}{\theta}^{N}\left({x}^{*}\right)\hfill \end{array}\right),
then, by the definition of θ^{ν}(·), we have
\begin{array}{c}\nabla F\left(x*\right)=\left(\begin{array}{cccc}\hfill {\nabla}_{{x}^{1}{x}^{1}}^{2}{\theta}^{1}\left(x*\right)\hfill & \hfill {\nabla}_{{x}^{1}{x}^{2}}^{2}{\theta}^{1}\left(x*\right)\hfill & \hfill \cdots \phantom{\rule{0.3em}{0ex}}\hfill & \hfill {\nabla}_{{x}^{1}{x}^{N}}^{2}{\theta}^{1}\left(x*\right)\hfill \\ \hfill {\nabla}_{{x}^{2}{x}^{1}}^{2}{\theta}^{2}\left(x*\right)\hfill & \hfill {\nabla}_{{x}^{2}{x}^{2}}^{2}{\theta}^{2}\left(x*\right)\hfill & \hfill \cdots \phantom{\rule{0.3em}{0ex}}\hfill & \hfill {\nabla}_{{x}^{2}{x}^{N}}^{2}{\theta}^{2}\left(x*\right)\hfill \\ \hfill \vdots \hfill & \hfill \vdots \hfill & \hfill \ddots \hfill \\ \hfill {\nabla}_{{x}^{N}{x}^{1}}^{2}{\theta}^{N}\left(x*\right)\hfill & \hfill {\nabla}_{{x}^{N}{x}^{2}}^{2}{\theta}^{N}\left(x*\right)\hfill & \hfill \cdots \phantom{\rule{0.3em}{0ex}}\hfill & \hfill {\nabla}_{{x}^{N}{x}^{N}}^{2}{\theta}^{N}\left(x*\right)\hfill \end{array}\right)\\ =\left(\begin{array}{c}\hfill {\nabla}_{{x}^{1}{x}^{1}}^{2}{f}^{1}\left({x}^{*,1}\right)\hfill \\ \hfill {\nabla}_{{x}^{2}{x}^{2}}^{2}{f}^{2}\left({x}^{*,2}\right)\hfill \\ \hfill \ddots \hfill \\ \hfill {\nabla}_{{x}^{N}{x}^{N}}^{2}{f}^{N}\left({x}^{*,N}\right)\hfill \end{array}\right)\end{array}
By the strong convexity of f^{ν}, we can conclude that ∇F(x*) is positive definite.
From {\lambda}_{i}^{*}\ge 0 and the convexity of g_{
i
}, we obtain that
{\nabla}_{x}\left(\nabla g\left({x}^{*}\right){\lambda}^{*}\right)=\sum _{i=1}^{m}{\lambda}_{i}^{*}{\nabla}^{2}{g}_{i}\left({x}^{*}\right)
is positive semidefinite, which together with ∇F(x*) is positive definite implies that
{\nabla}_{x}L\left({\omega}^{*}\right)=\nabla F\left({x}^{*}\right)+\sum _{i=1}^{m}{\lambda}_{i}^{*}{\nabla}^{2}{g}_{i}\left({x}^{*}\right)
is positive definite. Thus, the strong secondorder sufficient condition for the VI(F, X) holds at x*. From Theorem 3.1, we obtain any element in ∂Φ(ω*) is nonsingular.
Proposition 3.2 Let{\omega}^{*}=\left({x}^{*},{\lambda}^{*}\right)\in {\Re}^{n+m}be such that Φ(ω*) = 0. Consider the case where the payoff functions are quadratic, i.e. for all ν = 1,...,N one has
{\theta}^{\nu}\left(x\right):=\frac{1}{2}{\left({x}^{\nu}\right)}^{T}{A}_{\nu \nu}{x}^{\nu}+\sum _{\mu =1,\mu \ne \nu}^{N}{\left({x}^{\nu}\right)}^{T}{A}_{\nu \mu}{x}^{\mu},
where the matrices{A}_{\nu \mu}\in {\Re}^{{n}_{\nu}}\times {\Re}^{{n}_{\mu}}and A_{
νν
}are symmetric. Suppose that LICQ holds at x*, and
\mathbf{B}:=\left[\begin{array}{cccc}\hfill {A}_{11}\hfill & \hfill {A}_{12}\hfill & \hfill \cdots \phantom{\rule{0.3em}{0ex}}\hfill & \hfill {A}_{1N}\hfill \\ \hfill {A}_{21}\hfill & \hfill {A}_{22}\hfill & \hfill \cdots \phantom{\rule{0.3em}{0ex}}\hfill & \hfill {A}_{2N}\hfill \\ \hfill \vdots \hfill & \hfill \vdots \hfill & \hfill \ddots \hfill & \hfill \vdots \hfill \\ \hfill {A}_{N1}\hfill & \hfill {A}_{N2}\hfill & \hfill \cdots \phantom{\rule{0.3em}{0ex}}\hfill & \hfill {A}_{NN}\hfill \end{array}\right]
is positive definite. Then all the elements in the generalized Jacobian ∂Φ(ω*) are nonsingular.
Proof. We show that ∇_{
x
}L(ω*) is positive definite, which implies that the strong second order sufficient condition for the VI(F,X) holds at x*, and then apply Theorem 3.1. To this end, first note that
F\left({x}^{*}\right)=\left(\begin{array}{c}\hfill {\nabla}_{{x}^{1}}{\theta}^{1}\left({x}^{*}\right)\hfill \\ \hfill {\nabla}_{{x}^{2}}{\theta}^{2}\left({x}^{*}\right)\hfill \\ \hfill \vdots \hfill \\ \hfill {\nabla}_{{x}^{N}}{\theta}^{N}\left({x}^{*}\right)\hfill \end{array}\right)=\left(\begin{array}{c}\hfill {\sum}_{\mu =1}^{N}{A}_{1\mu}{x}^{\mu}\hfill \\ \hfill {\sum}_{\mu =1}^{N}{A}_{2\mu}{x}^{\mu}\hfill \\ \hfill \vdots \hfill \\ \hfill {\sum}_{\mu =1}^{N}{A}_{N\mu}{x}^{\mu}\hfill \end{array}\right).
Moreover,
\nabla F\left({x}^{*}\right)=\left(\begin{array}{cccc}\hfill {A}_{11}\hfill & \hfill {A}_{12}\hfill & \hfill \cdots \phantom{\rule{0.3em}{0ex}}\hfill & \hfill {A}_{1N}\hfill \\ \hfill {A}_{21}\hfill & \hfill {A}_{22}\hfill & \hfill \cdots \phantom{\rule{0.3em}{0ex}}\hfill & \hfill {A}_{21}\hfill \\ \hfill \cdots \phantom{\rule{0.3em}{0ex}}\hfill \\ \hfill {A}_{N1}\hfill & \hfill {A}_{N1}\hfill & \hfill \cdots \phantom{\rule{0.3em}{0ex}}\hfill & \hfill {A}_{NN}\hfill \end{array}\right),
which together with {\lambda}_{i}^{*}\ge 0 and the convexity of g_{
i
}implies that
{\nabla}_{x}\left(\nabla g\left({x}^{*}\right){\lambda}^{*}\right)=\sum _{i=1}^{m}{\lambda}_{i}^{*}{\nabla}^{2}{g}_{i}\left({x}^{*}\right)
is positive semidefinite. Hence, we obtain that ∇_{
x
}L(ω*) is positive definite. The statement therefore follows from Theorem 3.1.