Generalized Nash Equilibrium Problem: existence,
uniqueness and reformulations
Didier Aussel
Univ. de Perpignan, France
CIMPA-UNESCO school, Delhi
November 25 - December 6, 2013
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Outline of the 7 lectures
Generalized Nash Equilibrium
a- Reformulations
b- Existence of equilibrium
Variational inequalities: motivations, defnitions
abcd-
Motivations, definitions
Existence of solutions
Stability
Uniqueness
Quasiconvex optimization
a- Classical subdifferential approach
b- Normal approach
Whenever a set-valued map is not really set-valued...
Quasivariational inequalities
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Outline of lecture 1
Variational inequalities: basic facts and existence
I- Some examples (motivation)
II- First Existence results
a- The linear case
b- The finite dimensional case
III- A general case
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Notations
X a topological vector space
X ∗ its topological dual (w ∗ -top.)
h·, ·i the duality product
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Notations
X a topological vector space
X ∗ its topological dual (w ∗ -top.)
h·, ·i the duality product
Stampacchia variational inequality :
∗
Let T : X → 2X be a map and C be a nonempty subset of X .
Find x̄ ∈ C such that there exists x̄ ∗ ∈ T (x̄) for which
hx̄ ∗ , y − x̄i ≥ 0,
∀ y ∈ C.
Notation : S(T , C ) set of solutions (⊂ C ).
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
First motivation = Optimality conditions
Let f : X → IR ∪ {+∞} and C ⊆ dom f be a convex subset.
(P)
find x̄ ∈ C : f (x̄) = inf f (x)
x∈C
Necessary condition: f is lsc + ...
if x̄ is a solution of (P) then
x̄ ∈ S(∂f (x̄), C ).
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
First motivation = Optimality conditions
Let f : X → IR ∪ {+∞} and C ⊆ dom f be a convex subset.
find x̄ ∈ C : f (x̄) = inf f (x)
(P)
x∈C
Necessary condition: f is lsc + ...
if x̄ is a solution of (P) then
x̄ ∈ S(∂f (x̄), C ).
Perfect case: f convex
f : X → IR ∪ {+∞} a proper convex function
C a nonempty convex subset of X , x̄ ∈ C + C.Q.
Then
f (x̄) = inf f (x) ⇐⇒ x̄ ∈ S(∂f , C )
x∈C
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
First motivation = Optimality conditions
Let f : X → IR ∪ {+∞} and C ⊆ dom f be a convex subset.
find x̄ ∈ C : f (x̄) = inf f (x)
(P)
x∈C
Necessary condition: f is lsc + ...
if x̄ is a solution of (P) then
x̄ ∈ S(∂f (x̄), C ).
Perfect case: f convex
f : X → IR ∪ {+∞} a proper convex function
C a nonempty convex subset of X , x̄ ∈ C + C.Q.
Then
f (x̄) = inf f (x) ⇐⇒ x̄ ∈ S(∂f , C )
x∈C
What about quasiconvex case? see in a future lecture
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Another motivation
Signorini’s frictionless contact problem
surface traction
f2 ∈ [L2 (Γ2 )]k
Γ1
Γ2
volume force
f0 ∈ [L2 (Ω)]k
ω
Γ3
@
I
@
@
@
@
@
Didier Aussel
Frictionless contact
Generalized Nash Equilibrium Problem: existence, uniqueness and
Notations
Functional spaces:
k
k×k
k×k
S = {σ = (σij )ij ∈ R
: σij = σji } = Rs
1
k
W = {v ∈ H (Ω) : v = 0 sur Γ1 }
2
k×k
Q = {q = (qij ) ∈ L (Ω)
: qij = qji , 1 ≤ i, j ≤ k} = L2 (Ω)k×k
s
W2 = {v ∈ W : vν ≤ 0 a.e. on Γ3 }
Deformation operator: ε : H 1 (Ω)k → Q
εij (u) =
Scalar product:
1
"
2
R
hp, qiQ = Ω pij (x)qij (x) dx
∂ui
∂xj
+
∂uj
∂xi
#
,
1 ≤ i, j ≤ k.
et hu, v iW = hε(u), ε(v )iQ
Elasticity operator: F : Ω × Sk → Sk
Stress function : σ : H 1 (Ω)k → Q defined by
σ(u) :
Ω
x
Didier Aussel
→
7
→
Sk
F (x, ε(u)(x)).
Generalized Nash Equilibrium Problem: existence, uniqueness and
Formulations of the problem:
Find a displacement field u : Ω → R such that
−Div σ(u) = f0
u=0
σ(u)ν = f2
uν ≤ 0, σ(u)ν ≤ 0, σ(u)ν uν = 0, σ(u)τ = 0
Didier Aussel
on
on
on
on
Ω
Γ1
Γ2
Γ3
Generalized Nash Equilibrium Problem: existence, uniqueness and
Formulations of the problem:
Find a displacement field u : Ω → R such that
−Div σ(u) = f0
u=0
σ(u)ν = f2
uν ≤ 0, σ(u)ν ≤ 0, σ(u)ν uν = 0, σ(u)τ = 0
on
on
on
on
Ω
Γ1
Γ2
Γ3
Variational formulation:
Find u ∈ W2 such that
hσ(u), ε(v ) − ε(u)iQ ≥ hf , v − uiW ,
∀ v ∈ W2
where f is an element of W defined by
Z
hf , v iW =
Z
f0 .v dx +
Ω
Didier Aussel
f2 .v da.
Γ2
Generalized Nash Equilibrium Problem: existence, uniqueness and
b - Minty and local Minty V. I.
∗
Let C ⊆ X and T : C → 2X be a map.
We denote by
• Sw (T , C ) set of weak solutions of Stampacchia V.I.
Find x̄ ∈ C such that
∀ y ∈ C , ∃ x̄ ∗ ∈ T (x̄) : hx̄ ∗ , y − x̄i ≥ 0
• M(T , C ) set of solutions of the Minty V.I.
Find x̄ ∈ C such that
∀ y ∈ C and ∀ y ∗ ∈ T (y ) : hy ∗ , y − x̄i ≥ 0.
• LM(T , C ) set of local solutions of Minty V.I., i.e.
x̄ ∈ LM(T , C ) if it exists a neighb. V of x̄ such that x̄ ∈ M(T , V ∩ C ).
If T is pseudomonotone, then LM (T , C ) = M(T , C ).
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
II - First Existence results
a- The linear case
b- The finite dimension case
c- V.I. under monotonicity
d- Other assumptions
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
II - a - The linear case
Let H be an Hilbert space, f an element of H ∗ = H and a : H × H → H
a bilinear application.
Theorem (Stampacchia 64)
Let C be a nonempty closed convex subset of H. If a is continuous and
satisfies the following property
∃ α > 0 such that a(v , v ) ≥ αkv k2 , ∀ v ∈ H,
then
there exists x̄ ∈ C such that a(x̄, y − x̄) ≥ hf , y − x̄i,
∀ y ∈ C.
D. Kinderlehrer and G. Stampacchia, An introduction to Variational Inequalities and Their Applications, 1980,
Academic Press
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Theorem (Lax-Milgram)
If a is continuous and satisfies the following property
∃ α > 0 such that a(v , v ) ≥ αkv k2 , ∀ v ∈ H,
then there exists a unique x̄ ∈ H such that
a(x̄, u) = hf , ui,
∀ u ∈ H.
Proof. From the Theorem of Stampacchia, there exists u ∈ H such that
a(u, v − u) ≥ hf , v − ui,
or in other words ϕu (v ) ≥ ϕu (u),
∀v ∈ H
∀ v ∈ H where
ϕu (v ) = a(u, v ) − hf , v i.
But ϕu is linear and therefore ϕu ≡ 0.
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
II - b - The finite dimension case
Let f : X → X ∗ be a function and C be a nonempty subset of X .
Find x̄ ∈ C such that
hf (x̄), y − x̄i ≥ 0,
∀ y ∈ C.
First property :
S(f , C ) ∩ int(C ) = {x ∈ int(C ) : f (x) = 0}
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
II - b - The finite dimension case
Proposition (Stampacchia 1966)
Let C be a nonempty convex compact subset of IRn and f : C → IRn a
continuous function. Then S(f , C ) is nonempty, that is,
there exists x̄ ∈ C such that hf (x̄), y − x̄i ≥ 0,
Didier Aussel
∀ y ∈ C.
Generalized Nash Equilibrium Problem: existence, uniqueness and
II - b - The finite dimension case
Proposition (Stampacchia 1966)
Let C be a nonempty convex compact subset of IRn and f : C → IRn a
continuous function. Then S(f , C ) is nonempty, that is,
there exists x̄ ∈ C such that hf (x̄), y − x̄i ≥ 0,
∀ y ∈ C.
Proof: Let us consider the application
ψ:
C
x
→
7
→
C
ψ(x) = PC ◦ (Id − f )(x) = PC (x − f (x))
ψ is continuous on the convex compact set C and therefore, according to the Brouwer fixed point theorem, there
exists a point x̄ ∈ C such that
x̄ = ψ(x̄)
⇔
⇔
⇔
x̄ = PC (x̄ − f (x̄))
hx̄ − f (x̄) − x̄, y − x̄i ≤ 0, ∀ y ∈ C
x̄ ∈ S(f , C ).
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Without compactness
The compactness hypothesis can be replaced by a ”coercivity condition”:
Proposition
Let C be a nonempty closed convex subset of IRn and f : C → IRn a
continuous function. The following conditions are equivalent:
- S(f , C ) is nonempty.
- ∃ r > 0 such that S(f , C ∩ B(0, r )) ∩ B(0, r ) 6= ∅,
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Without compactness 2
The compactness hypothesis can be replaced by a ”coercivity condition”
in term of the function:
Proposition
Let C be a nonempty closed convex subset of IRn and f : C → IRn a
continuous function. If f is coercive, that is,
∃ x0 ∈ C such that lim
hf (x) − f (x0 ), x − x0 i
= +∞
kx − x0 k
then S(f , C ) is nonempty.
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Multivalued form
Theorem
n
Let C be a nonempty convex compact subset of IRn and F : C → 2IR a
upper semicontinuous map with convex compact values. Then S(F , C ) is
nonempty, that is,
there exist x̄ ∈ C and x̄ ∗ ∈ F (x̄)
such that hx̄ ∗ , y − x̄i ≥ 0,
Didier Aussel
∀ y ∈ C.
Generalized Nash Equilibrium Problem: existence, uniqueness and
Multivalued form
Theorem
n
Let C be a nonempty convex compact subset of IRn and F : C → 2IR a
upper semicontinuous map with convex compact values. Then S(F , C ) is
nonempty, that is,
there exist x̄ ∈ C and x̄ ∗ ∈ F (x̄)
such that hx̄ ∗ , y − x̄i ≥ 0,
∀ y ∈ C.
can be found, e.g., Borwein, J. and Lewis, A., Convex Analysis and Nonlinear Optimization, 2000
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Multivalued form
Theorem
n
Let C be a nonempty convex compact subset of IRn and F : C → 2IR a
upper semicontinuous map with convex compact values. Then S(F , C ) is
nonempty, that is,
there exist x̄ ∈ C and x̄ ∗ ∈ F (x̄)
such that hx̄ ∗ , y − x̄i ≥ 0,
∀ y ∈ C.
can be found, e.g., Borwein, J. and Lewis, A., Convex Analysis and Nonlinear Optimization, 2000
F is upper semicontinuous at x if
∀ V neighb. of F (x), ∃ U neighb. of x such that F (U) ⊂ V .
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Multivalued form: proof
Proof. Since F is cusco, its range F (C ) is compact and the map
T = (Id − F ) ◦ PC is also cusco since PC is continuous (Exercise).
Now one can observe that T (co(C − F (C ))) = co(C − F (C )). Therefore T
has a fixed point (say x0 ) on C − F (C ), that is
x0 ∈ T (x0 )
⇔
⇔
⇔
x0 ∈ (Id − F ) ◦ PC (x0 )
x̄ = PC (x0 )
x̄ ∗ = x̄ − x0 ∈ F (x̄)
hx0 − x̄, y − x̄i ≤ 0,
x̄ ∗ = x̄ − x0 ∈ F (x̄)
∀y ∈ C
and thus x̄ ∈ S(F , C ).
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Equivalence
Actually the above existence theorem 6 is equivalent to the Kakutani Fixed point theorem. Indeed, let C be a
nonempty convex compact subset of IRn and T : C → 2C be a cusco map. Observing that the set-valued map
F = Id − T is also cusco, Theorem 6 implies the solvability of the set-valued Stampacchia variational inequality
defined by F and C , that is there exists x̄ ∈ C such that
x̄ ∈ S(Id − T , C )
∃ u ∗ ∈ (Id − T )(x̄) such that
hu ∗ , y − x̄i ≥ 0, ∀ y ∈ C
∃ x̄ ∗ ∈ T (x̄) such that
hx̄ − x̄ ∗ , y − x̄i ≥ 0, ∀ y ∈ C
⇔
⇔
Taking y = x̄ ∗ , one has
∗
hx̄ − x̄ , x̄
∗
− x̄i ≥ 0
showing that x̄ is a fixed point of T .
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
II
Existence of solution
c - V.I. under monotonicity
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Monotone case
Theorem (1966)
Let X be a reflexive Banach space, C be a weakly compact convex
nonempty subset of X . If f : C → X ∗ is a monotone function,
continuous on finite dimension subsets, then S(f , C ) is nonempty.
Hartmann, G. and Stampacchia, G., Acta Mathematica, 115 (1966), 271-310
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Monotone case
Theorem (1966)
Let X be a reflexive Banach space, C be a weakly compact convex
nonempty subset of X . If f : C → X ∗ is a monotone function,
continuous on finite dimension subsets, then S(f , C ) is nonempty.
Theorem
Let X be a reflexive Banach space, C be a nonempty convex and weakly
∗
compact subset of X . If F : C → 2X is a monotone map with convex
weakly compact values and continuous on finite dimension subsets, then
S(F , C ) is nonempty.
can be found, e.g., in Shih, M.H. and Tan, K.K., JMAA, 134 (1988), 431-440
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Monotonicities
T is monotone iff
∀ x, y ∈ X , ∀ x ∗ ∈ T (x) and ∀ y ∗ ∈ T (x)
hy ∗ − x ∗ , y − xi ≥ 0.
T is pseudomonotone iff
∃ x ∗ ∈ T (x) : hx ∗ , y − xi ≥ 0 ⇒ hy ∗ , y − xi ≥ 0, ∀ y ∗ ∈ T (y ).
T is quasimonotone iff
∃ x ∗ ∈ T (x) : hx ∗ , y − xi > 0 ⇒ hy ∗ , y − xi ≥ 0, ∀ y ∗ ∈ T (y ).
monotone
⇓
pseudomonotone
⇓
quasimonotone
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Weakening monotonicity: why?
Let f : X → IR be differentiable
f is convex iff df is monotone
iff df (x)(y − x) > 0 ⇒ df (y )(y − x) ≥ 0
f is pseudoconvex iff df is pseudomonotone
iff df (x)(y − x) > 0 ⇒ df (y )(y − x) ≥ 0
f is quasiconvex iff df is quasimonotone
iff df (x)(y − x) > 0 ⇒ df (y )(y − x) ≥ 0
Let f : X → IR ∪ {+∞} be lsc
f is convex iff ∂f is monotone
iff ∀ x ∗ ∈ ∂f (x), y ∗ ∈ ∂f (y ), hy ∗ − x ∗ , y − xi ≥ 0
f is pseudoconvex iff ∂f is pseudomonotone
iff ∃ x ∗ ∈ ∂f (x) : hx ∗ , y − xi ≥ 0
⇒ ∀ y ∗ ∈ ∂f (y ), hy ∗ , y − xi ≥ 0
f is quasiconvex iff ∂f is quasimonotone
iff ∃ x ∗ ∈ ∂f (x) : hx ∗ , y − xi > 0
⇒ ∀ y ∗ ∈ ∂f (y ), hy ∗ , y − xi ≥ 0
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
monotone
⇓
pseudomonotone
⇓
quasimonotone
Some previous attempts:
J.C. Yao (1994) with pseudomonotone map on reflexive space
N. Hadjisavvas and S. Schaible (1996) with quasimonotone +
existence of inner points
Dinh The Luc (2001) with singlevalued densely pseudomonotone
maps
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Existence: simplified form
Theorem
C nonempty convex compact subset of X .
∗
T : C → 2X quasimonotone
+ upper hemi-continuous
+ T (x) 6= ∅ convex w∗ -compact.
Then S(T , C ) 6= ∅.
T is upper hemicontinuous on X if the restriction of T to any line segment is
usc with respect to the w ∗ -topology
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Knaster-Kuratowski-Mazurkiewicz (1929) :
C nonempty subset of X .
ϕ : C × C → IR is said to be a KKM-application if
∀ x1 , . . . , xn ∈ C , ∀ x ∈ co{x1 , . . . , xn }
∃ i ∈ 1, . . . , n such that ϕ(xi , x) ≥ 0.
Theorem
C nonempty closed convex subset of X .
ϕ : C × C → IR KKM-application
+ ϕ(x, .) usc quasiconcave, ∀ x
+ ∃ x̃ ∈ C with {y ∈ C : ϕ(x̃, y ) ≥ 0} compact.
Then
∃ ȳ ∈ C
such that ϕ(x, ȳ ) ≥ 0, ∀ x ∈ C .
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Existence: simplified form
Theorem
C nonempty convex compact subset of X .
∗
T : C → 2X quasimonotone
+ upper hemi-continuous
+ T (x) 6= ∅ convex w∗ -compact.
Then S(T , C ) 6= ∅.
Theorem
C nonempty closed convex subset of X .
ϕ : C × C → IR KKM-application
+ ϕ(x, .) usc quasiconcave, ∀ x
+ ∃ x̃ ∈ C with {y ∈ C : ϕ(x̃, y ) ≥ 0} compact.
Then
∃ ȳ ∈ C
such that ϕ(x, ȳ ) ≥ 0, ∀ x ∈ C .
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Upper sign-continuity
∗
• T : X → 2X is said to be upper sign-continuous on C iff for any
x, y ∈ C , one have :
∀ t ∈ ]0, 1[,
inf
x ∗ ∈T (xt )
=⇒
hx ∗ , y − xi ≥ 0
sup hx ∗ , y − xi ≥ 0
x ∗ ∈T (x)
where xt = (1 − t)x + ty .
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Upper sign-continuity
∗
• T : X → 2X is said to be upper sign-continuous on C iff for any
x, y ∈ C , one have :
∀ t ∈ ]0, 1[,
inf
x ∗ ∈T (xt )
=⇒
hx ∗ , y − xi ≥ 0
sup hx ∗ , y − xi ≥ 0
x ∗ ∈T (x)
where xt = (1 − t)x + ty .
upper semi-continuous
⇓
upper hemicontinuous
⇓
upper sign-continuous
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
locally upper sign continuity
Definition
∗
Let T : C → 2X be a set-valued map.
T est called locally upper sign-continuous on C if, for any x ∈ C there
exist a neigh. Vx of x and a upper sign-continuous set-valued map
∗
Φx : Vx → 2X with nonempty convex w ∗ -compact values such that
Φx (y ) ⊆ T (y ) \ {0}, ∀ y ∈ Vx
upper semi-continuous
⇓
upper hemicontinuous
⇓
upper sign-continuous
⇓ (∗)
locally upper sign-continuous
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Links ?
Proposition
∗
C nonempty convex subset of X and T : C → 2X a map.
i) If T is pseudomonotone, then LM (T , C ) = M(T , C ).
ii) If T is locally upper sign-continuous at x then
LM(T , C ) ⊆ Sw (T , C ).
iii) If, additionally to ii), the maps Sx are convex valued,
then LM(T , C ) ⊆ Sw (T , C ) = S(T , C ).
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Existence: complete form
Theorem
C nonempty convex compact subset of X .
∗
T : C → 2X quasimonotone
+ T (x) 6= ∅ convex w∗ -compact
+ upper hemicontinuous
Then
S(T , C ) 6= ∅.
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Existence: complete form
Theorem
C nonempty convex compact subset of X .
∗
T : C → 2X quasimonotone
+upper sign-continuous
Then
S(T , C ) 6= ∅.
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Existence: complete form
Theorem
C nonempty convex compact subset of X .
∗
T : C → 2X quasimonotone
+ T (x) 6= ∅ convex w∗ -compact
+ locally upper sign-continuous on C
Then
S(T , C ) 6= ∅.
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Existence: complete form
Theorem
C nonempty convex subset of X .
∗
T : C → 2X quasimonotone
+ locally upper sign-continuous on C
+ coercivity condition:
∃ ρ > 0, ∀ x ∈ C \ B(0, ρ), ∃ y ∈ C with ky k < kxk
such that ∀ x ∗ ∈ T (x), hx ∗ , x − y i ≥ 0
and there exists ρ0 > ρ such that C ∩ B(0, ρ0 ) is weakly
compact (6= ∅).
Then
S(T , C ) 6= ∅.
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
II - d - Other assumptions
Hartman and Stampacchia : sum of monotone operator and
continuous function
J.C. Yao, N.D. Yen, B.T. Kien and many others: continuous maps
with degre technics
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
References
1- D. A. & N. Hadjisavvas, On Quasimonotone Variational Inequalities, JOTA
121 (2004), 445-450.
2- D. A. & D.T. Luc, Existence Conditions in General Quasimonotone
Variational Inequalities, Bull. Austral. Math. Soc. 71 (2005), 285-303.
3- J.-C. Yao, Multivalued variational inequalities with K -pseudomonotone
operators, JOTA (1994).
4- J.-P. Crouzeix, Pseudomonotone Variational Inequalities Problems:
Existence of Solutions, Math. Programming (1997).
5- D. T. Luc, Existence results for densely pseudomonotone variational
inequality, J. Math. Anal. Appl. (2001).
6- R. John, A note on Minty Variational Inequality and Generalized
Monotonicity, Proc. GCM7 (2001).
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and