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, definitions
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 2
I- Classical approaches of quasiconvex analysis
II- Normal approach
a- First definitions
b- Adjusted sublevel sets and normal operator
III- Quasiconvex optimization
a- Optimality conditions
b- Convex constraint case
c- Nonconvex constraint case
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
I - Introduction to quasiconvex optimization
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Quasiconvexity
A function f : X → IR ∪ {+∞} is said to be quasiconvex on K if,
for all x, y ∈ K and all t ∈ [0, 1],
f (tx + (1 − t)y ) ≤ max{f (x), f (y )}.
or
for all λ ∈ IR, the sublevel set
Sλ = {x ∈ X : f (x) ≤ λ} is convex.
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Quasiconvexity
A function f : X → IR ∪ {+∞} is said to be quasiconvex on K if,
for all x, y ∈ K and all t ∈ [0, 1],
f (tx + (1 − t)y ) ≤ max{f (x), f (y )}.
or
for all λ ∈ IR, the sublevel set
Sλ = {x ∈ X : f (x) ≤ λ} is convex.
f differentiable
f is quasiconvex iff df is quasimonotone
iff df (x)(y − x) > 0 ⇒ df (y )(y − x) ≥ 0
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Quasiconvexity
A function f : X → IR ∪ {+∞} is said to be quasiconvex on K if,
for all x, y ∈ K and all t ∈ [0, 1],
f (tx + (1 − t)y ) ≤ max{f (x), f (y )}.
or
for all λ ∈ IR, the sublevel set
Sλ = {x ∈ X : f (x) ≤ λ} is convex.
f differentiable
f is quasiconvex iff df is quasimonotone
iff df (x)(y − x) > 0 ⇒ df (y )(y − x) ≥ 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
Quasiconvexity
A function f : X → IR ∪ {+∞} is said to be quasiconvex on K if,
for all x, y ∈ K and all t ∈ [0, 1],
f (tx + (1 − t)y ) ≤ max{f (x), f (y )}.
or
for all λ ∈ IR, the sublevel set
Sλ = {x ∈ X : f (x) ≤ λ} is convex.
A function f : X → IR ∪ {+∞} is said to be semistrictly quasiconvex
on K if, f is quasiconvex and for any x, y ∈ K ,
f (x) < f (y ) ⇒ f (z) < f (y ),
∀ z ∈ [x, y [.
convex ⇒ semistrictly quasiconvex ⇒ quasiconvex
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Why not a subdifferential for quasiconvex programming?
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Why not a subdifferential for quasiconvex programming?
No (upper) semicontinuity of ∂f if f is not supposed to be
Lipschitz
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Why not a subdifferential for quasiconvex programming?
No (upper) semicontinuity of ∂f if f is not supposed to be
Lipschitz
No sufficient optimality condition
H
x̄ ∈ Sstr (∂f , C )=⇒
H x̄ ∈ arg min f
H
C
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
II - Normal approach
a- First definitions
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
A first approach
Sublevel set:
Sλ = {x ∈ X : f (x) ≤ λ}
Sλ> = {x ∈ X : f (x) < λ}
Normal operator:
∗
Define Nf (x) : X → 2X by
Nf (x)
= N(Sf (x) , x)
= {x ∗ ∈ X ∗ : hx ∗ , y − xi ≤ 0, ∀ y ∈ Sf (x) }.
With the corresponding definition for Nf> (x)
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
But ...
Nf (x) = N(Sf (x) , x) has no upper-semicontinuity properties
Nf> (x) = N(Sf>(x) , x) has no quasimonotonicity properties
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
But ...
Nf (x) = N(Sf (x) , x) has no upper-semicontinuity properties
Nf> (x) = N(Sf>(x) , x) has no quasimonotonicity properties
Example
Define f : R2 → R by
f (a, b) =
|a| + |b| ,
1,
if |a| + |b| ≤ 1
.
if |a| + |b| > 1
Then f is quasiconvex.
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
But ...
Nf (x) = N(Sf (x) , x) has no upper-semicontinuity properties
Nf> (x) = N(Sf>(x) , x) has no quasimonotonicity properties
Example
Define f : R2 → R by
f (a, b) =
|a| + |b| ,
1,
if |a| + |b| ≤ 1
.
if |a| + |b| > 1
Then f is quasiconvex.
Consider x = (10, 0), x ∗ = (1, 2), y = (0, 10) and y ∗ = (2, 1).
We see that x ∗ ∈ N < (x) and y ∗ ∈ N < (y ) (since |a| + |b| < 1 implies (1, 2) · (a − 10, b) ≤ 0 and
(2, 1) · (a, b − 10) ≤ 0) while x ∗ , y − x > 0 and y ∗ , y − x < 0. Hence N < is not quasimonotone.
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
But ...another example
Nf (x) = N(Sf (x) , x) has no upper-semicontinuity properties
Nf> (x) = N(Sf>(x) , x) has no quasimonotonicity properties
Example
Then f is quasiconvex.
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
But ...another example
Nf (x) = N(Sf (x) , x) has no upper-semicontinuity properties
Nf> (x) = N(Sf>(x) , x) has no quasimonotonicity properties
Example
Then f is quasiconvex.
We easily see that N(x) is not upper semicontinuous....
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
But ...another example
Nf (x) = N(Sf (x) , x) has no upper-semicontinuity properties
Nf> (x) = N(Sf>(x) , x) has no quasimonotonicity properties
Example
Then f is quasiconvex.
We easily see that N(x) is not upper semicontinuous....
These two operators are essentially adapted to the class of semi-strictly
quasiconvex functions. Indeed in this case, for each x ∈ dom f \ arg min f ,
the sets Sf (x) and Sf<(x) have the same closure and Nf (x) = Nf< (x).
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
II - Normal approach
b- Adjusted sublevel sets
and
normal operator
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Definition
Adjusted sublevel set
For any x ∈ dom f , we define
Sfa (x) = Sf (x) ∩ B(Sf<(x) , ρx )
where ρx = dist(x, Sf<(x) ), if Sf<(x) 6= ∅
and Sfa (x) = Sf (x) if Sf<(x) = ∅.
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Definition
Adjusted sublevel set
For any x ∈ dom f , we define
Sfa (x) = Sf (x) ∩ B(Sf<(x) , ρx )
where ρx = dist(x, Sf<(x) ), if Sf<(x) 6= ∅
and Sfa (x) = Sf (x) if Sf<(x) = ∅.
Sfa (x) coincides with Sf (x) if cl(Sf>(x) ) = Sf (x)
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Definition
Adjusted sublevel set
For any x ∈ dom f , we define
Sfa (x) = Sf (x) ∩ B(Sf<(x) , ρx )
where ρx = dist(x, Sf<(x) ), if Sf<(x) 6= ∅
and Sfa (x) = Sf (x) if Sf<(x) = ∅.
Sfa (x) coincides with Sf (x) if cl(Sf>(x) ) = Sf (x)
e.g. f is semistrictly quasiconvex
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Definition
Adjusted sublevel set
For any x ∈ dom f , we define
Sfa (x) = Sf (x) ∩ B(Sf<(x) , ρx )
where ρx = dist(x, Sf<(x) ), if Sf<(x) 6= ∅
and Sfa (x) = Sf (x) if Sf<(x) = ∅.
Sfa (x) coincides with Sf (x) if cl(Sf>(x) ) = Sf (x)
e.g. f is semistrictly quasiconvex
Proposition
Let f : X → IR ∪ {+∞} be any function, with domain dom f . Then
f is quasiconvex ⇐⇒ Sfa (x) is convex , ∀ x ∈ dom f .
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Proof
Let us suppose that Sfa (u) is convex for every u ∈ dom f . We will show that for any x ∈ dom f , Sf (x) is convex.
If x ∈ arg min f then Sf (x) = Sfa (x) is convex by assumption.
Assume now that x ∈
/ arg min f and take y , z ∈ Sf (x) .
, ρx , then y , z ∈ Sfa (x) thus [y , z] ⊆ Sfa (x) ⊆ Sf (x) .
If both y and z belong to B Sf<
(x)
, ρx , then
If both y and z do not belong to B Sf<
(x)
f (x) = f (y ) = f (z), Sf<
= Sf<
= Sf<
(z)
(y )
(x)
and ρy , ρz are positive. If, say, ρy ≥ ρz then y , z ∈ B Sf<
, ρy thus
(y )
a
a
y , z ∈ Sf (y ) and [y , z] ⊆ Sf (y ) ⊆ Sf (y ) = Sf (x) .
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Proof
Finally, suppose that only one of y , z, say z, belongs to B(Sf<
, ρx ) while y ∈
/ B(Sf<
, ρx ). Then
(x)
(x)
f (x) = f (y ), Sf<
= Sf<
and ρy > ρx
(y )
(x)
so we have z ∈ B Sf<
, ρx ⊆ B Sf<
, ρy and we deduce as before that [y , z] ⊆ Sfa (y ) ⊆ Sf (y ) = Sf (x) .
(x)
(y )
The other implication is straightforward.
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Adjusted normal operator
Adjusted sublevel set:
For any x ∈ dom f , we define
Sfa (x) = Sf (x) ∩ B(Sf<(x) , ρx )
where ρx = dist(x, Sf<(x) ), if Sf<(x) 6= ∅.
Ajusted normal operator:
Nfa (x) = {x ∗ ∈ X ∗ : hx ∗ , y − xi ≤ 0, ∀ y ∈ Sfa (x)}
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Example
x
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Example
x
B(Sf<(x) , ρx )
Sfa (x) = Sf (x) ∩ B(Sf<(x) , ρx )
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Example
Sfa (x) = Sf (x) ∩ B(Sf<(x) , ρx )
Nfa (x) = {x ∗ ∈ X ∗ : hx ∗ , y − xi ≤ 0,
Didier Aussel
∀ y ∈ Sfa (x)}
Generalized Nash Equilibrium Problem: existence, uniqueness and
An exercice.........
Let us draw the normal operator value Nfa (x, y ) at the points
(x, y ) = (0.5, 0.5), (x, y ) = (0, 1), (x, y ) = (1, 0), (x, y ) = (1, 2),
(x, y ) = (1.5, 0) and (x, y ) = (0.5, 2).
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
An exercice.........
Let us draw the normal operator value Nfa (x, y ) at the points
(x, y ) = (0.5, 0.5), (x, y ) = (0, 1), (x, y ) = (1, 0), (x, y ) = (1, 2),
(x, y ) = (1.5, 0) and (x, y ) = (0.5, 2).
Operator Nfa provide information at any point!!!
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Subdifferential vs normal operator
One can have
Nfa (x) ⊆
/ cone(∂f (x))
or
a
cone(∂f (x)) ⊆
/ Nf (x)
Proposition
f is quasiconvex and x ∈ dom f
If there exists δ > 0 such that 0 6∈ ∂ L f (B(x, δ)) then
h
i
L
∞
a
cone(∂ f (x)) ∪ ∂ f (x) ⊂ Nf (x).
Proposition
f is lsc semistrictly quasiconvex and 0 6∈ ∂f (X ) then
a
cone(∂f (x)) ⊂ Nf (x).
If additionally ∂ = ∂ ↑ and f is Lipschitz then
a
Nf (x) = cone(∂f (x)).
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Basic properties of Nfa
Nonemptyness:
Proposition
Let f : X → IR ∪ {+∞} be lsc. Assume that rad. continuous on dom f
or dom f is convex and intSλ 6= ∅, ∀ λ > inf X f . Then
f is quasiconvex ⇔ Nfa (x) \ {0} =
6 ∅, ∀ x ∈ dom f \ arg min f .
Quasimonotonicity:
The normal operator Nfa is always quasimonotone
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Upper sign-continuity
∗
• T : X → 2X is said to be upper sign-continuous on K iff for any
x, y ∈ K , 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 : K → 2X be a set-valued map.
T is called locally upper sign-continuous on K if, for any x ∈ K 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
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
locally upper sign continuity
Definition
∗
Let T : K → 2X be a set-valued map.
T is called locally upper sign-continuous on K if, for any x ∈ K 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
Continuity of normal operator
Proposition
Let f be lsc quasiconvex function such that int(Sλ ) 6= ∅, ∀ λ > inf f .
Then Nf is locally upper sign-continuous on dom f \ arg min f .
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Proposition
If f is quasiconvex such that int(Sλ ) 6= ∅, ∀ λ > inf f and f is lsc
at x ∈ dom f \ arg min f ,
Then Nfa is norm-to-w∗ cone-usc at x.
∗
A multivalued map with conical valued T : X → 2X is said to be
cone-usc at x ∈ dom T if there exists a neighbourhood U of x and a
base C (u) of T (u), u ∈ U, such that u → C (u) is usc at x.
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Integration of Nfa
Let f : X → IR ∪ {+∞} quasiconvex
Question: Is it possible to characterize the functions g : X → IR ∪ {+∞}
quasiconvex such that Nfa = Nga ?
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Integration of Nfa
Let f : X → IR ∪ {+∞} quasiconvex
Question: Is it possible to characterize the functions g : X → IR ∪ {+∞}
quasiconvex such that Nfa = Nga ?
A first answer:
Let C = {g : X → IR ∪ {+∞} cont. semistrictly quasiconvex
such that argmin f is included in a closed hyperplane}
Then, for any f , g ∈ C,
Nfa = Nga
⇔ g is Nfa \ {0}-pseudoconvex
⇔ ∃ x ∗ ∈ Nfa (x) \ {0} : hx ∗ , y − xi ≥ 0 ⇒ g (x) ≤ g (y ).
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Integration of Nfa
Let f : X → IR ∪ {+∞} quasiconvex
Question: Is it possible to characterize the functions g : X → IR ∪ {+∞}
quasiconvex such that Nfa = Nga ?
A first answer:
Let C = {g : X → IR ∪ {+∞} cont. semistrictly quasiconvex
such that argmin f is included in a closed hyperplane}
Then, for any f , g ∈ C,
Nfa = Nga
⇔ g is Nfa \ {0}-pseudoconvex
⇔ ∃ x ∗ ∈ Nfa (x) \ {0} : hx ∗ , y − xi ≥ 0 ⇒ g (x) ≤ g (y ).
General case: open question
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
IV
Quasiconvex programming
a- Optimality conditions
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Quasiconvex programming
Let f : X → IR ∪ {+∞} and K ⊆ dom f be a convex subset.
(P)
find x̄ ∈ K : f (x̄) = inf f (x)
x∈K
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Quasiconvex programming
Let f : X → IR ∪ {+∞} and K ⊆ dom f be a convex subset.
(P)
find x̄ ∈ K : f (x̄) = inf f (x)
x∈K
Perfect case: f convex
f : X → IR ∪ {+∞} a proper convex function
K a nonempty convex subset of X , x̄ ∈ K + C.Q.
Then
f (x̄) = inf f (x) ⇐⇒ x̄ ∈ Sstr (∂f , K )
x∈K
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Quasiconvex programming
Let f : X → IR ∪ {+∞} and K ⊆ dom f be a convex subset.
(P)
find x̄ ∈ K : f (x̄) = inf f (x)
x∈K
Perfect case: f convex
f : X → IR ∪ {+∞} a proper convex function
K a nonempty convex subset of X , x̄ ∈ K + C.Q.
Then
f (x̄) = inf f (x) ⇐⇒ x̄ ∈ Sstr (∂f , K )
x∈K
What about f quasiconvex case?
H
x̄ ∈ Sstr (∂f (x̄), K )H
=⇒
Hx̄ ∈ arg min f
K
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Sufficient optimality condition
Theorem
f : X → IR ∪ {+∞} quasiconvex, radially cont. on dom f
C ⊆ X such that conv (C ) ⊂ dom f .
Suppose that C ⊂ int(dom f ) or AffC = X .
Then x̄ ∈ S(Nfa \ {0}, C )
=⇒
∀ x ∈ C , f (x̄) ≤ f (x).
where x̄ ∈ S(Nfa \ {0}, K ) means that there exists x̄ ∗ ∈ Nfa (x̄) \ {0} such that
hx̄ ∗ , c − xi ≥ 0,
Didier Aussel
∀ c ∈ C.
Generalized Nash Equilibrium Problem: existence, uniqueness and
Lemma
Let f : X → IR ∪ {+∞} be a quasiconvex function, radially continuous
on dom f . Then f is Nfa \ {0}-pseudoconvex on int(dom f ), that is,
∃ x ∗ ∈ Nfa (x) \ {0} : hx ∗ , y − xi ≥ 0 ⇒ f (y ) ≥ f (x).
Proof.
Let x, y ∈ int(dom f ). According to the quasiconvexity of f , Nfa (x) \ {0} is nonempty.
Let us suppose that hx ∗ , y − xi ≥ 0 with x ∗ ∈ Nfa (x) \ {0}. Let d ∈ X be such that hx ∗ , yn − xi > 0 for any
n, where yn = y + n1 d (∈ dom f for n large enough).
This implies that yn 6∈ Sf< (x) since x ∗ ∈ Nfa (x) ⊂ Nf< (x).
It follows by the radial continuity of f that f (y ) ≥ f (x).
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Necessary and Sufficient conditions
Proposition
Let C be a closed convex subset of X , x̄ ∈ C and f : X → IR be
continuous semistrictly quasiconvex such that int(Sfa (x̄)) 6= ∅ and
f (x̄) > inf X f .
Then the following assertions are equivalent:
i) f (x̄) = minC f
ii) x̄ ∈ Sstr (Nfa \ {0}, C )
iii) 0 ∈ Nfa (x̄) \ {0} + NK (C , x̄).
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Proposition
Let C be a closed convex subset of an Asplund space X and f : X → IR
be a continuous quasiconvex function. Suppose that either f is
sequentially normally sub-compact or C is sequentially normally compact.
If x̄ ∈ C is such that
0 6∈ ∂ L f (x̄)
and
0 6∈ ∂ ∞ f (x̄) \ {0} + NK (C , x̄)
then the following assertions are equivalent:
i) f (x̄) = minC f
ii) x̄ ∈ ∂ L f (x̄) \ {0} + NK (C , x̄)
iii) 0 ∈ Nfa (x̄) \ {0} + NK (C , x̄)
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
References
D. A. & N. Hadjisavvas, Adjusted sublevel sets, normal operator and
quasiconvex programming, SIAM J. Optim., 16 (2005), 358367.
D. Aussel & J. Ye, Quasiconvex minimization on locally finite union of
convex sets, J. Optim. Th. Appl., 139 (2008), no. 1, 116.
D. Aussel & J. Ye, Quasiconvex programming with starshaped constraint
region and application to quasiconvex MPEC, Optimization 55 (2006),
433-457.
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Appendix 5 - Normally compactness
A subset C is said to be sequentially normally compact at x ∈ C if for
any sequence (xk )k ⊂ C converging to x and any sequence (xk∗ )k ,
xk∗ ∈ N F (C , xk ) weakly converging to 0, one has kxk∗ k → 0.
Examples:
X finite dimensional space
C epi-Lipschitz at x
C convex with nonempty interior
A function f is said to be sequentially normally subcompact at x ∈ C if
the sublevel set Sf (x) is sequentially normally compact at x.
Examples:
X finite dimensional space
f is locally Lipschitz around x and 0 6∈ ∂ L f (x)
f is quasiconvex with int(Sf (x) ) 6= ∅
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Appendix 2 - Limiting Normal Cone
Let C be any nonempty subset of X and x ∈ C̄ . The limiting normal
cone to C at x, denoted by N L (C , x) is defined by
N L (C , x) = lim sup N F (C , x 0 )
x 0 →x
where the Frèchet normal cone N F (C , x 0 ) is defined by
(
)
hx ∗ , u − x 0 i
F
0
∗
∗
N (C , x ) = x ∈ X : lim sup
≤0 .
0
u→x 0 , u∈C ku − x k
The limiting normal cone is closed (but in general non convex)
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Appendix - Limiting subdifferential
One can define the Limiting subdifferential (in the sense of
Mordukhovich), and its asymptotic associated form, of a function
f : X → IR ∪ {+∞} by
∂ L f (x)
∂ ∞ f (x)
= Limsup f ∂ F f (y )
y →x
= x ∗ ∈ X ∗ : (x ∗ , −1) ∈ N L (epi f , (x, f (x)))
= Limsup
= x∗ ∈ X
f
y →x
λ&0
∗
λ∂ F f (y )
: (x ∗ , 0) ∈ N L (epi f , (x, f (x))) .
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and