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
Single-valuedness and single-directionality.....
or whenever a set-valued map is not really one
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
This talk is based on the following works:
D. A., N. Hadjisavvas & Y. Garcia, Single-directional property of multivalued maps and variational
systems, SIAM Optimization, 20 (2009), 1274–1285
D. A. & A. Eberhard, Maximal quasimonotonicity and dense single-directional properties of quasimonotone
operators, Math. Programming, (2013), 28 pp.
D. A. & M. Fabian Single-directional properties of quasi-monotone operators, submitted (2012), 11 pp.
D. A. & Y. Garcia, On extensions of the Kenderov’s single-valuedness result, preprint (2012), 16pp
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Setting
X Banach space
X ∗ its topological dual (w ∗ -top.)
T : X ⇒ X ∗ a set-valued map
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Setting
X Banach space
X ∗ its topological dual (w ∗ -top.)
T : X ⇒ X ∗ a set-valued map
Question: under what monotonicity assumptions T is single-valued? In
which sense?
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
I - The monotone case
Let T : X ⇒ X ∗ be a map
T is monotone iff
∀ x, y ∈ X , ∀ x ∗ ∈ T (x) and ∀ y ∗ ∈ T (x)
hy ∗ − x ∗ , y − xi ≥ 0.
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Pointwise single-valuedness
Theorem (Kenderov 1975)
Let T : X ⇒ X ∗ be a monotone set-valued map.
If T is lower semicontinuous at a point x0 ,
then T is single-valued at x0 .
T is lsc at x0 if, for every open V such that V ∩ T (x0 ) 6= ∅, there exists a
neigh. U of x0 such that V ∩ T (x) 6= ∅, for any x ∈ U.
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Local single-valuedness
Theorem (Dontchev and Hager 1994)
Let T : X ⇒ X ∗ be a monotone set-valued map. If T is
Lipschitz-like around (x0 , y ) ∈ gr T ,
then T is single-valued on a neigh. of x0 .
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Local single-valuedness
Theorem (Dontchev and Hager 1994)
Let T : X ⇒ X ∗ be a monotone set-valued map. If T is
Lipschitz-like around (x0 , y ) ∈ gr T ,
then T is single-valued on a neigh. of x0 .
• T : X ⇒ X ∗ is Lipschitz-like around (x, y ) ∈ gr T if it exist a neighb. U of x, V
neigh of y and l > 0 such that
T (u) ∩ V ⊂ T (u 0 ) + lku 0 − ukB x (0, 1),
∀ u, u 0 ∈ U
où B Y (0, 1) est la boule unité de Y .
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Local single-valuedness
Theorem (Dontchev and Hager 1994)
Let T : X ⇒ X ∗ be a monotone set-valued map. If T is
Lipschitz-like around (x0 , y ) ∈ gr T ,
then T is single-valued on a neigh. of x0 .
• T : X ⇒ X ∗ is Lipschitz-like around (x, y ) ∈ gr T if it exist a neighb. U of x, V
neigh of y and l > 0 such that
T (u) ∩ V ⊂ T (u 0 ) + lku 0 − ukB x (0, 1),
∀ u, u 0 ∈ U
où B Y (0, 1) est la boule unité de Y .
• also called Aubin property or pseudo-Lipschitzianity.
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Local single-valuedness
Theorem (Dontchev and Hager 1994)
Let T : X ⇒ X ∗ be a monotone set-valued map. If T is
Lipschitz-like around (x0 , y ) ∈ gr T ,
then T is single-valued on a neigh. of x0 .
• T : X ⇒ X ∗ is Lipschitz-like around (x, y ) ∈ gr T if it exist a neighb. U of x, V
neigh of y and l > 0 such that
T (u) ∩ V ⊂ T (u 0 ) + lku 0 − ukB x (0, 1),
∀ u, u 0 ∈ U
où B Y (0, 1) est la boule unité de Y .
• also called Aubin property or pseudo-Lipschitzianity.
• ⇒ T is nonempty valued on U.
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Local single-valuedness
Theorem (Dontchev and Hager 1994)
Let T : X ⇒ X ∗ be a monotone set-valued map. If T is
Lipschitz-like around (x0 , y ) ∈ gr T ,
then T is single-valued on a neigh. of x0 .
• T : X ⇒ X ∗ is Lipschitz-like around (x, y ) ∈ gr T if it exist a neighb. U of x, V
neigh of y and l > 0 such that
T (u) ∩ V ⊂ T (u 0 ) + lku 0 − ukB x (0, 1),
où
•
•
•
∀ u, u 0 ∈ U
B Y (0, 1) est la boule unité de Y .
also called Aubin property or pseudo-Lipschitzianity.
⇒ T is nonempty valued on U.
If T is single-valued
T is Lipschitz-like around (x, T (x)) ⇔ T is loc. Lipschitz at x
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Dense single-valuedness
if X is an Asplund space and T : X ⇒ X ∗ is a minimal w ∗ -cusco
operator on an open set D of the domain of T then, there exists a
Gδ -dense subset of D on which T is single-valued.
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Dense single-valuedness
if X is an Asplund space and T : X ⇒ X ∗ is a minimal w ∗ -cusco
operator on an open set D of the domain of T then, there exists a
Gδ -dense subset of D on which T is single-valued.
if T : X ⇒ X ∗ is a maximal monotone operator and D is an open
subset of the domain of T then, the restriction of T to D is
minimal w ∗ -cusco on D.
where
T w ∗ -cusco means that T is USC with nonempty convex w ∗ -compact values
and
A is said to be a Gδ -dense subset of B if A is dense in B and is a countable intersection
of open subsets of B.
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
I - The Quasimonotone case
Let T : X ⇒ X ∗ be a map
T is quasimonotone iff
∀ x, y ∈ X , ∀ x ∗ ∈ T (x) and ∀ y ∗ ∈ T (x)
hx ∗ , y − xi > 0
Didier Aussel
⇒ hy ∗ , y − xi ≥ 0.
Generalized Nash Equilibrium Problem: existence, uniqueness and
No hope in the quasimonotone case
No similar result (single-valued property) can be obtained in the
quasimonotone case!!.
Consider T : R ⇒ R defined by
T (x) = R++ .
then
• T is quasimonotone
T is Lipschitz-like at each point of its graph
(U, V any neigh., l any positive real)
T is multivalued everywhere!!!!
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Definition
A set-valued map T : X ⇒ X ∗ is said to be
- single-directional at x ∈ dom T if, T (x) ⊆ R+ {x ∗ } for some
x ∗ ∈ T (x).
- locally single-directional at x ∈ dom T if it exists a neigh. U of x
such that, for any x 0 ∈ U, T (x 0 ) is single-directional at x 0 .
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Definition
A set-valued map T : X ⇒ X ∗ is said to be
- single-directional at x ∈ dom T if, T (x) ⊆ R+ {x ∗ } for some
x ∗ ∈ T (x).
- locally single-directional at x ∈ dom T if it exists a neigh. U of x
such that, for any x 0 ∈ U, T (x 0 ) is single-directional at x 0 .
- strictly single-directional at x if T (x) ⊆]0, +∞[{x ∗ } for some
x ∗ 6= 0 .
- locally strictly single-directional at x if T is strictly single-directional
at any point of a neigh. of x
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
continuity concept
A map T : X ⇒ X ∗ is said to be
- lower semi-continuous at x ∈ Dom T (for the strong topology on X
and the weak∗ topology on X ∗ ) if for every weak∗ - open set V
intersecting T (x) there is a neighbourhood U of x such that
V ∩ T (x 0 ) 6= ∅ for all x 0 ∈ U.
- lower sign-continuous at x on a convex subset K if, for any
x, y ∈ K , the following implication holds:
∗
∀t ∈ ]0, 1[, ∗ inf hxt , y − xi ≥ 0 ⇒ ∗ inf hx ∗ , y − xi ≥ 0
xt ∈T (xt )
x ∈T (x)
where xt = tx + (1 − t)y .
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Upper sign-continuity
Let X be a real Banach space, X ∗ its topological dual and h·, ·i the
duality pairing. The topological dual X ∗ of X will be equipped with the
weak∗ topology.
∗
Let T : X → 2X be a set-valued map and C be a convex subset of X .
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Upper sign-continuity
Let X be a real Banach space, X ∗ its topological dual and h·, ·i the
duality pairing. The topological dual X ∗ of X will be equipped with the
weak∗ topology.
∗
Let T : X → 2X be a set-valued map and C be a convex subset of X .
N.
Hadjisavvas, JCA 10 (2003)
• T is said to be upper sign-continuous on C iff for any x, y ∈ C , one
have :
∀t ∈]0, 1[
inf
xt∗ ∈T (xt )
hxt∗ , y − xi ≥ 0 ⇒
sup hx ∗ , y − xi ≥ 0
x ∗ ∈T (x)
• T is said to be lower sign-continuous on C if, for any x, y ∈ C , the
following implication holds:
∀t ∈]0, 1[
inf
xt∗ ∈T (xt )
hxt∗ , y − xi ≥ 0 ⇒
inf hx ∗ , y − xi ≥ 0
x ∗ ∈T (x)
where xt = (1 − t)x + ty .
Lower sign-continuous ⇔ Upper sign-continuous
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
upper semi-continuous
lower semicontinuous
↓
↓
upper hemicontinuous
lower sign-continuous
&
.
upper sign-continuous
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Proposition
∗
Let T : X → 2X be a set-valued map with w ∗ -compact values and
x ∈ dom T . If T is upper semicontinuous at x then coT is upper
semicontinuous x.
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Proposition
∗
Let T : X → 2X be a set-valued map with w ∗ -compact values and
x ∈ dom T . If T is upper semicontinuous at x then coT is upper
semicontinuous x.
The reciprocal of Proposition 4 is not true in general, consider
{−1, 1} if x = 0,
T (x) =
{0}
if x 6= 0
this set-valued map is not upper semicontinuous but its convex hull map
is upper semicontinuous.
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Proposition
∗
Let T : X → 2X be a set-valued map with w ∗ -compact values and
x ∈ dom T . If T is upper semicontinuous at x then coT is upper
semicontinuous x.
The reciprocal of Proposition 4 is not true in general, consider
{−1, 1} if x = 0,
T (x) =
{0}
if x 6= 0
this set-valued map is not upper semicontinuous but its convex hull map
is upper semicontinuous.
Proposition
∗
Let T : X → 2X be a set-valued map and x ∈ dom T . Then T is upper
(lower) sign-continuous at x if and only if coT is upper (lower)
sign-continuous at x.
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Solution concepts for V.I
For V.I., classical solution concepts are:
- Stampacchia solutions
S(T , K ) = {x ∈ K ; ∃ x ∗ ∈ T (x) with hx ∗ , y − xi ≥ 0, ∀ y ∈ K }
- star Stampacchia solutions
S ∗ (T , K ) = {x ∈ K ; ∃ x ∗ ∈ T (x) \ {0}
with hx ∗ , y − xi ≥ 0, ∀ y ∈ K }
- strict Stampacchia solutions
S < (T , K ) = {x ∈ K ; ∃ x ∗ ∈ T (x)
with hx ∗ , y − xi > 0, ∀ y ∈ K \ {x}}.
Clearly one always has S < (T , K ) ⊆ S ∗ (T , K ) ⊆ S(T , K ).
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Links between solution concepts
Lemma
∗
Let K be a convex subset of X and T : K → 2X be a set-valued map.
i) If T is locally upper sign-continuous then, M(T , K ) ⊆ S ∗ (T , K ).
ii) If T is upper sign-continuous and convex w ∗ -compact valued then,
M(T , K ) ⊆ S(T , K ).
iii) If T is quasimonotone then, S < (T , K ) ⊆ M(T , K ).
iv) If K has nonempty interior and T is quasimonotone and lower
semicontinuous then, S ∗ (T , K ) ⊆ M(T , K ).
v) If K has nonempty interior and T is locally upper sign-continuous,
lower semicontinuous and quasimonotone then,
S ∗ (T , K ) = M(T , K ).
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Pointwise quasimonotone case
Theorem
Let T : X → X ∗ be a quasimonotone set-valued map and x be an
element of int(dom T ). If one of the following assumptions holds:
i) T is lower sign-continuous at x;
ii) there exists a localization TU×V of T at x which is lower
sign-continuous at x and nontrivial at x (i.e. TU×V (x) 6= {0});
then T is single-directional at x.
Corollary
Let T : X → X ∗ be a quasimonotone set-valued map. If T is lower
semi-continuous at x ∈ int(dom T ), then T is single-directional at x.
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Recovering Kenderov’73 (pointwise monotone case)
Proposition (D.A., J.-N. Corvellec & M. Lassonde ’94)
A map T : X ⇒ X ∗ is monotone if and only if, for any α∗ ∈ X ∗ , the map
T + {α∗ } is quasimonotone
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Recovering Kenderov’73 (pointwise monotone case)
Proposition (D.A., J.-N. Corvellec & M. Lassonde ’94)
A map T : X ⇒ X ∗ is monotone if and only if, for any α∗ ∈ X ∗ , the map
T + {α∗ } is quasimonotone
Proposition
Let T : X ⇒ X ∗ be a set-valued map and x be a point of its domain
dom T be such that, for any α∗ ∈ X ∗ , the map T + α∗ is
single-directional at x, then T is single-valued at x.
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Recovering Kenderov’73 (pointwise monotone case)
Hyp : T is monotone and lsc at x.
⇒ ∀ α∗ ∈ X ∗ , T + {α∗ } is quasimonotone lsc at x
⇒ ∀ α∗ ∈ X ∗ , T + {α∗ } is single-directional at x
⇒ T is single-valued at x.
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Recovering Kenderov’73 (pointwise monotone case)
Hyp : T is monotone and lsc at x.
⇒ ∀ α∗ ∈ X ∗ , T + {α∗ } is quasimonotone lsc at x
⇒ ∀ α∗ ∈ X ∗ , T + {α∗ } is single-directional at x
⇒ T is single-valued at x.
Theorem (Kenderov 1975)
Let T : X ⇒ X ∗ be a monotone set-valued map.
If T is lower semicontinuous at a point x0 ,
then T is single-valued at x0 .
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Dense single-directionality: space assumption
A topological space S is called fragmentable if there exists a metric ρ on
it (equivalently, a function ρ : S × S → [0, +∞) with ρ(x, y ) 6= 0, if and
only if x, y ∈ S are distinct) such that, for every ε > 0 and every
∅=
6 M ⊂ S, there exists an open set Ω ⊂ S such that the intersection
M ∩ Ω is non-empty and has ρ-diameter less than ε.
An important particular example of this concept comes from Asplund
spaces: If X is such, then the dual X ∗ provided with the weak∗ topology
is fragmentable (by the metric generated with the dual norm) where for
Ω one can take a weak∗ open halfspace.
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Dense single-directionality
Theorem
Let (X , k · k) be a Banach space such that X ∗ , in the weak∗ topology, is
fragmentable. Let ∅ =
6 U ⊂ X be an open set, let 0 6∈ C ⊂ X ∗ be a
∗
∗
weak compact (not necessarily convex) set, and let Q : U → 2X be a
quasi-monotone multivalued mapping such that Q(x) ∩ C 6= ∅ for every
x ∈ U. Then there exists a dense Gδ subset D ⊂ U such that Q(x) is
single-directional and lies in [0, +∞)C for every x ∈ D.
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
special case
Let us now consider a special kind of cone-valued operators.
Corollary
Let X be a Banach space such that X ∗ , in the weak∗ topology, is
∗
fragmentable. Let ∅ =
6 U ⊂ X be an open set. Let Q : U → 2X be a
quasi-monotone and cone-valued mapping. Assume that there exists
s ∈ X such that
Q(x) ⊂ {x ∗ ∈ X ∗ : kx ∗ k ≤ hx ∗ , si} for every
x ∈ U.
(1)
Then there exists a dense Gδ subset D ⊂ U such that Q(x) is
single-directional for every x ∈ D.
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Generic semicontinuity
Actually, from the developments above, some generic
upper-semi-continuity can be obtained.
Let us recall that a Banach space X is said to be Asplund generated if X = Y , where Y is an Asplund space and
BY ⊂ BX (more exactly, if there are an Asplund space Y and a continuous linear injection ι : Y ,→ X with dense
range). Note that then X ∗ , with the weak∗ topology, is fragmented by the metric
X ∗ × X ∗ 3 (x1∗ , x2∗ ) 7→ sup hx1∗ − x2∗ , BY i.
Proposition
Let (X , k · k) be a Banach space, be an open set, let 0 6∈ C ⊂ X ∗ be a
∗
weak∗ compact (not necessarily convex) set, and let Q : U → 2X be a
quasi-monotone multivalued mapping such that Q(x) ∩ C 6= ∅ for every
x ∈ U.
(i) If X ∗ , with the weak∗ topology, is fragmentable, then Q is
norm-to-weak∗ cone-upper-semi-continuous at any point of a dense
Gδ subset of U.
(ii) If X is an Asplund generated space, with Y witnessing for that,
then Q is even norm-to-k.kY cone-upper-semi-continuous at any
point of a dense Gδ subset of U, where kx ∗ kY = supy ∈BY hx ∗ , y i.
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Application to normal operator
Theorem
Let (X , k · k) be a Banach space, let ∅ =
6 V ⊂ X be an open set, let
f : V → R be a quasiconvex lower semicontinuous and solid function,
and let x ∈ V be such that argminV f is a proper subset of Sf<(x) . Then
there exists an open set x ∈ U ⊂ V such that:
(i) If X ∗ , with the weak∗ topology, is fragmentable, then the adjusted
∗
normal operator Nfa : U −→ 2X is single-directional and
∗
norm-to-weak cone-upper-semi-continuous at any point of a dense
Gδ subset of U.
(ii) If X is an Asplund generated space, with Y witnessing for that,
then Nfa is single-directional and even norm-to-k.kY
cone-upper-semicontinuous at any point of a dense Gδ subset of U,
where kx ∗ kY = supy ∈BY hx ∗ , y i.
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Application to solution maps
we will consider the solution map of the following general variational
system, i.e., the set-valued map R : X → 2Y defined by
R(x) = {y ∈ Y : 0 ∈ f (x, y ) + T (y )} ,
(2)
where X , Y are Banach spaces, f : X × Y → Y ∗ is a differentiable
∗
function, and T : Y → 2Y is a set-valued map.
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Theorem
Let us suppose that f : X × Y → Y ∗ is strictly differentiable at
(x̄, ȳ ) ∈ gr R and satisfies the ample parametrization condition:
∗
∇x f (x̄, ȳ ) is surjective. If the set-valued map T : Y → 2Y satisfies the
hypotheses
(i) T is quasi-monotone;
(ii) f (x̄, ȳ ) 6= 0 or T is convex-valued in a neighborhood of y ;
(iii) the solution map R is metrically regular around (x̄, ȳ ),
then T is locally single-directional at ȳ .
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Proof based on...
Theorem
Let X , Y , and Z be Banach spaces. Suppose that f : X × Y → Z is
strictly differentiable at (x̄, ȳ ) and satisfies the ample parametrization
condition:
∇x f (x̄, ȳ ) is surjective.
Consider a set-valued map T : Y → 2Z such that (ȳ , −f (x̄, ȳ )) is an
element of its graph. Then the set-valued map S defined by
S(y ) = x ∈ X : 0 ∈ f (x, y ) + T (y )
satisfies the Aubin property around (x̄, ȳ ) if and only if T satisfies the
Aubin property around (ȳ , −f (x̄, ȳ )).
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Proof
The metric regularity of the set-valued map R around (x̄, ȳ ) is equivalent
to the fact that the set-valued map R −1 satisfies the Aubin property
around (ȳ , x̄). But, according to the previous Theorem, this also
equivalently expresses the fact that the map T has the Aubin property
around (ȳ , −f (x̄, ȳ )). The conclusion follows from the single directional
propositions in the quasimonotone case.
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Solution map of quasi-variational inequalities
Y stands for a reflexive Banach space, equipped with a norm such that
both Y and Y ∗ are strictly convex. We will focus our interest on the
particular case of the perturbed quasi-variational inequality problem (Px )
where, for any y , the constraint set K (y ) is the sublevel set of a given
quasiconvex function g .
we will consider the following problems:
(Px≤ )
Find ȳ ∈ Y such that
hf (x, ȳ ), y − ȳ i ≥ 0
∀ y ∈ Y such that g (y ) ≤ g (ȳ ),
and
(Pxa )
find ȳ ∈ Y such that
hf (x, ȳ ), y − ȳ i ≥ 0
∀ y ∈ Sga (ȳ ),
where f : X × Y → Y ∗ is a differentiable function and
g : Y → IR ∪ {+∞} is a lower semicontinuous quasi-convex function.
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
The solution maps associated with those problems will be denoted,
respectively, by R ≤ and R a :
R≤ : X
x
→ Y,
7
→
R ≤ (x) := y ∈ Y : y solution of (Px≤ )
Ra : X
x
→ Y,
7→ R a (x) := y ∈ Y : y solution of (Pxa ) .
and
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Strict sublevel case
Theorem
Let us suppose that f : X × Y → Y ∗ is strictly differentiable at
(x̄, ȳ ) ∈ gr R ≤ with ∇x f (x̄, ȳ ) surjective and that the lower
semicontinuous semistrictly quasi-convex function g : Y → IR ∪ {+∞} is
such that Sg (ȳ ) is a polyhedron and card(I (ȳ )) > 1.
Then the solution map R ≤ is not metrically regular around (x̄, ȳ ).
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Adjusted sublevel case
Theorem
Let us suppose that f : X × Y → Y ∗ is strictly differentiable at
(x̄, ȳ ) ∈ gr R a with ∇x f (x̄, ȳ ) surjective, and the lower semicontinuous
quasi-convex function g : Y → IR ∪ {+∞} is such that the sublevel
sets
Sg (ȳ ) and cl(S < g (ȳ ) ) are polyhedra and card I (ȳ ) 6= card I < (ȳ ) .
Then the solution map R a is not metrically regular around (x̄, ȳ ).
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and