Existence and uniqueness results for
variational inequalities
Lecture 4
Various complements
Didier Aussel
Univ. de Perpignan
– p. 1
Outline of lecture 4
I- About gap function
II- Remarks about QVI
II- Generalized Nash Equilibrium problem
a-
Description
b-
Reformulations of GNEP
Classical ones
An extented reformulation
A special important case
– p. 2
I - About gap functions
X a Banach space
X ∗ its topological dual (w ∗ -top.)
h·, ·i the duality product
Stampacchia variational inequality (strong):
∗
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).
– p. 3
Definition
Let us recall that a function θ : IRn → IR ∪ {+∞} is called a gap
function of the variational inequality V I(T, C) (respectively
V Iw (T, C)) if
θ(x) ≥ 0 for all x ∈ C
θ(x) = 0 if and only if x is a solution of V I(T, C) (resp.
V Iw (T, C)).
Motivations :
Stopping rule for algorithms
Estimation of speed of convergence
– p. 4
Single-valued case
Given a function f : IRn → IRn and a convex subset C of IRn , then
the variational inequality problem V I(f, C) is the problem of
finding x̄ ∈ C such that
hf (x̄), y − x̄i ≥ 0,
∀y ∈ C.
Gap function:
For the problem V I(f, C), a gap function is given by
fP (x) = suphf (x), x − yi.
y∈C
It is easy to see that fP is nonnegative and fP (x) = 0 if and only if x
solves V I(f, C).
– p. 5
Set-valued case: normalization
n
IRn
Given a set-valued map T : IR → 2 , let us introduce, the
normalized operator T̃ defined by:


if 0 ∈ T (x)
 {0}
T̃ (x) =
1
∗
∗


x
:
x
∈ T (x)
otherwise.
∗
1 + kx k
It is clear from the definition of T̃ that, for any x such that 0 6∈ T (x),
kx∗ k ≤ 1,
∀ x∗ ∈ T̃ (x).
– p. 6
Lemma 1
Let C be a nonempty convex subset of IRn and T be any set-valued
map from IRn to IRn . Then S(T, C) = S(T̃ , C) and
Sw (T, C) = Sw (T̃ , C).
n
A set-valued map T : IRn → 2IR is said to be directionally closed if,
for any x ∈ IRn , T̃ (x) is closed.
Lemma 2
n
IRn
Any bounded map T : IR → 2 with nonempty closed values is
directionally closed.
– p. 7
Gap for set-valued case
Let us introduce, for any α > 0, the following function
gP (x) = sup ∗ inf hx∗ , x − yi.
y∈C x ∈T (x)
Proposition 3
Assume that the set-valued map T is non-empty closed valued. Then,
the function gP is a gap function of the weak Stampacchia
variational inequality V Iw (T, C). Moreover if C is closed convex
and T is convex-valued then gP is also a gap function for the
Stampacchia variational inequality V I(T, C).
– p. 8
Error bound
Let us first recall that an operator T is strongly monotone if there
exists µ > 0 such that, for any x, y ∈ dom T and any x∗ ∈ T (x),
y ∗ ∈ T (y), one has
hy ∗ − x∗ , y − xi ≥ µky − xk2 .
If T is µ-strongly monotone, then one has the following error bound:
for any x ∈ C (x 6= x̄),
s
gP (x)
||x − x̄|| ≤
.
µ
– p. 9
But...
Proposition 4 Let C be a bounded subset of IRn and
n
IRn
T : IR → 2 be a set-valued map with nonempty values on C.
Then, gP is finite-valued on C.
– p. 10
Regularized gap function
We will consider, for any β > 0, the function:
GβP (x) =
∗
inf
kx
−
p
(x
−
βx
)k
C
∗
x ∈T (x)
where pC (u) denotes the projection of the point u on the nonempty
closed convex subset C. The function GβP is non negative by
definition.
Proposition 5
Let C be a nonempty closed convex subset of IRn and
n
T : IRn → 2IR be a directionally closed map with nonempty values.
Then, for any β > 0, GβP is a gap function for the Stampacchia
variational inequality V I(T, C).
– p. 11
Proposition 6
Let C be a nonempty bounded convex subset of IRn and
n
IRn
T : IR → 2 be a (µ)-strongly monotone set-valued. If T is
bounded on C (by K > 0) and if x̄ ∈ C is the (unique) minimum of
V IP (T, C) then, at any point x ∈ C and for any β > 0, the
regularized gap function GβP satisfies the following error bound
formula
s
diam(C) + 2βK β
kx − x̄k ≤
GP (x).
µβ
– p. 12
II
Remarks about QVI
– p. 13
Introduction to quasivariational inequalities (QVI)
n
IRn
n
IRn
Let S : IR −→ 2 , T : IR −→ 2 . The quasivariational
inequality problem is to find some x ∈ S(x) such that
∃x∗ ∈ T (x), hx∗ , y − xi ≥ 0,
∀y ∈ S(x).
This problem is denoted QV I(T, S).
If S(x) = C, for all x ∈ Rn , then we obtain a Stampacchia
variational inequality, denoted V I(T, C).
We denote: F P (S) = {x ∈ Rn | x ∈ S(x)}
– p. 14
Existence for QVI
Proposition 7 (Kien-Wong-Yao-07’)
Let X be a nonempty compact convex set in IRn , let T : X → 2X
and S : X → 2X be two multifunctions. Assume that:
The multifunction S is lower semicontinuous with convex
values and the set F P (S) is closed.
The set T (x) is nonempty, compact for each x ∈ X and convex
for each x ∈ F P (S).
For each y ∈ X, the set
{x ∈ F P (S) : inf z∈T (x) hz, x − yi ≤ 0} is closed.
Then, the quasivariational QV I(T, S) admits a solution.
– p. 15
Uniqueness for QVI
If T is strongly monotone and S satisfies the following property :
∀ x ∈ F P (S), F P (S) ⊂ S(x).
then QV I(T, S) admits at most a solution.
– p. 16
III
Generalized Nash Equilibrium
problem
a- The model
b- Reformulations of GNEP
Classical ones
An extented reformulation
A special important case
– p. 17
Noncooperative multi-leader-follower games
The generalized Nash equilibrium problem (GNEP) is a
noncooperative game is which each player’s admissible strategy set
depends on the other players’ strategies.
More precisely, assume that there are N players and each player ν
controls variables xν ∈ Rnν . In fact xν is a strategy of the player ν.
Let us denote by x the following vector


(x1 )
 . 
.. 
x=


(xN )
and let us set n = n1 + n2 + · · · + nN . Thus x ∈ Rn .
– p. 18
Denote by x−ν the vector formed of all players decision variables
except the one of the player ν. So we can also write
x = (xν , x−ν ).
The strategy of the player ν belongs to a strategy set
xν ∈ Xν (x−ν )
which depends on the decision variables of the other players.
– p. 19
Aim of the player ν, given the strategy x−ν , is to choose a strategy
xν such that xν solves the following optimization problem
(Pν )
ν
−ν
min
θ
(x
,
x
),
ν
ν
x
subject to
xν ∈ Xν (x−ν ),
where θν (·, x−ν ) : IRν → IR is the decision function for player ν.
In fact, θν (xν , x−ν ) denotes the loss the player ν suffers when the
rival players have chosen the strategy x−ν .
– p. 20
For any given strategy vector x−ν of the rival players the solution set
of the problem (Pν ) is denoted by Sν (x−ν ).
Thus a vector x̄ is a solution of the Generalized Nash Equilibrium if
for any ν,
x̄ν ∈ Sν (x̄−ν ).
– p. 21
A particular case
Whenever the strategy set of each player does not depend on the
choice of the rival players, that is,
Xν (x−ν ) is constant := Xν
for any ν,
then the noncooperative game reduces to
Find x̄ ∈
Q
ν
Xν such that
∀ ν, θν (x̄ν , x̄−ν ) = min θν (u, x−ν )
s.t.
u ∈ Xν
– p. 22
A particular case
Whenever the strategy set of each player does not depend on the
choice of the rival players, that is,
Xν (x−ν ) is constant := Xν
for any ν,
then the noncooperative game reduces to
Find x̄ ∈
Q
ν
Xν such that
∀ ν, θν (x̄ν , x̄−ν ) = min θν (u, x−ν )
s.t.
u ∈ Xν
that is, a Nash game.
– p. 22
II - GNEP
b - Reformulations of GNEP
– p. 23
a- Classical reformulations
−ν
n−ν
Suppose that for any ν and any x ∈ IR , function θν (·, x−ν ) is
continuously differentiable and convex and Xν (x−ν ) is convex.
– p. 24
a- Classical reformulations
−ν
n−ν
Suppose that for any ν and any x ∈ IR , function θν (·, x−ν ) is
continuously differentiable and convex and Xν (x−ν ) is convex.
Denoting by
and
X(x) =
Y
Xν (x−ν ),
∀ x ∈ IRn
ν
F (x) = (∇x1 θ1 (x), . . . , ∇xN θN (x))
∈ IRn
we have the reformulation

 x̄ ∈ X(x̄) and
x̄ gene. Nash equil. ⇔
 hF (x̄), y − x̄i ≥ 0,
∀ y ∈ X(x̄)
– p. 24
a- Classical reformulations
−ν
n−ν
Suppose that for any ν and any x ∈ IR , function θν (·, x−ν ) is
continuously differentiable and convex and Xν (x−ν ) is convex.
Denoting by
and
X(x) =
Y
Xν (x−ν ),
∀ x ∈ IRn
ν
F (x) = (∇x1 θ1 (x), . . . , ∇xN θN (x))
∈ IRn
we have the reformulation

 x̄ ∈ X(x̄) and
x̄ gene. Nash equil. ⇔
 hF (x̄), y − x̄i ≥ 0,
∀ y ∈ X(x̄)
that is a quasi-variational inequality.
– p. 24
The jointly convex case
Let us consider a special form of the sets Xν (x−ν ). This form was
originally used by Rosen in his fundamental paper (1965):
Given a nonempty convex subset X of Rn , for any ν, the set
Xν (x−ν ) is given as
Xν (x−ν ) = {xν ∈ Rnν : (xν , x−ν ) ∈ X}.
– p. 25
The jointly convex case
Let us consider a special form of the sets Xν (x−ν ). This form was
originally used by Rosen in his fundamental paper (1965):
Given a nonempty convex subset X of Rn , for any ν, the set
Xν (x−ν ) is given as
Xν (x−ν ) = {xν ∈ Rnν : (xν , x−ν ) ∈ X}.
Define the Nikaido-Isoda (or Ky Fan) function Ψ : IRn × IRn → IR
defined by
Ψ(x, y) =
N
X
i=1
ν
−ν
ν
−ν
θν (x , x ) − θν (y , x )
– p. 25
Reformulation by V.I.
Then the GNEP can be reformulated in the following form
min V (x)
x∈X
where V (x) = supy∈X Ψ(x, y).
Proposition 9 If the decision functions θν are C 1 , then any
solution of the Stampacchia variational inequality defined by X and
F (x) = (∇x1 θ1 (x), . . . , ∇xN θN (x))
∈ IRn
is a solution of the GNEP.
– p. 26
b- An extended reformulation
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)}
– p. 27
Example
x
– p. 28
Example
x
B(Sf<(x) , ρx )
Sfa (x) = Sf (x) ∩ B(Sf<(x) , ρx )
– p. 28
Example
Sfa (x) = Sf (x) ∩ B(Sf<(x) , ρx )
Nfa (x) = {x∗ ∈ X ∗ : hx∗ , y − xi ≤ 0, ∀ y ∈ Sfa (x)}
– p. 28
To simplify the notations, we will denote, for any ν and any x ∈ Rn , by
Sν (x) and Aν (x−ν ) the subsets of Rnν
Sν (x) = Sθaν (·,x−ν ) (xν )
and
−ν
θ
(·,
x
).
Aν (x−ν ) = arg min
ν
n
R
ν
In order to construct the variational inequality problem we define the
n
Rn
a
following set-valued map Nθ : R → 2 which is described, for any
x = (x1 , . . . , xp ) ∈ Rn1 × . . . × Rnp , by
Nθa (x) = F1 (x) × . . . × Fp (x),
where Fν (x) =

 B ν (0, 1)

co(Nθaν (xν ) ∩ Sν (0, 1))
if xν ∈ Aν (x−ν )
otherwise
The set-valued map Nθa has nonempty convex compact values.
– p. 29
Lemma 11 Let ν ∈ {1, . . . , p}. If the function θν is continuous
quasiconvex with respect to the ν-th variable, then,
0 ∈ Fν (x̄) ⇐⇒ x̄ν ∈ Aν (x̄−ν ).
Proof. It is sufficient to consider the case of a point x̄ such that x̄ν 6∈ Aν (x̄−ν ). Since θν (·, x̄−ν ) is
continuous at x̄ν , the interior of Sν (x̄) is nonempty. Let us denote by Kν the convex cone
Kν = Nθaν (x̄ν ) = (Sν (x̄) − x̄ν )◦ .
By quasiconvexity of θν , Kν is not reduced to {0}. Let us first observe that, since Sν (x̄) has a
nonempty interior, Kν is a pointed cone, that is Kν ∩ (−Kν ) = {0}.
Now let us suppose that 0 ∈ Fν (x̄). By Caratheodory theorem, there exist vectors
vi ∈ [Kν ∩ Sν (0, 1)], i = 1, . . . , n + 1 and scalars λi ≥ 0, i = 1, . . . , n + 1 with
n+1
X
i=1
λi = 1 and 0 =
n+1
X
λi vi .
i=1
– p. 30
Since there exists at least one r ∈ {1, . . . , n + 1} such that λr > 0 we have
vr = −
n+1
X
i=1,i6=r
λi
vi
λr
which clearly shows that vr is an element of the convex cone −Kν . But vr ∈ Sν (0, 1) and thus
vr 6= 0. This contradicts the fact that Kν is pointed and the proof is complete.
– p. 31
Sufficient condition
In the following we assume that X is a given nonempty subset X of Rn , such that for any ν, the set
Xν (x−ν ) is given as
Xν (x−ν ) = {xν ∈ Rnν : (xν , x−ν ) ∈ X}.
Theorem 13 Let us assume that, for any ν, the function θν is
continuous and quasiconvex with respect to the ν-th variable. Then
every solution of S(Nθa , X) is a solution of the GNEP.
– p. 32
Sufficient condition
In the following we assume that X is a given nonempty subset X of Rn , such that for any ν, the set
Xν (x−ν ) is given as
Xν (x−ν ) = {xν ∈ Rnν : (xν , x−ν ) ∈ X}.
Theorem 15 Let us assume that, for any ν, the function θν is
continuous and quasiconvex with respect to the ν-th variable. Then
every solution of S(Nθa , X) is a solution of the GNEP.
Note that the link between GNEP and variational inequality is valid
even if the constraint set X is neither convex nor compact.
– p. 32
Sufficient optimality condition
Proposition 17
f : X → IR ∪ {+∞} quasiconvex, radially cont. on dom f
C ⊆ X such that conv(C) ⊂ dom f .
Suppose that C ⊂ int(dom f ).
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,
∀ c ∈ C.
– p. 33
Proof. Let us consider x̄ to be a solution of S(Nθa , X). There exists v ∈ Nθa (x̄) such that
hv, y − x̄i ≥ 0,
∀ y ∈ X.
(∗)
Let ν ∈ {1, . . . , p}.
If x̄ν ∈ Aν (x̄−ν ) then obviously x̄ν ∈ Solν (x̄−ν ).
Otherwise v ν ∈ Fν (x̄) = co(Nθaν (x̄ν ) ∩ Sν (0, 1)). Thus, according to the preceding Lemma, there
exist λ > 0 and uν ∈ Nθaν (x̄ν ) \ {0} satisfying v ν = λuν .
´
`
Now for any xν ∈ Xν (x̄−ν ), consider y = x̄1 , . . . , x̄ν−1 , xν , x̄ν+1 , . . . , x̄p .
From (∗) one immediately obtains that huν , xν − x̄ν i ≥ 0. Since xν is an arbitrary element of
Xν (x̄−ν ), we have that x̄ν is a solution of S(Nθaν \ {0}, Xν (x̄−ν )) and therefore, according to
Prop. 17,
x̄ν ∈ Solν (x̄−ν )
Since ν was arbitrarily chosen we conclude that x̄ solves the GNEP.
– p. 34
Necessary and sufficient condition
Theorem 19 Let us suppose that, for any ν, the loss function θν is
continuous and semistrictly quasiconvex with respect to the ν-th
variable. Further assume that the set X is a nonempty convex subset
of Rn .
Then x̄ be a solution of the GNEP if and only if x̄ is a solution of the
variational inequality S(Nθa , X).
– p. 35
Existence result 1
Theorem 21 (Ichiishi 83) Assume that for every player ν, the loss
function θν is continuous on Rn and quasiconvex with respect to the
ν-th variable. If the set-valued map Xν is continuous (usc and lsc)
and for any ν there exists N compact convex sets Kν of IRnν such
that for any x ∈ IRn with, for any ν, xν ∈ Kν , the subset Xµ (x−ν ) is
a nonempty convex closed subset of Kν , then the generalized Nash
equilibrium problem admits a solution.
– p. 36
Existence result 2
Theorem 23 Assume that for every player ν, the loss function θν
is continuous on Rn and semistrictly quasiconvex with respect to the
ν-th variable. If the set X is nonempty, convex and compact, then
the generalized Nash equilibrium problem admits a solution.
– p. 37
without the Rosen’s law
n
Nθa : Rn → 2R which is described, for any x = (x1 , . . . , xp ) ∈ Rn1 × . . . × Rnp , by
Nθa (x) = F1 (x) × . . . × Fp (x),
where Fν (x) =
8
< B ν (0, 1)
:
co(Nθaν (xν ) ∩ Sν (0, 1))
if xν ∈ Aν (x−ν )
otherwise
Theorem 24
If for any ν ∈ {1, · · · , N }, θν is continous on Rn and quasiconvex
with respect to the ν-th variable then:
x is a solution of QV I(Nθa , S) =⇒ x is a solution of GN EP
Moreover, if θν is continous on Rn and semistrictly quasiconvex with
respect to the ν-th variable, then we have the equivalence.
– p. 38
An application
Theorem 25 We suppose that :
(i) There exists a convex compact K such that S(K) ⊂ K.
(ii) For all x ∈ K, S(x) is convex, S is lsc, and the set F P (S) is
closed.
(iii) For all ν ∈ {1, ..., p}, θν is continous on Rn and semistrictly
quasiconvex with respect to the ν-th variable.
Then GNEP admits at least one solution.
For this result from B.T Kien, N.C. Wong, J.C Yao, JOTA (2007).
– p. 39
III -c
A special important case
– p. 40
Let us consider the leader-follower game with a unique leader. This
problem corresponds to the following bilevel programming problem:
(BL)
inf fu (x, y)
n
s.t.
gk (x, y) ≤ 0, k = 1, . . . , p
where y is a solution of the lower level problem
(P Lx )
inf′ fl (x, y ′ )
y

 (x, y ′ ) ∈ C
s.t.
 hj (x, y ′ ) ≤ 0, j = 1, . . . , q
– p. 41
Let us consider the set-valued map S : X → 2Y which associates to
any point x the (possibly empty) solution set of the lower level
problem (P Lx ) defined by x, that is
S(x) = arg min fl (x, ·)
Ω(x)
where Ω(x) is the feasible region of the lower level problem (P Lx ),
namely
Ω(x) = {(x, y) ∈ X×Y : (x, y) ∈ C and hj (x, y) ≤ 0, j = 1, . . . , q}.
– p. 42
Then the problem becomes:
(BL)
inf fu (x, y)

 g (x, y) ≤ 0, k = 1, . . . , p
k
s.t.
 (x, y) ∈ Gr (S).
– p. 43
Then the problem becomes:
(BL)
inf fu (x, y)

 g (x, y) ≤ 0, k = 1, . . . , p
k
s.t.
 (x, y) ∈ Gr (S).
Thus a Variational Inequality
– p. 43
Existence for bilevel
Theorem 26 Let fu : X × Y → IR ∪ {+∞} be a lsc quasiconvex function,
radially continuous on dom fu . Assume that
a) for every λ > inf X×Y fu , int(Sλ (fu )) 6= ∅;
b) Gr (S) is a locally finite union of a family {Kα : α ∈ A} of closed convex
sets of X × Y ;
c) the functions gk (k = 1, . . . , p) are lsc quasiconvex on X × Y ;
d) one of the following assumption holds:
i) for any α ∈ A, Kα is weakly compact and bounded;
ii) the set M = {(x, y) : gk (x, y) ≤ 0, k = 1, . . . , p} is weakly
compact and bounded;
e) the feasible region of problem (BL) is nonempty.
Then if A is finite the bilevel programming problem (BL) admits a global
solution; Then if A is not finite but there exists a local mapping
M = {(ρw , Aw ) : w ∈ Gr (S)} of Gr (S) such that the set
{w ∈ Gr (S) : card(Aw ) > 1} is included in a weakly compact subset of Gr (S),
– p. 44
The linear Bilevel problem
Let us consider the following particular case of the bilevel problem
(BL_Lin)
inf fu (x, y)

 y ∈ S(x)
s.t.
 L1 (x, y) ≤ 0
where S(x) is the solution set of the linear lower level problem
(P Lx _Lin)
inf′ L2 (x, y ′ )
y
s.t. L3 (x, y ′ ) ≤ 0
where L1 , L2 and L3 are linear continuous functions defined on
X × Y with, respectively, values in IRp , IR and IRq .
– p. 45
Existence for linear Bilevel problem
Corollary 28 Let fu : X × Y → IR ∪ {+∞} be a lower
semicontinuous quasiconvex function, radially continuous on
dom fu . Assume that
a) for every λ > inf X×Y fu , int(Sλ (fu )) 6= ∅;
c) the functions L1 , L2 and L3 are linear continuous functions;
d) the set M = {(x, y) : L3 (x, y) ≤ 0, k = 1, . . . , p} is weakly
compact and bounded;
e) the feasible region of problem (BL) is nonempty.
Then the bilevel programming problem (BL_Lin) admits a global
solution.
– p. 46
References
J.-S. Pang & M. Fukushima, Quasivariational enequalities,
Generalized Nash Equilibrium, and multi-leader-follower games
CMS (2005).
F. Facchinei & C. Kanzow, Generalized Nash Equilibrium Problem
4OR5 (2007).
D. Aussel & J. Dutta, Generalized Nash Equilibrium Problem,
Variational inequalities and Quasiconvexity, Oper. Reasearch Letter
(2008).
D. Aussel & J. Ye, Quasiconvex programming on locally finite union
of convex sets, JOTA, (2008).
D. Aussel & J. Ye, Quasiconvex programming with starshaped
constraint region and application to quasiconvex MPEC,
Optimization 55 (2006), 433-457.
– p. 47