Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 4, Issue 2, Article 28, 2003
GENERALIZED QUASI-VARIATIONAL INEQUALITIES AND DUALITY
JACQUELINE MORGAN AND MARIA ROMANIELLO
D IPARTIMENTO DI M ATEMATICA E S TATISTICA
U NIVERSITÀ DI NAPOLI F EDERICO II
V IA C INTHIA , 80126 NAPOLI , I TALY.
morgan@unina.it
D IPARTIMENTO DI O RGANIZZAZIONE A ZIENDALE E A MMINISTRAZIONE P UBBLICA
U NIVERSITÀ DELLA C ALABRIA
V IA P IETRO B UCCI , 87036 C OSENZA , I TALY.
mromanie@unina.it
Received 18 November, 2002; accepted 4 February, 2003
Communicated by A.M. Rubinov
A BSTRACT. We present a scheme which associates to a generalized quasi-variational inequality
a dual problem and generalized Kuhn-Tucker conditions. This scheme allows to solve the primal
and the dual problems in the spirit of the classical Lagrangian duality for constrained optimization problems and extends, in non necessarily finite dimentional spaces, the duality approach
obtained by A. Auslender for generalized variational inequalities. An application to social Nash
equilibria is presented together with some illustrative examples.
Key words and phrases: Generalized quasi-variational inequality, Primal and dual problems, Generalized Kuhn-Tucker conditions, Banach space, Social Nash equilibrium, Subdifferential.
2000 Mathematics Subject Classification. 65K10, 49N15, 91A10.
1. I NTRODUCTION
Let X be a real Banach space with dual X ∗ or, more generally, let X and X ∗ be two real
locally convex topological vector spaces, duals with respect to a product of duality h·, ·i (see
[14, p. 336]).
If A and K are two set-valued operators from X to X ∗ and from X to X, respectively, we
are interested to the following variational problem (in short (V P )):
(V P )
find x∗ ∈ X such that x∗ ∈ K (x∗ ) and there exists
z ∗ ∈ A(x∗ ) satisfying hz ∗ , x − x∗ i ≥ 0,
for all x ∈ K (x∗ ) .
This problem, called Generalized Quasi-Variational Inequality ([16], [8], [12], ...), generalizes
the following problems:
– variational inequalities as introduced by G. Stampacchia [17] (see also [2], [6], [11], ...)
ISSN (electronic): 1443-5756
c 2003 Victoria University. All rights reserved.
126-02
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JACQUELINE M ORGAN AND M ARIA ROMANIELLO
– generalized variational inequalities ([2], [5], [11], ...)
– quasi-variational inequalities ([6], [12], ...)
and describes various economic and engineering problems (see Section 3 and, for example, [1],
[7], [10]).
Existence results for solutions of such a problem have been given in [8] and [16], while
stability of the following problem (equivalent to (V P ) under suitable assumptions):
(V P )0 : find x∗ ∈ X such that x∗ ∈ K (x∗ ) and
inf
z ∗ ∈A(x∗ )
hz ∗ , x − x∗ i ≥ 0,
for all x ∈ K (x∗ )
has been investigated in [12].
Differently, to our knowledge there exists no results concerning a duality scheme or a numerical method which solves a generalized quasi-variational inequality. Nevertheless, in the case
of generalized variational inequalities, for constraints defined by a finite number of inequalities
and in finite dimensional spaces, A. Auslender introduced in [2] a duality scheme which associates to the Primal Problem another generalized variational inequality (with only constraints of
positivity) for which an algorithm has been developed (see [3]).
In this paper, we extend to generalized quasi-variational inequalities in non necessarily finite dimensional spaces the duality approach obtained by Auslender for generalized variational
inequalities. More precisely we present a scheme which associates to the variational problem
(V P ):
– a dual problem, called (DV P )
– Generalized Kuhn-Tucker Conditions
which allows us to solve (V P ) and (DV P ) in the spirit of the classical Lagrangian duality for
constrained optimization problems. From a numerical point of view, we point out that the dual
problem (DV P ) has a special structure which allows to apply the algorithm introduced in [3]
for generalized variational inequalities.
In Section 2, we present the duality scheme and the connections between the primal and the
dual problems through the Generalized Kuhn-Tucker Conditions. In Section 3, we apply this
method to find Social Nash Equilibria for nonzero-sum games with coupled constraints defined
by a finite number of inequalities and we give some illustrative examples.
2. D UALITY S CHEME FOR (V P )
The scheme presented in this section takes advantage of the particular structure of the setvalued operator K defined by a finite number of inequalities. More precisely, we assume that
for all x ∈ X:
K (x) = {z ∈ X/fj (x, z) ≤ 0, for all j = 1, 2, . . . , m}
where:
(H1)
fj (x, ·) : X → R ∪ {+∞} is a proper, closed and
convex function ([18]) for all j = 1, . . . , m.
Now, for all u ∈ Rm
+ , let
F (x, y) = (f1 (x, y) , . . . , fm (x, y))
(2.1)
and
(
(2.2)
G (u) =
−F (x, x) / 0 ∈ A (x) +
m
X
)
uj ∂2 fj (x, x)
j=1
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G ENERALIZED Q UASI -VARIATIONAL I NEQUALITIES AND D UALITY
3
where ∂2 fj (x, t) is the subdifferential of the function fj (x, ·) at the point t, that is:
∂2 fj (x, t) = {z ∈ X ∗ /fj (x, y) ≥ fj (x, t) + hz, y − ti ∀y ∈ X}
Definition 2.1. The Dual Problem of the problem (V P ) (in short (DV P )), is the following
generalized variational inequality:
(DV P )
∗
∗
to find u∗ ∈ Rm
+ such that there exists d ∈ G(u )
satisfying hd∗ , u − u∗ i ≥ 0,
for all u ∈ Rm
+.
The problem (DV P ) is termed a Dual Problem because we have:
Theorem 2.1. Assume that (H1) is satisfied and that x∗ is a point of X such that E (x∗ ) =
∗
∗
∗
∗
m
∩m
j=1 dom(fj (x , ·)) is an open subset of X. If (x , u ), with u ∈ R+ , satisfies the following
conditions, called ”Generalized Kuhn-Tucker Conditions”:
(KT )1 : x∗ ∈ K (x∗ ); P
∗
∗
∗
(KT )2 : 0 ∈ A (x∗ ) + m
j=1 uj ∂2 fj (x , x );
(KT )3 : F (x∗ , x∗ ) ∈ NRm
(u∗ );
+
then
(i) x∗ is a solution to (V P )
(ii) u∗ is a solution to (DV P ).
Proof. First, to prove (i) we observe that:
“ (x∗ , z ∗ ) , with z ∗ ∈ A (x∗ ) , solves (V P )”
is equivalent to
“x∗ is a solution to the optimization problem (OP )”
where (OP ) is:
(OP )
min∗ hz ∗ , x − x∗ i .
x∈K(x )
The problem (OP ) admits as classical Lagrangian the
defined by:

m
P
∗
∗

hz
,
x
−
x
i
+
uj fj (x∗ , x)



j=1
L (x, u) =
−∞




+∞
function L, from E (x∗ ) × Rm to R,
if x ∈ E (x∗ ) and u ∈ Rm
+
u∈
/ Rm
+
otherwise.
So to prove (i), it is sufficient to apply the Theorem 7.5.1 in ([14]) to the problem (OP ),
taking into account that NE(x∗ ) (x∗ ) = {0} (since E (x∗ ) is open) and ∂ (hz ∗ , x − x∗ i) =
z ∗ + NE(x∗ ) (x∗ ).
Now we prove (ii). In light of the assumption (KT )2 , it follows that −F (x∗ , x∗ ) ∈ G (u∗ ),
where F and G are defined, respectively, by (2.1) and (2.2). So, since F (x∗ , x∗ ) ∈ NRm
(u∗ )
+
by assumption (KT )3 , and
( ∗
m
v ∈ Rm
if u∗ ∈ Rm
+ / hv, u − u i ≤ 0 ∀u ∈ R+
+
∗
NR m
(u ) =
+
∅
otherwise,
then u∗ solves the problem (DV P ) defined in Definition 2.1.
Theorem 2.2. Assume that (H1) is satisfied. If x∗ is a solution to (V P ) and if:
∗
(i) E (x∗ ) = ∩m
j=1 dom(fj (x , ·)) is an open subset of X
(ii) ∃y ∈ X such that fj (x∗ , y) < 0 for all j = 1, . . . , m
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4
JACQUELINE M ORGAN AND M ARIA ROMANIELLO
∗
∗
then, there exists a point u∗ ∈ Rm
+ such that (x , u ) satisfies the Generalized Kuhn-Tucker
Conditions (KT )1 to (KT )3 (and therefore u∗ solves (DV P ) following Theorem 2.1).
Proof. Let x∗ be a solution to (V P ) and z ∗ ∈ A (x∗ ) such that hz ∗ , x − x∗ i ≥ 0 for all x ∈
∗
∗
K (x∗ ). By Theorem 7.5.2 in [14], there exists a point u∗ ∈ Rm
+ such that (x , u ) is a saddle
point for the Lagrangian L above defined. So, it results that:
0 ∈ ∂x L (x∗ , u∗ ) = z ∗ +
m
X
u∗j ∂2 fj (x∗ , x∗ )
j=1
which implies that 0 ∈ A (x∗ ) +
for all u ∈ Rm
+:
m
X
Pm
j=1
u∗j ∂2 fj (x∗ , x∗ ). Moreover, since L (x∗ , u) ≤ L (x∗ , u∗ )
uj − u∗j fj (x∗ , x∗ ) = hF (x∗ , x∗ ) , u − u∗ i ≤ 0 ∀u ∈ Rm
+
j=1
that is F (x∗ , x∗ ) ∈ NRm
(u∗ ). Therefore (x∗ , u∗ ) satisfies (KT )1 to (KT )3 and u∗ solves
+
(DV P ).
In light of Theorems 2.1 and 2.2, the variational problem (DV P ) can be considered as a dual
problem associated to (V P ).
Remark 2.3. If X = Rn , for all x ∈ X:
K (x) = C = {z ∈ X/fj (z) ≤ 0, for all j = 1, 2, . . . , m}
and the generalized quasi-variational inequality comes from an optimization problem defined
by a convex and differentiable function, then the previous theorems reduce to the classical
theorems of Convex Mathematical Programming (Theorems 3.2 and 3.3 in [2]).
Remark 2.4. Let us observe that the condition
∗
E (x∗ ) = ∩m
j=1 dom(fj (x , ·)) is an open set of X
has been needed to properly handle convex programs within the formalism of extended valued
functions ([14]).
By the previous theorems it follows that, to solve (V P ), one can solve the dual problem
(DV P ) and then, using the generalized Kuhn-Tucker condition (KT )2 , one can find the solutions of problem (V P ) proceeding as in the following example.
Example 2.1. If
K (x) = {y ∈ R/y − 2x ≤ 0 and x − y ≤ 0}
and
 x − 31 , 0 if 0 < x < 13



A (x) =
[x, 1]
if 31 ≤ x ≤ 1



∅
otherwise
then the dual problem (DV P ) associated to the primal problem (V P ) is the easier generalized
variational inequality:
to find u∗ ∈ R2+ such that there exists d∗ ∈ G(u∗ )
satisfying hd∗ , u − u∗ i ≥ 0,
J. Inequal. Pure and Appl. Math., 4(2) Art. 28, 2003
for all u ∈ R2+
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G ENERALIZED Q UASI -VARIATIONAL I NEQUALITIES AND D UALITY
where
5
 
0, u2 − u1 + 13 × {0} if − 13 < u2 − u1 < 0


 1
G (u1 , u2 ) =
,
u
−
u
× {0}
if 13 ≤ u2 − u1 ≤ 1
2
1
3



 ∅
otherwise.
The solutions to the problem (DV P ) are all the points (0, u2 ) such that 1/3 ≤ u2 ≤ 1 , so,
using the Generalized Kuhn-Tucker Condition (KT )2 , we find that all the points x∗ such that
1/3 ≤ x∗ ≤ 1 are solutions to (V P ).
3. A PPLICATION TO S OCIAL N ASH E QUILIBRIA
Let us consider a n-person noncooperative game with coupled constraints, as considered by
G. Debreu in [7]. Let Yi be a Banach space (or, more generally, a real locally convex topological
vector space) and, for the player i, let Xi ⊆ Yi be the strategy set, Ji from X = X1 × · · · × Xn
to R be the payoff function, and
Ki (x−i ) = yi ∈ Xi /fji (yi , x−i ) ≤ 0, for all j = 1, 2, . . . , mi
be the constraints depending on the strategies of the other players, where x−i is a shorthand for
(xj )j∈N \{i} . We assume that the players want to minimize their payoff function and play a Social
Nash Equilibrium [7] (also called Generalized Nash Equilibrium [10], which is a generalization
of the concept of Nash Equilibria [15]). We recall that a Social Nash Equilibrium of the game
Γ = {Xi , Ji , Ki } is a point x∗ ∈ X such that no player can uniterally decrease his payoff given
the constraints imposed on him by the other players; that is a point such that:
(SN E)
Ji (x∗ ) ≤ Ji xi , x∗−i for all xi ∈ Ki x∗−i and for all i = 1, . . . n.
It is well known that, under suitable assumptions, the Social Nash Equilibrium problem can be
put into the form of a generalized quasi-variational inequality (see for example [6, 4, 11]). More
precisely, if we assume that the following condition is satisfied:
(H2) for every x−i ∈ X−i the function Ji (·, x−i ) is convex and bounded from below on Xi ,
for all i = 1, . . . , n
then, a point x∗ is a solution to the problem (SN E) if and only if x∗ solves the following system
of generalized quasi-variational inequalities:

∗
∗
∗
∗
find
x
∈
X
such
that
x
∈
K
x
×
·
·
·
×
K
x

1
n
−1
−n






and there exist z1∗ ∈ ∂x1 J1 (x∗ ) , · · · , zn∗ ∈ ∂xn Jn (x∗ ) satisfying




hz1∗ , x1 − x∗1 i ≥ 0, for all x1 ∈ K1 x∗−1
(SN E)




..


.





hzn∗ , xn − x∗n i ≥ 0, for all xn ∈ Kn x∗−n
where ∂xi Ji is the subdifferential of Ji (·, x−i ) for all i = 1, . . . , n.
Now, if we considered the set-valued operator defined on X by:
A (x) = ∂x1 J1 (x) × · · · × ∂xn Jn (x)
and
K (x) = {y ∈ X / yi ∈ Ki (x−i ) ∀i = 1, . . . , n}
= {y ∈ X / fj (x, y) ≤ 0 j = 1, . . . , m}
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JACQUELINE M ORGAN AND M ARIA ROMANIELLO
where m = m1 + · · · + mn and
 1
fj (y1 , x−1 )






..


.




fji (yi , x−i )
fj (x, y) =




..


.





 n
fj (yn , x−n )
if j = 1, . . . , m1
if j =
Pi−1
mr + 1, . . . ,
if j =
Pn−1
mr + 1, . . . , m
r=1
r=1
Pi−1
r=1
mr + mi
then x∗ is a Social Nash Equilibrium for Γ if and only if it solves the following generalized
quasi-variational inequality:
(SN E)
find x∗ ∈ X such that x∗ ∈ K (x∗ ) and there exists
z ∗ ∈ A(x∗ ) satisfying hz ∗ , x − x∗ i ≥ 0, for all x ∈ K (x∗ ) .
If the problem (SN E) satisfies the assumptions (H1) and (H2), we can define the dual problem:
(DSN E)
∗
∗
find u∗ ∈ Rm
+ such that there exists d ∈ G(u )
satisfying hd∗ , u − u∗ i ≥ 0, for all u ∈ Rm
+,
where G is the set-valued operator defined by:
(
G (u) =
−F (x, x) / 0 ∈ ∂xh Jh (x) +
m
X
)
uj ∂xh fj (x, x) , for all h = 1, . . . , n .
j=1
Therefore, we can find the Social Nash equilibria of Γ using the method introduced in section
2, as one can see in the following example:
Example 3.1. Let us consider a two-player game Γ with
J1 (x, y) = x2 + 2x − y 2
J2 (x, y) = y 2 + 2xy
and
K1 (y) = {x ∈ R / x − y ≤ 0}
K2 (x) = {y ∈ R / 2x − y ≤ 0} .
The Social Nash Equilibrium problem associated to this game is equivalent to the following
generalized quasi-variational inequality:
(SN E)
find (x∗ , y ∗ ) ∈ K (x∗ , y ∗ )
such that (2x∗ + 2) (x − x∗ ) + (2y ∗ + 2x∗ ) (y − y ∗ ) ≥ 0
for all (x, y) ∈ K (x∗ , y ∗ ) .
Since
G (u1 , u2 ) = {(2u1 + u2 + 4/2, 3u1 + u2 + 6/2)} ,
the dual of (SN E) is the easier problem:
find u∗ ∈ R2+ such that
(DSN E)
(2u∗1 + u∗2 + 4/2) (u1 − u∗1 ) + (3u∗1 + u∗2 + 6/2) (u2 − u∗2 ) ≥ 0
for all u ∈ R2+ .
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G ENERALIZED Q UASI -VARIATIONAL I NEQUALITIES AND D UALITY
7
The unique solution of (DSN E) is (u∗1 , u∗2 ) = (0, 0) and so, by the Generalized Kuhn-Tucker
Condition (KT )2 , we have that the point (x∗ , y ∗ ) = (−1, 1) is a Social Nash Equlibrium for the
game Γ.
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