www.elsevier.com/locate/dsw
Relatively monotone variational inequalities over product sets
Igor V. Konnov
Department of Applied Mathematics, Kazan University, UL Lenina, Kazan 420008, Russia
Received 1 February 2000; received in revised form 1 September 2000
Abstract
New concepts of monotonicity which extend the usual ones are introduced for variational inequality problems over product sets. They enable us to strengthen existence results and extend the eld of applications of the solution methods. c2001 Elsevier Science B.V. All rights reserved.
Keywords:Variational inequalities; Product sets; Relative monotonicity; Existence results
1. Introduction
Variational inequalities (VIs) proved to be a very useful tool for investigation and solutions of various equilibrium-type problems arising in Operations Re-search, Economics, Mathematical Physics and other elds. It is well known that most of such problems aris-ing in game theory, transportation and network eco-nomics have a decomposable structure, i.e., they can be formulated as VIs over Cartesian product sets; see e.g. Nagurney [11] and Ferris and Pang [4]. At the same time, most existence results for such VIs were es-tablished under either compactness of the feasible set in the norm topology or monotonicity-type assump-tions regardless of the decomposable structure of the VI; see e.g. [1,6]. In fact, Bianchi [2] considered the corresponding extensions of concepts ofP-mappings and noticed that they are not sucient to derive exis-tence results with the help of Fan’s Lemma.
The main goal of this paper is to present other monotonicity-type concepts which are suitable for
de-E-mail address:ikonnov@ksu.ru (I.V. Konnov).
composable VIs and to obtain new existence theorems for such VIs by employing Fan’s Lemma. In addition, we show that iterative methods which are convergent for usual (pseudo) monotone VIs can be, in fact, ap-plied to solve much more general problems, thus ex-tending the eld of their applications.
In our notation,Rm
¿ denotes the set of all positive
vectors inRm, i.e.,Rm
¿={a∈Rm|ai¿0i=1; : : : ; m},
edenotes the unit vector inRm, i.e.,e= (1; : : : ;1). We
use superscripts to denote dierent vectors. Subscripts are used to denote dierent scalars or components of vectors.
LetEbe a real Banach space andE∗
its dual. Then
hx; fidenotes the duality pairing betweenx∈Eand f∈E∗
,kxkdenotes the norm ofx. Also, for each set B⊆E, we denote by Bwthe weak closure ofB.
2. Basic denitions
Let M be the set of indices {1; : : : ; m}. For each s∈M, letXsbe a real Banach space andXs∗its dual.
Set X =Q
s∈MXs, so that for eachx ∈ X, we have
x= (xs|s∈M) wherexs∈Xs. Analogously, for each
s∈M, letKs be a nonempty, convex and closed set
inXsand let
K=Y
s∈M
Ks: (1)
Next, for eachs∈M, letGs:K→Xs∗be a mapping.
Then we can dene the following VI: Find x∗ ∈
K such that
X
s∈M
hGs(x∗); xs−x∗si¿0 ∀xs∈Ks; s∈M: (2)
Of course, if we dene the mapping G:K →X∗
as follows:
G(x) = (Gs(x)|s∈M); (3)
then (2) can be equivalently rewritten as the usual VI: Findx∗∈
K such that
hG(x∗
); x−x∗i
¿0 ∀x∈K:
We denote byKvthe solution set of VI (2). Now, we
recall the known monotonicity properties forG.
Denition 1. The mappingG:K→X∗
is said to be (a) monotone, if for allx; y∈K, we have
hG(x)−G(y); x−yi¿0;
andstrictly monotone, if the inequality is strict for allx6=y;
(b) pseudomonotone, if for allx; y∈K, we have
hG(y); x−yi¿0 ⇒ hG(x); y−xi¿0;
andstrictly pseudomonotone, if the second in-equality is strict for allx6=y.
It is clear that (strict) monotonicity implies (strict) pseudomonotonicity and that the reverse assertions are not true. We now give other concepts of monotonicity which extend those in Denition 1.
Denition 2. We say that the mappingG:K →X∗,
dened by (3), is
(a) relatively monotone, if there exist vectors; ∈
Rm
¿, such that for allx; y∈K, we have
X
s∈M
hsGs(x)−sGs(y); xs−ysi¿0;
andrelatively strictly monotone, if the inequality is strict for allx6=y;
(b) relatively pseudomonotone, if there exist vectors ; ∈Rm
¿ such that for allx; y∈K, we have
X
s∈M
shGs(y); xs−ysi¿0
⇒ X
s∈M
shGs(x); xs−ysi¿0;
and relatively strictly pseudomonotone, if the second inequality is strict for allx6=y.
It what follows, we reserve the symbols and for parameters associated to relative (pseudo) mono-tonicity ofG. From the denitions, it follows that rela-tive (strict) monotonicity implies relarela-tive (strict) pseu-domonotonicity, but the reverse assertions are not true in general. Moreover, relative monotonicity type prop-erties obviously extend the usual monotonicity ones.
We also need the following continuity assumption.
Denition 3. The mappingG:K→X∗
is said to be
hemicontinuous, if for anyx ∈ K; y ∈ K and ∈
[0;1], the mapping→ hG(x+z); ziwithz=y−x is continuous.
3. Existence results
Fix an element∈Rm
¿. Then we can consider two
problems associated with. The rst is to nd a point x∗∈
K such that
X
s∈M
shGs(x∗); ys−x∗si¿0 ∀ys∈Ks; s∈M: (4)
The second is to nd an elementx∗∈Ksuch that
X
s∈M
shGs(y); ys−x∗si¿0 ∀ys∈Ks; s∈M: (5)
It is clear that (4) coincides with VI (2) when=eand that (5) represents the so-called dualVI. We denote byKv() andKd() the solution sets of problems (4) and (5), respectively.
The following lemma justies the coincidence of solution sets for all VIs of form (4).
Lemma 1. (i)For each∈Rm
¿;VI (4)is equivalent
to the problem of nding an elementx∗∈Ksuch that
hGs(x∗); ys−xs∗i¿0 ∀ys∈Ks; s∈M: (6)
(ii)For each∈Rm
Proof. Obviously, (6) implies (4). The reverse asser-tion is true due to (1). Thus, (i) holds and (ii) follows immediately from (i).
We also recall the known relationship between (4) and (5).
Lemma, see e.g. Kinderlehrer and Stampacchia [7, Chapter 3, Lemma 1.5].
From Lemmas 1 and 2 we obtain the following im-mediately.
Lemma 3. If G is hemicontinuous and relatively pseudomonotone;thenKv=Kv() =Kd() =Kv().
We are now ready to establish an existence result for relatively pseudomonotone VIs.
Theorem 1. Suppose that G is hemicontinuous and relatively pseudomonotone and thatKis weakly com-pact. Then there exists a solution toVI (2).
Proof. Dene set-valued mappingsA; B:K →2K by
We divide the proof into the following three steps. (i) T
a contradiction. SinceB(y)wis weakly compact, we conclude thatT
y∈KB(y) w
6
=∅due to Fan’s Lemma [3, Lemma 10].
(ii) T
y∈KA(y) 6= ∅. From relative
pseudomono-tonicity of G it follows that B(y)⊆A(y), but A(y) is obviously weakly closed. Therefore, (i) now implies (ii).
(iii) Kv6=∅. On account of (ii), we haveKd()6=∅.
The result follows now from Lemma 3. Thus, VI (2) is solvable.
Corollary 1. Suppose thatGis hemicontinuous and relatively strictly pseudomonotone and that K is weakly compact. ThenVI (2)has a unique solution.
Proof. Due to Theorem 1, if suces to show that VI (2) has at most one solution. Assume, for contra-diction, that x′
; x′′ ∈
By relative strict pseudomonotonicity, we obtain
X
In order to obtain existence results on unbounded sets we need certain coercivity conditions.
Corollary 2. Suppose thatGis hemicontinuous and relatively pseudomonotone and that there exist a weakly compact subsetCofX and a pointy˜∈C∩K
such that
X
s∈M
shGs(x);y˜s−xsi¡0 for all x∈K\C:
ThenVI (2)is solvable.
The new concepts enable us to strengthen the pre-vious existence results, as the following example il-lustrates.
Example 1. Let Q:K → X∗
be a hemicontinu-ous monotone mapping of the form (3) and let K be weakly compact. Denote by F(Q) the family of all “weighted” mappings Q() of the form (3),
where
Qs()(x) =sQs(x); s¿0 for alls∈M:
Clearly, each element ofF(Q) is relatively monotone. Hence, due to Theorem 1, we now conclude that VI of form (2) with any cost mapping belonging toF(Q) will be solvable, whereas the general theory guaran-tees the existence of solutions only for VI with the cost mappingQ. The dierence between these results is appreciable even in the simplest case where m= 2 andG is an ane monotone mapping. For instance, setQ(x) =Ax, where
Since A is positive denite, Q is monotone. Next, set= (10=31;1) and choose Q() ∈F(Q). Clearly,
positive semidenite, hence Q() is not monotone.
Moreover, Q() is not pseudomonotone either, since
taking x = (1:2;−1:1) and y = (−1:1;1:2) yields
hQ()(y); x−yi= 0:023¿0 and hQ()(x); x−yi= −0:506¡0. At the same time,G=Q()is relatively monotone with== (3:1;1).
In a reexive Banach space setting, we can use strong monotonicity-type assumptions.
Denition 4. We say that the mappingG, dened by (3), is
(a) relatively strongly monotone with constant ¿0, if there exist vectors; ∈Rm
(b) relatively strongly pseudomonotone with con-stant ¿0, if there exist vectors; ∈Rm
It is clear that relative strong monotonicity implies relative strong pseudomonotonicity and that relative strong (pseudo) monotonicity implies relative strict (pseudo) monotonicity.
Theorem 2. LetXs; s∈M be real reexive Banach
spaces and let G be hemicontinuous and relatively strongly pseudomonotone. ThenVI (2)has a unique solution.
Proof. Due to Theorem 1 and Corollary 1 it suces to show that VI (2) is solvable. LetBs(r) denote the
closed ball with center at 0 and radiusr ¿0 inXsfor
each s∈ M. If Ks∩Bs(r)6=∅ for all s∈ M, there
exists the unique solutionxrof the following VI: Find
xr
From the relative strong pseudomonotonicity ofGis follows that the set {xr|r ¿0} is bounded. In fact,
contradiction. Hence, there exists anr′
Remark 1. The approach to establish existence re-sults in this paper can be viewed as an application of general approaches of Oettli [12] and Konnov and Schaible [10]. However, in this case, the cost map-ping of the problem under consideration, in contrast to those in [12,10], need not be of pseudomonotone type with respect to any mapping.
4. Applications in convergence theory
Most algorithms for VIs of (2) and (1) are known to be convergent under (strict) monotonicity assump-tions on G; see e.g. Patriksson [13]. Obviously, one can apply such algorithms to VI (2) in the case where Gis relatively (strictly) monotone with=. It suf-ces to replaceGwith the “weighted” mappingG(),
where
Gs()(x) =sGs(x) for alls∈M: (7)
Recently, Konnov in [8,9] proposed several algorithms which converge to a solution of VI (2) if there exists a solution of the dual VI (5) with =e. Note that Kd() 6= ∅ if G is relatively pseudomonotone and Kv() =Kv 6= ∅. Therefore, replacing G withG(), dened by (7), we can apply these algorithms as well. On the other hand, algorithms [8,9] can be applied under the pseudo P-monotonicity assumption.
Denition 5(M. Bianchi [2]). The mapping G, de-ned by (3), is said to bepseudo P-monotone, if for allx; y∈K, we have
min
s∈MhGs(y); xs−ysi
¿0 ⇒ X
s∈M
hGs(x); xs−ysi¿0:
It was noticed in [2] that the pseudo P-monotonicity is not sucient to derive existence results based on applying Fan’s Lemma. Nevertheless, ifGis pseudo P-monotone andKv 6=∅, then we see that the dual VI
(5) with=eis solvable; hence, we can apply algo-rithms [8,9] to nd a solution of the initial VI (2). It is clear that relative pseudomonotonicity with =e implies pseudo P-monotonicity, but the correspond-ing classes of mappcorrespond-ings do not contain each other in general.
Let us now turn to the simplest case whereXs=R
for s∈ M. Then VI (2) is called a box-constrained VI and involves, in particular, the well-known
com-plementarity problem. Hence, one can verify relative (pseudo) monotonicity with=by considering posi-tive deniteness of the symmetric part of∇G()(x). In
particular, aneM-mappings (see [5,14]) satisfy this property. In fact, it is known that for everyM-matrix there exists a diagonal matrix with positive diagonal entries such that the symmetric part of their product is positive denite; see [15]. Hence, each ane map-pingG of the formG(x) =Ax+b with Abeing an M-matrix is relatively monotone, but it need not be monotone in general.
It would be of particular interest to obtain similar rules to verify relative (pseudo) monotonicity in gen-eral case.
5. Conclusions
In this paper, we have suggested new concepts of (pseudo) monotonicity which are expected to be use-ful in investigating VIs over product sets. Namely, these properties allow us to strengthen existence re-sults for such VIs and enlarge the eld of applications of existing iterative algorithms.
Also, it would be worthwhile to extend the above results to multivalued [9] and vector [1] decomposable VIs. The author intends to consider these questions in forthcoming work.
Acknowledgements
This research is supported in part by RFBR Grant No. 98-01-00200 and by the R.T. Academy of Sci-ences. The author would like to thank an anonymous referee for his=her valuable comments.
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