339
Internat. J. Math. and Math. Sci.
Vol. i0 No. 2 (1987) 339-344
ITERATIVE METHODS FOR NONLINEAR QUASI
COMPLEMENTARITY PROBLEMS
MUHAMMAD ASLAM NOOR
Mathematics Department,College of Science
King Saud University,Riyadh l1451,Saudi Arabia.
(Received April 13, 1985 and in revised form July i, 1986)
ABSTRACT.
In this paper, we consider and study an iterative algorithm
for finding the approximate solution of the nonlinear quasi complementa
rity problem of finding ueK(u)
such that
TueK*(u)
and
where m is a point-to-point mapping,
(u-m(u) ,Tu)=0
T is a
(nonlinear)
from a real Hilbert space H into itself and K* (u)
convex cone K(u)
in H.
veral special cases,
KEY WORDS AND PHRASES.
continuous mapping
is the polar cone of the
We also discuss the convergence criteria and se-
which can be obtained from our main results.
Complementarity problems, Variational inequali-
ties, Iterative methods, Converaence criteria.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES.
i.
90C33,
49D20,
65KI0.
INTRODUCTION.
The relationship between variational inequalities and complementarity
Variational
is well known.
inequalities have been extended and generalized
in various directions to study a wide class of difficulty problems arising
in different branches of mathematical and engineering sciences.
san and Lions
[I]
introduced a general class of variational inequalities
known as quasi variational
inequality,
which has many applications
imulse control and genral theory of equilibrium.
fluid dynamics,
quasi variational inequality formulation,
the solution.
in
In a
the convex set also depends on
Related to the variational inequality problem,
also a complementarity problem,
research,
Bensous-
there is
which has many applications in operations
economics equilibrium and management sciences.
known that if the underlying set in both these problems
In fact,
it is
is a convex cone,
then complementarity problem and variational inequality problem are equivalent.
This equivalent has been used quite effectively in suggesting
unified and general algorithms for solving complementarity problems,
Ahn
[2]
and Noor
see
[3,4,5]..
Motivated and inpired by the research activities in these fields,
we
introduce a new class of complementarity problems, which is the natural
340
M.A. N00R
generalization of linear and nonlinear complementarity problems studied
extensively in recent years.
After introducing the problem,
we show
We also
that its solution can be obtained from an iterative scheme.
discuss the convergence criteria for the approximate solution.
special cases, which can be obtained from our main results,
Several
are disucssed.
Our results are an extension and improvements of the previously known
results.
2.
PRELIMINARIES AND BASIC RESULTS.
Let H be a real Hilbert space with its dual H’, whose inner product
and norm are denoted by
(.,.)
II- II
and
on H,
We identify the
respectively.
Hilbert space H with its dual space H’.
We define by < partil ordering
which is consistent with the topological vector structure of H that
is the cone of all non-negative elements
v>0}
{veH
is a closed convex
cone.
Let K be a closed convex set in H.
ping
then we consider the problem of finding uEK such that
(Tu,v-u)>0,
Inequality of type
(i.I)
for all veK.
is called quasi variational inequality.
(Tu,v-u)>0,
In many important applications,
K(u)
K(u)
To be more specific,
the
such that
(1.2)
for all veK(u)
has the following form
re(u)
where m is a point-to-point mapping.
If the
then variational inequality
quasi variational inequality problem is to find uEK(u)
m(u)K
[,,,].
(i.I)
is known as variational inequality.
convex set K also depends upon the solution,
(i.i)
H is a continuous map-
If T:K
+ K,
(1.3)
In this case K* (u)
(m (u) +K)
see
For applications of quasi variational inequalities,
Related to quasi variational inequalities, we now introduce a new
class of complementarity problems,
which will be called quasi complement-
arity problem of finding u such that
u-m(u)>0,
Tu>0,
If T is a nonlinear transformation,
linear quasi complementarity problem
mation of the form u
problem
(1.4)
(u-m(u) ,Tu)
then problem
(NQCP)
(1.4)
is called the non-
If T is an affine transfor-
Mu+q, where M is a nxn matrix, q is a n-vector,
is equivalent to finding u such that
u-m(u)>0,
Mu+q>0,
(1.5)
(u-m(u) ,Mu+q)=0,
which is called linear quasi complementarity problem.
(1.5)
(1.4)
0
was introduced and studied by Dolcetta
[i0]
Problem of type
Mosco
[7]
and Pang
[,].
In particular,
(1.5)
if the mapping m is zero,
then problems
(1.4)
are equivalent to the following complementarity problems;
and
then
ITERATIVE METHODS FOR NONLINEAR QUASI COMPLEMENTARITY PROBLEMS
u>0,
Tu>0,
u>0,
Mu+q>0,
(u,Tu)
341
0
(1.6)
and
(u,Mu+q)
which have been extensively studied by Cottle
[16]
Mangasarian
vex cone K in H.
self,
and Ahn
[17]
[13,14],
Karamardian
[15],
in recent years.
(u,v)>0 for all veK}
{uEH,
Let K*
(1.7)
0,
be the polar cone of the con-
If the convex cone also depends upon the solution it-
then we introduce the generalized quasi complementarity problem of
finding ueK(u)
such that
TuEK (u) and (u-m(u) ,Tu)=0,
(1.8)
It is clear that the nonlinear quasi complementarity problem
(1.4)
is a
special case of the generalized nonlinear quasi complementarity problme
(1.8).
(1.8)
Furthermore,
if the point-to-point mapping m is zero,then problem
is equivalent to finding uEK such that
TuEK
and
(u,Tu)
0,
which is known as the generalized nonlinear complementarity problem con-
sidered and studied by Karamardian
[15]
Cottle
[14]
Noor
[3]
and
many others.
3.
MAIN RESULTS.
We need the following results,
result of Karamardian
LEMMA 3.1.
the first is a generalization of a
[ii].
and Pang
If K is the positive cone in H,
is a solution of
(1.8)
then ueK(u),
defined by
(1.3),
if and only if u satisfies the quasi variational
(1.2).
inequality
PROOF.
[15
The proof is similar to that of lemma 3.1 in
[19,20].
THEOREM 3.1
For K(u)
quasi variational inequality
given by
(1.2)
(1.3), ueK(u)
[18,11].
is a solution of
if and only if u satisfies the follow-
ing relation
u=m (u)
for some 0>0,
mE’
the positive cone in H.
is
K
For the description
[8].
see
From lemma 3.1 and theorem 3.1
problem
(3.1)
where m is an arbitrary point-to-point mapping and P
the projection of H into K,
of
+PK [u-0 (Tu) -m (u)
we conclude that the solution to
(1.8) may be obtained by computing the fixed point of the function
(3.1).
defined by
solution u
n
In fact, theorem 3.1
enables us to find the approximate
by the following iterative scheme:
Un+l
m(Un)+PK [un-PT(un)-m(un
]’
n=0,1,2,
We also need the following concepts.
DEFINITION 3.1.
i.
A mapping T:H+H is said to be
Strongly Monoton, if there exists
<Tu-Tv,
u-v>>
u-v
a constant >0 such that
for all u,vEH.
(3 2)
M.A. NOOR
342
Lipschitz Continuous,
ii.
II Tu-Tvll <811 u-vll
In particular,
8>0
if there exists a constant
it follows that
such that
for all u,vH.
e<8.
In the next theorem, we study the conditions under which the approximate
(3.2)
solution obtained from
THEOREM 3.2.
tinuous.
1.8).
Let the mapping T be strongly monotone and Lipschitz con-
If the arbitrary mapping m is Lipschitz continuous,u
solutions satisfying
H,
converges to the exact solution u of quasi
(QCP,
complementarity problem
(3.1)
and
(1.8),
then u
and u are
n
u
to
converges
strongly in
n
for
and y<i/2,
where
y is the Lipschitz constant of m.
PROOF.
The method of proof is similar to the one given in
lemma 3.1 and theorem 3.1
[5]
we observe that the solution u of
be characterized by the relation
(3.1).
Hence from
(3.1)
and
From
(1.8)
can
(3.2), we
have
II Un+l-ull <ll m(Un)-m(u)+PK[Un-P(Tu n )-m(u n )]-P K [u-p(Tu)-m(u)]ll
<_ll
m(u
since
-’ll
<
n
)-m(u)II /11
PK
u
n
u
n
-u-pu -Tu)-m(u
n
is non-expansive,
-nil /ll
u
n
see
-u-u n -u
-
n
)+m(u)il
[21,6].
II
Now by the strongly monotonicity and Lipschitz continuity of T, we obtain
II
Un-U-p (TUn-U
II -<
=+s = = ll
u -u
n
If, =
Thus using the above inequality, we have
n+l
n
where
@
2)"
=[/(I-2P+82Q
Y<2I8<1,
Since
and
forlP-
+ 2Y] < 1
> 28/y-y 2
so the fixed point problem
and consequently the Picard iterate u
n+l
(3.1)
- -182 </2-48282 (y_y2)
has a unique solution u
converges to u;
see
[22]
which
is the required result.
It is clear that the use of algorithms as constructive methods for
proving the existence of solution brings theory and computation closer
together.
Theorem 3.2 enables us to find the approximate solution of
by the following iterative scheme:
i.
ii.
Given
u
n+l
u0eH.
m(u )+P [u
n
K
n-
(Tu)-m(u )],
n
n
n=0,i,2,3
(1.8)
ITERATIVEMETHODS FOR NONLINEAR QUASI COMPLEMENTARITY PROBLEMS
where P
p>0 is
is the projection of H into K,
K
required to satisfy the following conditions:
10
<
/2-42(y-2)
82
the Lipschitz constant y is zero.
as considered by Fang
[23]
and Gana
duces to a result of Noor and Noor
inequalities.
that T is
[15]
< 1
2-an d
>2
If the point-to-point mapping is zero,
REMARK 3.1.
ional
343
m is a point-to-point mapping and
/$
2
then it implies that
Thus our algorithm is exactly the same
[24]
[21,25]
Furthermore,
theorem 3.1 re-
for the corresponding variat-
When T is of the form Tu=Mu+q,
one can easily prove
strongly monotone if and only if M is positive definite,
Obviously T is also Lipschitz continuous.
for proof.
quasi complementarity problem,
see
[18]
see
For the linear
where one can find a general algo-
rithm for finding the approximate solution along with the necessary con-
vergence criteria.
I wish to express my gratitude to Prof.
ACKNOWLEDGEMENT.
Dr. W. Oettli
for his valuable suggestions and comments.
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