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On Two Iterative Methods for Mixed Monotone Variational Inequalities
Fixed Point Theory and Applications volume 2010, Article number: 291851 (2009)
Abstract
A mixed monotone variational inequality (MMVI) problem in a Hilbert space is formulated to find a point
such that
for all
, where
is a monotone operator and
is a proper, convex, and lower semicontinuous function on
. Iterative algorithms are usually applied to find a solution of an MMVI problem. We show that the iterative algorithm introduced in the work of Wang et al., (2001) has in general weak convergence in an infinite-dimensional space, and the algorithm introduced in the paper of Noor (2001) fails in general to converge to a solution.
1. Introduction
Let be a real Hilbert space with inner product
and norm
and let
be an operator with domain
and range
in
. Recall that
is monotone if its graph
is a monotone set in
. This means that
is monotone if and only if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F291851/MediaObjects/13663_2009_Article_1252_Equ1_HTML.gif)
A monotone operator is maximal monotone if its graph
is not properly contained in the graph of any other monotone operator on
.
Let be a proper, convex, and lower semicontinuous functional. Thesubdifferential of
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F291851/MediaObjects/13663_2009_Article_1252_Equ2_HTML.gif)
It is well known (cf. [1]) that is a maximal monotone operator.
Themixed monotone variational inequality (MMVI) problem is to find a point with the property
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F291851/MediaObjects/13663_2009_Article_1252_Equ3_HTML.gif)
where is a monotone operator and
is a proper, convex, and lower semicontinuous function on
.
If one takes to be the indicator of a closed convex subset
of
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F291851/MediaObjects/13663_2009_Article_1252_Equ4_HTML.gif)
then the MMVI (1.3) is reduced to the classical variational inequality (VI):
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F291851/MediaObjects/13663_2009_Article_1252_Equ5_HTML.gif)
Recall that theresolvent of a monotone operator is defined as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F291851/MediaObjects/13663_2009_Article_1252_Equ6_HTML.gif)
If , we write
for
It is known that
is monotone if and only of for each
the resolvent
is nonexpansive, and
is maximal monotone if and only of for each
, the resolvent
is nonexpansive and defined on the entire space
. Recall that a self-mapping of a closed convex subset
of
is said to be
(i)nonexpansive if for all
;
(ii)firmly nonexpansive if for
. Equivalently,
is firmly nonexpansive if and only of
is nonexpansive. It is known that each resolvent of a monotone operator is firmly nonexpansive.
We use to denote the set of fixed points of
; that is,
.
Variational inequalities have extensively been studied; see the monographs by Baiocchi and Capelo [2], Cottle et al. [3], Glowinski et al. [4], Giannessi and Maugeri [5], and Kinderlehrer and Stampacciha [6].
Iterative methods play an important role in solving variational inequalities. For example, if is a single-valued, strongly monotone (i.e.,
for all
and some
), and Lipschitzian (i.e.,
for some
and all
) operator on
, then the sequence
generated by the iterative algorithm
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F291851/MediaObjects/13663_2009_Article_1252_Equ7_HTML.gif)
where is the identity operator and
is the metric projection of
onto
, and the initial guess
is chosen arbitrarily, converges strongly to the unique solution of VI (1.5) provided,
is small enough.
2. An Inexact Implicit Method
In this section we study the convergence of an inexact implicit method for solving the MMVI (1.3) introduced by Wang et al. [7] (see also [8, 9] for related work).
Let and
be two sequences of nonnegative numbers such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F291851/MediaObjects/13663_2009_Article_1252_Equ8_HTML.gif)
Let and
. The inexact implicit method introduced in [7] generates a sequence
defined in the following way. Once
has been constructed, the next iterate
is implicitly constructed satisfying the equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F291851/MediaObjects/13663_2009_Article_1252_Equ9_HTML.gif)
where is a sequence of nonnegative numbers such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F291851/MediaObjects/13663_2009_Article_1252_Equ10_HTML.gif)
for , and for
and
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F291851/MediaObjects/13663_2009_Article_1252_Equ11_HTML.gif)
and where is such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F291851/MediaObjects/13663_2009_Article_1252_Equ12_HTML.gif)
with given as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F291851/MediaObjects/13663_2009_Article_1252_Equ13_HTML.gif)
We note that is a solution of the MMVI (1.3) if and only if, for each
,
satisfies the fixed point equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F291851/MediaObjects/13663_2009_Article_1252_Equ14_HTML.gif)
Before discussing the convergence of the implicit algorithm (2.2), we look at a special case of (2.2), where . In this case, the MMVI (1.3) reduces to the problem of finding a
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F291851/MediaObjects/13663_2009_Article_1252_Equ15_HTML.gif)
in another word, finding an absolute minimizer of
over
. This is equivalent to solving the inclusion
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F291851/MediaObjects/13663_2009_Article_1252_Equ16_HTML.gif)
and the algorithm (2.2) is thus reduced to a special case of the Eckastein-Bertsekas algorithm [10]
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F291851/MediaObjects/13663_2009_Article_1252_Equ17_HTML.gif)
where If
, then algorithm (2.2) is reduced to a special case of Rockafellar's proximal point algorithm [11]
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F291851/MediaObjects/13663_2009_Article_1252_Equ18_HTML.gif)
Rockafellar's proximal point algorithm for finding a zero of a maximal monotone operator has received tremendous investigations; see [12–14] and the references therein.
Remark 2.1.
Theorem  5.1 of Wang et al. [7] holds true only in the finite-dimensional setting. This is because in the infinite-dimensional setting, a bounded sequence fails, in general, to have a norm-convergent subsequence. As a matter of fact, in the infinite-dimensional case, the special case of (2.2) where and
corresponds to Rockafellar's proximal point algorithm (2.11) which fails to converge in the norm topology, in general, in the infinite-dimensional setting; see Güler's counterexample [15]. This infinite-dimensionality problem occurred in several papers by Noor (see, e.g., [16–26]).
In the infinite-dimensional setting, whether or not Wang et al.'s implicit algorithm (2.2) converges even in the weak topology remains an open question. We will provide a partial answer by showing that if the operator is weak-to-strong continuous (i.e.,
takes weakly convergent sequences to strongly convergent sequences), then the implicit algorithm (2.2) does converge weakly.
We next collect the (correct) results proved in [7].
Proposition 2.2.
Assume that is generated by the implicit algorithm (2.2).
(a)For ,
is a nondecreasing function of
.
-
(b)
If
is a solution to the MMVI (1.3),
and
, then
(2.12)
(c)For any solution to the MMVI (1.3),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F291851/MediaObjects/13663_2009_Article_1252_Equ20_HTML.gif)
where satisfies
.
(d) is bounded.
(e)There is a such that
Since algorithm (2.2) is, in general, not strongly convergent, we turn to investigate its weak convergence. It is however unclear if the algorithm is weakly convergent (if the space is infinite dimensional). We present a partial answer below. But first recall that an operator is said to be weak-to-strong continuous if the weak convergence of a sequence
to a point
implies the strong convergence of the sequence
to the point
.
Theorem 2.3.
Assume that is generated by algorithm (2.2). If
is weak-to-strong continuous, then
converges weakly to a solution of the MMVI (1.3).
Proof.
Putting
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F291851/MediaObjects/13663_2009_Article_1252_Equ21_HTML.gif)
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F291851/MediaObjects/13663_2009_Article_1252_Equ22_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F291851/MediaObjects/13663_2009_Article_1252_Equ23_HTML.gif)
This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F291851/MediaObjects/13663_2009_Article_1252_Equ24_HTML.gif)
So, if weakly (hence
strongly since
is weak-to-strong continuous), it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F291851/MediaObjects/13663_2009_Article_1252_Equ25_HTML.gif)
Thus, is a solution.
To prove that the entire sequence of is weakly convergent, assume that
weakly. All we have to prove is that
. Passing through further subsequences if necessary, we may assume that
and
both exist.
For , since
strongly and since
and
are bounded, there exists an integer
such that, for
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F291851/MediaObjects/13663_2009_Article_1252_Equ26_HTML.gif)
It follows that for ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F291851/MediaObjects/13663_2009_Article_1252_Equ27_HTML.gif)
This implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F291851/MediaObjects/13663_2009_Article_1252_Equ28_HTML.gif)
However,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F291851/MediaObjects/13663_2009_Article_1252_Equ29_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F291851/MediaObjects/13663_2009_Article_1252_Equ30_HTML.gif)
Similarly, by repeating the argument above we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F291851/MediaObjects/13663_2009_Article_1252_Equ31_HTML.gif)
Adding these inequalities, we get .
3. A Counterexample
It is not hard to see that solves MMVI (1.3) if and only of
solves the inclusion
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F291851/MediaObjects/13663_2009_Article_1252_Equ32_HTML.gif)
which is in turn equivalent to the fixed point equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F291851/MediaObjects/13663_2009_Article_1252_Equ33_HTML.gif)
where is the resolvent of
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F291851/MediaObjects/13663_2009_Article_1252_Equ34_HTML.gif)
Recall that if is the indicator of a closed convex subset
of
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F291851/MediaObjects/13663_2009_Article_1252_Equ35_HTML.gif)
then MMVI (1.3) is reduced to the classical variational inequality (VI)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F291851/MediaObjects/13663_2009_Article_1252_Equ36_HTML.gif)
In [27], Noor introduced a new iterative algorithm [27, Algorithm  3.3, page 36] as follows. Given , compute
by the iterative scheme
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F291851/MediaObjects/13663_2009_Article_1252_Equ37_HTML.gif)
where and
are constant, and
is given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F291851/MediaObjects/13663_2009_Article_1252_Equ38_HTML.gif)
Noor [27] proved a convergence result for his algorithm (3.6) as follows.
Theorem 3.1 (see [27, page 38]).
Let be a finite-dimensional Hilbert space. Then the sequence
generated by algorithm (3.6) converges to a solution of MMVI (1.3).
We however found that the conclusion stated in the above theorem is incorrect. It is true that solves MMVI (1.3) if and only if
solves the fixed point equation (3.2). The reason that led Noor to his mistake is his claim that
solves MMVI (1.3) if and only if
solves the following iterated fixed point equation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F291851/MediaObjects/13663_2009_Article_1252_Equ39_HTML.gif)
As a matter of fact, the two fixed point equations (3.2) and (3.8) are not equivalent, as shown in the following counterexample which also shows that the convergence result of Noor [27] is incorrect.
Example 3.2.
Take . Define
and
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F291851/MediaObjects/13663_2009_Article_1252_Equ40_HTML.gif)
Notice that (Clarke [28])
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F291851/MediaObjects/13663_2009_Article_1252_Equ41_HTML.gif)
It is easily seen that is the unique solution to the MMVI
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F291851/MediaObjects/13663_2009_Article_1252_Equ42_HTML.gif)
Observe that equation is equivalent to the fixed point equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F291851/MediaObjects/13663_2009_Article_1252_Equ43_HTML.gif)
Now since for all
, we get that
solves (3.12) if and only if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F291851/MediaObjects/13663_2009_Article_1252_Equ44_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F291851/MediaObjects/13663_2009_Article_1252_Equ45_HTML.gif)
It follows from (3.13) that . Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F291851/MediaObjects/13663_2009_Article_1252_Equ46_HTML.gif)
But, since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F291851/MediaObjects/13663_2009_Article_1252_Equ47_HTML.gif)
we deduce that the solution set of the fixed point equation (3.12) is given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F291851/MediaObjects/13663_2009_Article_1252_Equ48_HTML.gif)
(We therefore conclude that equation is not equivalent to MMVI (1.3), as claimed by Noor [27].)
Now take the initial guess for
. Then
and we have that algorithm (3.6) generates a constant sequence
for all
. However,
is not a solution of MMVI (3.11). This shows that algorithm (3.6) may generate a sequence that fails to converge to a solution of MMVI (1.3) and Noor's result in [27] is therefore false.
Remark 3.3.
Noor has repeated his above mistake in a number of his recent articles. A partial search found that articles [20, 21, 26, 29–32] contain the same error.
References
Brezis H: Operateurs Maximaux Monotones et Semi-Groups de Contraction dans les Espaces de Hilbert. North-Holland, Amsterdam, The Netherlands; 1973.
Baiocchi C, Capelo A: Variational and Quasivariational Inequalities: Applications to Free Boundary Problems, A Wiley-Interscience Publication. John Wiley & Sons, New York, NY, USA; 1984:ix+452.
Cottle RW, Giannessi F, Lions JL: Variational Inequalities and Complementarity Problems: Theory and Applications. John Wiley & Sons, New York, NY, USA; 1980.
Glowinski R, Lions J-L, Trémolières R: Numerical Analysis of Variational Inequalities, Studies in Mathematics and Its Applications. Volume 8. North-Holland, Amsterdam, The Netherlands; 1981:xxix+776.
Giannessi F, Maugeri A: Variational Inequalities and Network Equilibrium Problems. Plenum Press, New York, NY, USA; 1995.
Kinderlehrer D, Stampacchia G: An Introduction to Variational Inequalities and Their Applications, Pure and Applied Mathematics. Volume 88. Academic Press, New York, NY, USA; 1980:xiv+313.
Wang SL, Yang H, He B: Inexact implicit method with variable parameter for mixed monotone variational inequalities. Journal of Optimization Theory and Applications 2001,111(2):431–443. 10.1023/A:1011942620208
He B: Inexact implicit methods for monotone general variational inequalities. Mathematical Programming 1999,86(1):199–217. 10.1007/s101070050086
Han D, He B: A new accuracy criterion for approximate proximal point algorithms. Journal of Mathematical Analysis and Applications 2001,263(2):343–354. 10.1006/jmaa.2001.7535
Eckstein J, Bertsekas DP: On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators. Mathematical Programming 1992,55(3):293–318. 10.1007/BF01581204
Rockafellar RT: Monotone operators and the proximal point algorithm. SIAM Journal on Control and Optimization 1976,14(5):877–898. 10.1137/0314056
Solodov MV, Svaiter BF: Forcing strong convergence of proximal point iterations in a Hilbert space. Mathematical Programming, Series A 2000,87(1):189–202.
Xu H-K: Iterative algorithms for nonlinear operators. Journal of the London Mathematical Society 2002,66(1):240–256. 10.1112/S0024610702003332
Marino G, Xu H-K: Convergence of generalized proximal point algorithms. Communications on Pure and Applied Analysis 2004,3(4):791–808.
Güler O: On the convergence of the proximal point algorithm for convex minimization. SIAM Journal on Control and Optimization 1991,29(2):403–419. 10.1137/0329022
Noor MA: Monotone mixed variational inequalities. Applied Mathematics Letters 2001,14(2):231–236. 10.1016/S0893-9659(00)00141-5
Noor MA: An implicit method for mixed variational inequalities. Applied Mathematics Letters 1998,11(4):109–113. 10.1016/S0893-9659(98)00066-4
Noor MA: A modified projection method for monotone variational inequalities. Applied Mathematics Letters 1999,12(5):83–87. 10.1016/S0893-9659(99)00061-0
Noor MA: Some iterative techniques for general monotone variational inequalities. Optimization 1999,46(4):391–401. 10.1080/02331939908844464
Noor MA: Some algorithms for general monotone mixed variational inequalities. Mathematical and Computer Modelling 1999,29(7):1–9. 10.1016/S0895-7177(99)00058-8
Noor MA: Splitting algorithms for general pseudomonotone mixed variational inequalities. Journal of Global Optimization 2000,18(1):75–89. 10.1023/A:1008322118873
Noor MA: An iterative method for general mixed variational inequalities. Computers & Mathematics with Applications 2000,40(2–3):171–176. 10.1016/S0898-1221(00)00151-6
Noor MA: Splitting methods for pseudomonotone mixed variational inequalities. Journal of Mathematical Analysis and Applications 2000,246(1):174–188. 10.1006/jmaa.2000.6776
Noor MA: A class of new iterative methods for general mixed variational inequalities. Mathematical and Computer Modelling 2000,31(13):11–19. 10.1016/S0895-7177(00)00108-4
Noor MA: Solvability of multivalued general mixed variational inequalities. Journal of Mathematical Analysis and Applications 2001,261(1):390–402. 10.1006/jmaa.2001.7533
Noor MA, Al-Said EA: Wiener-Hopf equations technique for quasimonotone variational inequalities. Journal of Optimization Theory and Applications 1999,103(3):705–714. 10.1023/A:1021796326831
Noor MA: Iterative schemes for quasimonotone mixed variational inequalities. Optimization 2001,50(1–2):29–44. 10.1080/02331930108844552
Clarke FH: Optimization and Nonsmooth Analysis, Classics in Applied Mathematics. Volume 5. 2nd edition. SIAM, Philadelphia, Pa, USA; 1990:xii+308.
Noor MA: An extraresolvent method for monotone mixed variational inequalities. Mathematical and Computer Modelling 1999,29(3):95–100. 10.1016/S0895-7177(99)00033-3
Noor MA: A modified extragradient method for general monotone variational inequalities. Computers & Mathematics with Applications 1999,38(1):19–24. 10.1016/S0898-1221(99)00164-9
Noor MA: Projection type methods for general variational inequalities. Soochow Journal of Mathematics 2002,28(2):171–178.
Noor MA: Modified projection method for pseudomonotone variational inequalities. Applied Mathematics Letters 2002,15(3):315–320. 10.1016/S0893-9659(01)00137-9
Acknowledgments
The authors are grateful to the anonymous referees for their comments and suggestions which improved the presentation of this manuscript. This paper is dedicated to Professor Wataru Takahashi on the occasion of his retirement. The second author supported in part by NSC 97-2628-M-110-003-MY3, and by DGES MTM2006-13997-C02-01.
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Lu, X., Xu, HK. & Yin, X. On Two Iterative Methods for Mixed Monotone Variational Inequalities. Fixed Point Theory Appl 2010, 291851 (2009). https://doi.org/10.1155/2010/291851
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DOI: https://doi.org/10.1155/2010/291851