On Two Iterative Methods for Mixed Monotone Variational Inequalities
© Xiwen Lu et al. 2010
Received: 22 September 2009
Accepted: 23 November 2009
Published: 7 December 2009
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.
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
It is well known (cf. ) that is a maximal monotone operator.
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
Variational inequalities have extensively been studied; see the monographs by Baiocchi and Capelo , Cottle et al. , Glowinski et al. , Giannessi and Maugeri , and Kinderlehrer and Stampacciha .
2. An Inexact Implicit Method
Let and . The inexact implicit method introduced in  generates a sequence defined in the following way. Once has been constructed, the next iterate is implicitly constructed satisfying the equation
and the algorithm (2.2) is thus reduced to a special case of the Eckastein-Bertsekas algorithm 
where If , then algorithm (2.2) is reduced to a special case of Rockafellar's proximal point algorithm 
Theorem 5.1 of Wang et al.  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 . 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 .
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 .
3. A Counterexample
which is in turn equivalent to the fixed point equation
then MMVI (1.3) is reduced to the classical variational inequality (VI)
Noor  proved a convergence result for his algorithm (3.6) as follows.
Theorem 3.1 (see [27, page 38]).
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:
(We therefore conclude that equation is not equivalent to MMVI (1.3), as claimed by Noor .)
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  is therefore false.
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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