- Open Access
Variant extragradient-type method for monotone variational inequalities
© Yao et al.; licensee Springer 2013
- Received: 25 May 2013
- Accepted: 26 June 2013
- Published: 12 July 2013
Korpelevich’s extragradient method has been studied and extended extensively due to its applicability to the whole class of monotone variational inequalities. In the present paper, we propose a variant extragradient-type method for solving monotone variational inequalities. Convergence analysis of the method is presented under reasonable assumptions on the problem data.
MSC:47H05, 47J05, 47J25.
We denote the solution set of this problem by . Under the monotonicity assumption, the solution set of is always closed and convex.
Convergence results for this method require some monotonicity properties of A. Note that for the method given by (3) there is no chance of relaxing the assumption on A to plain monotonicity. The typical example consists of taking and A, a rotation with a angle. A is monotone and the unique solution of is . However, it is easy to check that for all and all , therefore the sequence generated by (3) moves away from the solution, independently of the choice of the stepsize .
where is a fixed number. The difference in (4) is that A is evaluated twice and the projection is computed twice at each iteration, but the benefit is significant, because the resulting algorithm is applicable to the whole class of monotone variational inequalities. However, we note that Korpelevich assumed that A is Lipschitz continuous and that an estimate of the Lipschitz constant is available. When A is not Lipschitz continuous, or it is Lipschitz but the constant is not known, the fixed parameter λ must be replaced by stepsizes computed through an Armijo-type search, as in the following method, presented in  (see also  for another related approach).
It is proved that if A is maximal monotone, point-to-point and uniformly continuous on bounded sets, and if is nonempty, then strongly converges to .
We now know that the difficult implementation of these methods is in computational respect. First, we note that in order to get , we have to compute , which may be time-consuming. At the same time, we observe that (6) involves two half-spaces and . If the sets C, and are simple enough, then , and are easily executed. But may be complicated, so that the projection is not easily executed. This might seriously affect the efficiency of the method.
The literature on the is vast and Korpelevich’s method has received great attention from many authors, who improved it in various ways; see, e.g., [33, 39–44] and references therein. It is known that Korpelevich’s method (4) has only weak convergence in the infinite-dimensional Hilbert spaces (please refer to a recent result of Censor et al.  and ). So, to obtain strong convergence, the original method was modified by several authors. For example, in [4, 43] it was proved that some very interesting Korpelevich-type algorithms strongly converge to a solution of . Very recently, Yao et al.  suggested modified Korpelevich’s method which converges strongly to the minimum norm solution of variational inequality (1) in infinite-dimensional Hilbert spaces.
Motivated by the works given above, in the present paper, we propose a variant extragradient-type method for solving monotone variational inequalities. Strong convergence analysis of the method is presented under reasonable assumptions on the problem data in the infinite-dimensional Hilbert spaces.
In this section, we present some definitions and results that are needed for the convergence analysis of the proposed method. Let C be a closed convex subset of a real Hilbert space H.
The following result is well known.
Proposition 1 
Let C be a bounded closed convex subset of a real Hilbert space H and let A be an α-inverse strongly monotone operator of C into H. Then is nonempty.
We denote by , where is called the metric projection of H onto C. The following is a useful characterization of projections.
for all .
It is well known that is nonexpansive.
Lemma 1 
In particular, if , then is nonexpansive.
Lemma 2 
Let and be bounded sequences in a Banach space X and let be a sequence in with .
for all ;
Lemma 3 
3 Algorithm and its convergence analysis
In this section, we present the formal statement of our proposal for the algorithm.
Variant extragradient-type method
- 2.Iterative step: Given , define(7)
Remark 1 Note that algorithm (7) includes Korpelevich’s method (4) as a special case.
Next, we shall perform a study on the convergence analysis of the proposed algorithm (7).
, and .
We shall prove our main result in several steps, included into the propositions given bellow.
Proposition 3 The sequences and are bounded. Therefore, the sequences , and are all bounded.
Proof From conditions (C1) and (C2), since and , we have , for n large enough. Without loss of generality, we may assume that, for all , . So, .
Then is bounded, and so are , , and . Therefore, the proof is complete. □
Next, we estimate .
and this concludes the proof. □
Proposition 5 , where .
As is bounded, we deduce that a subsequence of converges weakly to z.
Then T is maximal monotone.
The proof of this proposition is now complete. □
Finally, by using Propositions 3-5, we prove Theorem 1.
We apply Lemma 3 to the last inequality to deduce that .
The proof of our main result is completed. □
Remark 2 Our algorithm (7) includes Korpelevich’s method (4) as a special case. However, it is well known that Korpelevich’s algorithm (4) has only weak convergence in the setting of infinite-dimensional Hilbert spaces. But our algorithm (7) has strong convergence in the setting of infinite-dimensional Hilbert spaces.
- 2.Iterative step: Given , define(11)
, and .
Then the sequence generated by (11) converges strongly to the minimum norm element in .
This indicates that is the minimum-norm element in . This completes the proof. □
Remark 3 Corollary 1 includes the main result in  as a special case.
Yonghong Yao was supported in part by NSFC 11071279 and NSFC 71161001-G0105. Yeong-Cheng Liou was partially supported by NSC 100-2221-E-230-012.
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