Strong convergence of extragradient method for generalized variational inequalities in Hilbert space
© Chen et al.; licensee Springer. 2014
Received: 15 December 2013
Accepted: 23 May 2014
Published: 3 June 2014
In this paper, we present a new type of extra-gradient method for generalized variational inequalities with multi-valued mapping in an infinite-dimensional Hilbert space. For this method, the generated sequence possesses an expansion property with respect to the initial point, and the existence of the solution to the problem can be verified through the behavior of the generated sequence. Furthermore, under mild conditions, we show that the generated sequence of the method strongly converges to the solution of the problem which is closest to the initial point.
where stands for the inner product of vectors in ℋ. If the multi-valued mapping F is a single-valued mapping from ℋ to ℋ, then the GVIP collapses to the classical variational inequality problem [1, 2].
Furthermore, we establish the strong convergence of the method in the case that the solution set is nonempty, and we show that the generated sequence diverges to infinity if the solution set is empty.
The rest of this paper is organized as follows. In Section 2, we give some related concepts and conclusions needed in the subsequent analysis. In Section 3, we present our designed algorithm and establish the convergence of the algorithm.
and we denote by . The well-known properties of the projection operator are as follows.
Lemma 2.1 
- (i)monotone if and only if
- (ii)pseudo-monotone if and only if, for any , , ,
To proceed, we present the definition of maximal monotone multi-valued mapping F.
is not properly contained in the graph of any other monotone operator.
It is clear that a monotone multi-valued mapping F is maximal if and only if, for any such that , , then .
upper semi-continuous at if for every open set V containing , there is an open set U containing x such that for all ;
lower semi-continuous at if given any sequence converging to x and any , there exists a sequence that converges to y;
continuous at if it is both upper semi-continuous and lower semi-continuous at x.
Throughout this paper, we assume that the multi-valued mapping is maximal monotone and continuous on X with nonempty compact convex values, where is a nonempty, closed, and convex set.
3 Main results
Then the projection residue can verify the solution set of problem (1.1).
Now, we give the description of the designed algorithm for problem (1.1), whose basic idea is as follows. At each step of the algorithm, compute the projection residue at iterate . If it is a zero vector for some , then stop with being a solution of problem (1.1); otherwise, find a trial point by a back-tracking search at along the residue , and the new iterate is obtained by projecting onto the intersection of X with two halfspaces, respectively, associated with and . Repeat this process until the projection residue is a zero vector.
Step 0: Choose , , .
Set and go to Step 1.
The following conclusion addresses the feasibility of the stepsize rule (3.1), i.e., the existence of point .
Lemma 3.1 If is not a solution of problem (1.1), then there exists a smallest non-negative integer m satisfying (3.1).
This completes the proof. □
where is a vector in . So, by the definition of and (3.3) it follows that .
the desired result follows. □
Regarding the projection step, we shall prove that the set is always nonempty, even when the solution set is empty. Therefore the whole algorithm is well defined in the sense that it generates an infinite sequence .
Lemma 3.3 If the solution set , then for all .
Thus . This shows that for all and the desired result follows. □
Lemma 3.4 Suppose that , then for all .
which implies that the solution set is nonempty. We arrive at a contradiction and the desired result follows. □
In order to establish the convergence of the algorithm, we first show the expansion property of the algorithm w.r.t. the initial point.
and the proof is completed. □
From Lemma 3.4, Algorithm 3.1 generates an infinite sequence if the solution set of problem (1.1) is empty. More precisely, we have the following conclusion.
if the solution set is empty.
for any . So, is a bounded sequence.
for some and is a solution of problem (1.1).
Otherwise, a similar argument to the one above leads to the conclusion that any weak accumulation point of is a solution of problem (1.1), which contradicts the emptiness of the solution set, and the conclusion follows. □
We are in a position to prove strong convergence of Algorithm 3.1.
Theorem 3.2 Suppose Algorithm 3.1 generates an infinite sequence . If the solution set is nonempty and the sequence is bounded away from zero, then the sequence converges strongly to a solution such that ; otherwise, . That is, the solution set of problem (1.1) is empty if and only if the sequence generated by Algorithm 3.1 diverges to infinity.
Since was taken as an arbitrary weak accumulation point of , it follows that is the unique weak accumulation point of this sequence. Since is bounded, the whole sequence weakly converges to . On the other hand, we have shown that every weakly convergent subsequence of converges strongly to . Hence, the whole sequence converges strongly to .
For the case that the solution set is empty, the conclusion can be obtained directly from Theorem 3.1. □
This work was supported by the Natural Science Foundation of China (Grant Nos. 11171180, 11101303), and the Specialized Research Fund for the Doctoral Program of Chinese Higher Education (20113705110002). The authors would like to thank the reviewers for their careful reading, insightful comments, and constructive suggestions, which helped improve the presentation of the paper.
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