Nonlinear Problems: Analytical and Computational Approach with ApplicationsView this Special Issue
Strong Convergence of a Modified Extragradient Method to the Minimum-Norm Solution of Variational Inequalities
We suggest and analyze a modified extragradient method for solving variational inequalities, which is convergent strongly to the minimum-norm solution of some variational inequality in an infinite-dimensional Hilbert space.
Let be a closed convex subset of a real Hilbert space . A mapping is called -inverse-strongly monotone if there exists a positive real number such that The variational inequality problem is to find such that The set of solutions of the variational inequality problem is denoted by . It is well known that variational inequality theory has emerged as an important tool in studying a wide class of obstacle, unilateral, and equilibrium problems, which arise in several branches of pure and applied sciences in a unified and general framework. Several numerical methods have been developed for solving variational inequalities and related optimization problems; see [1–36] and the references therein.
It is well known that variational inequalities are equivalent to the fixed point problem. This alternative formulation has been used to study the existence of a solution of the variational inequality as well as to develop several numerical methods. Using this equivalence, one can suggest the following iterative method.
Algorithm 1.1. For a given , calculate the approximate solution by the iterative scheme It is well known that the convergence of Algorithm 1.1 requires that the operator must be both strongly monotone and Lipschitz continuous. These restrict conditions rules out its applications in several important problems. To overcome these drawbacks, Korpelevič suggested in  an algorithm of the form Noor  further suggested and analyzed the following new iterative methods for solving the variational inequality (1.2).
Algorithm 1.2. For a given , calculate the approximate solution by the iterative scheme
which is known as the modified extragradient method. For the convergence analysis of Algorithm 1.2, see Noor [1, 2] and the references therein. We would like to point out that Algorithm 1.2 is quite different from the method of Korpelevič . However, Algorithm 1.2 fails, in general, to converge strongly in the setting of infinite-dimensional Hilbert spaces.
In this paper, we suggest and consider a very simple modified extragradient method which is convergent strongly to the minimum-norm solution of variational inequality (1.2) in an infinite-dimensional Hilbert space. This new method includes the method of Noor  as a special case.
Let be a real Hilbert space with inner product and norm , and let be a closed convex subset of . It is well known that, for any , there exists a unique such that We denote by , where is called the metric projection of onto . The metric projection of onto has the following basic properties:(i) for all ;(ii) for every ;(iii) for all , .
We need the following lemma for proving our main results.
Lemma 2.1 (see ). Assume that is a sequence of nonnegative real numbers such that where is a sequence in and is a sequence such that(1);(2) or .Then .
3. Main Result
In this section we will state and prove our main result.
Theorem 3.1. Let be a closed convex subset of a real Hilbert space . Let be an -inverse-strongly monotone mapping. Suppose that . For given arbitrarily, define a sequence iteratively by where is a sequence in and is a constant. Assume the following conditions are satisfied:(): ;(): ; (): . Then the sequence generated by (3.1) converges strongly to which is the minimum-norm element in .
We will divide our detailed proofs into several conclusions.
Proof. Take . First we need to use the following facts: (1); in particular, (2) is nonexpansive and for all
From (3.1), we have
Therefore, is bounded and so are , and .
From (3.1), we have where is a constant such that . Hence, by Lemma 2.1, we obtain From (3.4), (3.5) and the convexity of the norm, we deduce Therefore, we have Since and as , we obtain as .
By the property (ii) of the metric projection , we have It follows that and hence which implies that Since , and , we derive .
Next we show that where . To show it, we choose a subsequence of such that As is bounded, we have that a subsequence of converges weakly to .
Next we show that . We define a mapping by Then is maximal monotone (see ). Let . Since and , we have . On the other hand, from , we have that is, Therefore, we have Noting that , , and is Lipschitz continuous, we obtain . Since is maximal monotone, we have , and hence . Therefore, Finally, we prove . By the property (ii) of metric projection , we have Hence, Therefore, We apply Lemma 2.1 to the last inequality to deduce that . This completes the proof.
Y. Yao was supported in part by Colleges and Universities Science and Technology Development Foundation (20091003) of Tianjin, NSFC 11071279 and NSFC 71161001-G0105. Y.-C. Liou was partially supported by the Program TH-1-3, Optimization Lean Cycle, of Sub-Projects TH-1 of Spindle Plan Four in Excellence Teaching and Learning Plan of Cheng Shiu University and was supported in part by NSC 100-2221-E-230-012.
G. M. Korpelevič, “An extragradient method for finding saddle points and for other problems,” Èkonomika i Matematicheskie Metody, vol. 12, no. 4, pp. 747–756, 1976.View at: Google Scholar
Y. Yao and N. Shahzad, “New methods with perturbations for non-expansive mappings in hilbert spaces,” Fixed Point Theory and Applications, vol. 2011, article 79, 2011.View at: Google Scholar
Y. Yao, M. A. Noor, K. I. Noor, and Y. -C. Liou, “On an iterative algorithm for variational inequalities in banach spaces,” Mathematical Communications, vol. 16, no. 1, pp. 95–104, 2011.View at: Google Scholar