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.
- Alber YI, Iusem AN: Extension of subgradient techniques for nonsmooth optimization in Banach spaces. Set-Valued Anal. 2001, 9: 315–335. 10.1023/A:1012665832688MathSciNetView ArticleGoogle Scholar
- Bao TQ, Khanh PQ: A projection-type algorithm for pseudomonotone non-Lipschitzian multivalued variational inequalities. Nonconvex Optim. Appl. 2005, 77: 113–129. 10.1007/0-387-23639-2_6MathSciNetView ArticleGoogle Scholar
- Bauschke HH: The approximation of fixed points of composition of nonexpansive mapping in Hilbert space. J. Math. Anal. Appl. 1996, 202: 150–159. 10.1006/jmaa.1996.0308MathSciNetView ArticleGoogle Scholar
- Bello Cruz JY, Iusem AN: A strongly convergent direct method for monotone variational inequalities in Hilbert space. Numer. Funct. Anal. Optim. 2009, 30: 23–36. 10.1080/01630560902735223MathSciNetView ArticleGoogle Scholar
- Bello Cruz JY, Iusem AN: Convergence of direct methods for paramonotone variational inequalities. Comput. Optim. Appl. 2010, 46: 247–263. 10.1007/s10589-009-9246-5MathSciNetView ArticleGoogle Scholar
- Bnouhachem A, Noor MA, Hao Z: Some new extragradient iterative methods for variational inequalities. Nonlinear Anal. 2009, 70: 1321–1329. 10.1016/j.na.2008.02.014MathSciNetView ArticleGoogle Scholar
- Bruck RE: On the weak convergence of an ergodic iteration for the solution of variational inequalities for monotone operators in Hilbert space. J. Math. Anal. Appl. 1977, 61: 159–164. 10.1016/0022-247X(77)90152-4MathSciNetView ArticleGoogle Scholar
- Cegielski A, Zalas R: Methods for variational inequality problem over the intersection of fixed point sets of quasi-nonexpansive operators. Numer. Funct. Anal. Optim. 2013, 34: 255–283. 10.1080/01630563.2012.716807MathSciNetView ArticleGoogle Scholar
- Censor Y, Gibali A, Reich S: Strong convergence of subgradient extragradient methods for the variational inequality problem in Hilbert space. Optim. Methods Softw. 2011, 26: 827–845. 10.1080/10556788.2010.551536MathSciNetView ArticleGoogle Scholar
- Censor Y, Gibali A, Reich S: Algorithms for the split variational inequality problem. Numer. Algorithms 2012, 59: 301–323. 10.1007/s11075-011-9490-5MathSciNetView ArticleGoogle Scholar
- Facchinei F, Pang JS: Finite-Dimensional Variational Inequalities and Complementarity Problems, vols. I and II. Springer, New York; 2003.MATHGoogle Scholar
- Glowinski R: Numerical Methods for Nonlinear Variational Problems. Springer, New York; 1984.MATHView ArticleGoogle Scholar
- He BS: A new method for a class of variational inequalities. Math. Program. 1994, 66: 137–144. 10.1007/BF01581141View ArticleGoogle Scholar
- He BS, Yang ZH, Yuan XM: An approximate proximal-extragradient type method for monotone variational inequalities. J. Math. Anal. Appl. 2004, 300: 362–374. 10.1016/j.jmaa.2004.04.068MathSciNetView ArticleGoogle Scholar
- Hirstoaga SA: Iterative selection methods for common fixed point problems. J. Math. Anal. Appl. 2006, 324: 1020–1035. 10.1016/j.jmaa.2005.12.064MathSciNetView ArticleGoogle Scholar
- Iiduka H, Takahashi W: Weak convergence of a projection algorithm for variational inequalities in a Banach space. J. Math. Anal. Appl. 2008, 339: 668–679. 10.1016/j.jmaa.2007.07.019MathSciNetView ArticleGoogle Scholar
- Iusem AN: An iterative algorithm for the variational inequality problem. Comput. Appl. Math. 1994, 13: 103–114.MathSciNetGoogle Scholar
- Khobotov EN: Modification of the extra-gradient method for solving variational inequalities and certain optimization problems. U.S.S.R. Comput. Math. Math. Phys. 1989, 27: 120–127.View ArticleGoogle Scholar
- Kinderlehrer D, Stampacchia G: An Introduction to Variational Inequalities and Their Applications. Academic Press, New York; 1980.MATHGoogle Scholar
- Korpelevich GM: An extragradient method for finding saddle points and for other problems. Èkon. Mat. Metody 1976, 12: 747–756.Google Scholar
- Lions JL, Stampacchia G: Variational inequalities. Commun. Pure Appl. Math. 1967, 20: 493–517. 10.1002/cpa.3160200302MathSciNetView ArticleGoogle Scholar
- Lu X, Xu HK, Yin X: Hybrid methods for a class of monotone variational inequalities. Nonlinear Anal. 2009, 71: 1032–1041. 10.1016/j.na.2008.11.067MathSciNetView ArticleGoogle Scholar
- Sabharwal A, Potter LC: Convexly constrained linear inverse problems: iterative least-squares and regularization. IEEE Trans. Signal Process. 1998, 46: 2345–2352. 10.1109/78.709518MathSciNetView ArticleGoogle Scholar
- Solodov MV, Svaiter BF: A new projection method for variational inequality problems. SIAM J. Control Optim. 1999, 37: 765–776. 10.1137/S0363012997317475MathSciNetView ArticleGoogle Scholar
- Solodov MV, Tseng P: Modified projection-type methods for monotone variational inequalities. SIAM J. Control Optim. 1996, 34: 1814–1830. 10.1137/S0363012994268655MathSciNetView ArticleGoogle Scholar
- Verma RU: General convergence analysis for two-step projection methods and applications to variational problems. Appl. Math. Lett. 2005, 18: 1286–1292. 10.1016/j.aml.2005.02.026MathSciNetView ArticleGoogle Scholar
- Xu HK, Kim TH: Convergence of hybrid steepest-descent methods for variational inequalities. J. Optim. Theory Appl. 2003, 119: 185–201.MathSciNetView ArticleGoogle Scholar
- Yamada I, Ogura N: Hybrid steepest descent method for the variational inequality problem over the fixed point set of certain quasi-nonexpansive mappings. Numer. Funct. Anal. Optim. 2004, 25: 619–655.MathSciNetView ArticleGoogle Scholar
- Yao JC: Variational inequalities with generalized monotone operators. Math. Oper. Res. 1994, 19: 691–705. 10.1287/moor.19.3.691MathSciNetView ArticleGoogle Scholar
- Yao Y, Chen R, Xu HK: Schemes for finding minimum-norm solutions of variational inequalities. Nonlinear Anal. 2010, 72: 3447–3456. 10.1016/j.na.2009.12.029MathSciNetView ArticleGoogle Scholar
- Yao Y, Liou YC, Kang SM: Two-step projection methods for a system of variational inequality problems in Banach spaces. J. Glob. Optim. 2013, 55(4):801–811. 10.1007/s10898-011-9804-0MathSciNetView ArticleGoogle Scholar
- Yao Y, Liou YC, Shahzad N: Construction of iterative methods for variational inequality and fixed point problems. Numer. Funct. Anal. Optim. 2012, 33: 1250–1267. 10.1080/01630563.2012.660796MathSciNetView ArticleGoogle Scholar
- Yao Y, Marino G, Muglia L: A modified Korpelevich’s method convergent to the minimum norm solution of a variational inequality. Optimization 2012. 10.1080/02331934.2012.674947Google Scholar
- Yao Y, Noor MA, Liou YC, Kang SM: Iterative algorithms for general multi-valued variational inequalities. Abstr. Appl. Anal. 2012., 2012: Article ID 768272Google Scholar
- Yao Y, Noor MA: On viscosity iterative methods for variational inequalities. J. Math. Anal. Appl. 2007, 325: 776–787. 10.1016/j.jmaa.2006.01.091MathSciNetView ArticleGoogle Scholar
- Zeng LC, Hadjisavvas N, Wong NC: Strong convergence theorem by a hybrid extragradient-like approximation method for variational inequalities and fixed point problems. J. Glob. Optim. 2010, 46: 635–646. 10.1007/s10898-009-9454-7View ArticleGoogle Scholar
- Zeng LC, Teboulle M, Yao JC: Weak convergence of an iterative method for pseudomonotone variational inequalities and fixed point problems. J. Optim. Theory Appl. 2010, 146: 19–31. 10.1007/s10957-010-9650-0MathSciNetView ArticleGoogle Scholar
- Zhu D, Marcotte P: New classes of generalized monotonicity. J. Optim. Theory Appl. 1995, 87: 457–471. 10.1007/BF02192574MathSciNetView ArticleGoogle Scholar
- Iusem AN, Svaiter BF: A variant of Korpelevich’s method for variational inequalities with a new search strategy. Optimization 1997, 42: 309–321. 10.1080/02331939708844365MathSciNetView ArticleGoogle Scholar
- Censor Y, Gibali A, Reich S: Extensions of Korpelevich’s extragradient method for the variational inequality problem in Euclidean space. Optimization 2010. 10.1080/02331934.2010.539689Google Scholar
- Censor Y, Gibali A, Reich S: The subgradient extragradient method for solving variational inequalities in Hilbert space. J. Optim. Theory Appl. 2011, 148: 318–335. 10.1007/s10957-010-9757-3MathSciNetView ArticleGoogle Scholar
- Iusem AN, Lucambio Peŕez LR: An extragradient-type algorithm for non-smooth variational inequalities. Optimization 2000, 48: 309–332. 10.1080/02331930008844508MathSciNetView ArticleGoogle Scholar
- Mashreghi J, Nasri M: Forcing strong convergence of Korpelevich’s method in Banach spaces with its applications in game theory. Nonlinear Anal. 2010, 72: 2086–2099. 10.1016/j.na.2009.10.009MathSciNetView ArticleGoogle Scholar
- Noor MA: New extragradient-type methods for general variational inequalities. J. Math. Anal. Appl. 2003, 277: 379–394. 10.1016/S0022-247X(03)00023-4MathSciNetView ArticleGoogle Scholar
- Takahashi W, Toyoda M: Weak convergence theorems for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl. 2003, 118: 417–428. 10.1023/A:1025407607560MathSciNetView ArticleGoogle Scholar
- Suzuki T: Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces. Fixed Point Theory Appl. 2005, 2005: 103–123.View ArticleGoogle Scholar
- Xu HK: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 2002, 66: 240–256. 10.1112/S0024610702003332View ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.