- Open Access
A Korpelevich-like algorithm for variational inequalities
© Wu et al.; licensee Springer 2013
- Received: 6 October 2012
- Accepted: 4 February 2013
- Published: 28 February 2013
A Korpelevich-like algorithm has been introduced for solving a generalized variational inequality. It is shown that the presented algorithm converges strongly to a special solution of the generalized variational inequality.
- Korpelevich-like algorithm
- sunny nonexpansive retraction
- generalized variational inequalities
- α-inverse-strongly accretive mappings
- Banach spaces
Motivated and inspired by the above algorithms (1.2), (1.4) and (1.5), in this paper, we suggest an extragradient-type method via the sunny nonexpansive retraction for solving the variational inequalities (1.3) in Banach spaces. It is shown that the presented algorithm converges strongly to a special solution of the variational inequality (1.3).
for all .
It is known that E is uniformly smooth if and only if . Let q be a fixed real number with . Then a Banach space E is said to be q-uniformly smooth if there exists a constant such that for all .
We need the following lemmas for the proof of our main results.
Lemma 2.1 
whenever for and . A mapping Q of C into itself is called a retraction if . If a mapping Q of C into itself is a retraction, then for every , where is the range of Q. A subset D of C is called a sunny nonexpansive retract of C if there exists a sunny nonexpansive retraction from C onto D. We know the following lemma concerning sunny nonexpansive retraction.
Lemma 2.2 
for all and .
Lemma 2.3 
Lemma 2.4 
In particular, if , then is nonexpansive.
It is clear that if , then is nonexpansive. □
Lemma 2.5 
Let C be a nonempty bounded closed convex subset of a uniformly convex Banach space E and let T be a nonexpansive mapping of C into itself. If is a sequence of C such that weakly and strongly, then x is a fixed point of T.
Lemma 2.6 
In this section, we present our Korpelevich-like algorithm and consequently we will show its strong convergence.
3.1 Conditions assumptions
(A1) E is a uniformly convex and 2-uniformly smooth Banach space with a weakly sequentially continuous duality mapping;
(A2) C is a nonempty closed convex subset of E;
(A3) is an α-strongly accretive and L-Lipschitz continuous mapping with ;
(A4) is a sunny nonexpansive retraction from E onto C.
3.2 Parameters restrictions
, for some a, b with ;
where K is the smooth constant of E.
(P2) is a sequence in such that and .
Theorem 3.2 The sequence generated by (3.1) converges strongly to , where is a sunny nonexpansive retraction of E onto .
Proof Let . First, from Lemma 2.2, we have for all . In particular, for all .
Since and , we get for enough large n. Without loss of generality, we may assume that for all , , i.e., . Hence, is nonexpansive.
Hence, is bounded.
By Lemma 2.5 and (3.8), we have , it follows from Lemma 2.3 that .
Applying Lemma 2.6 to the last inequality, we conclude that converges strongly to . This completes the proof. □
Yonghong Yao was supported in part by NSFC 11071279 and NSFC 71161001-G0105. Yeong-Cheng Liou was supported in part by NSC 101-2628-E-230-001-MY3 and NSC 101-2622-E-230-005-CC3.
- Korpelevich GM: An extragradient method for finding saddle points and for other problems. Ekon. Mat. Metod. 1976, 12: 747–756.Google 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
- Iusem AN, Lucambio Peŕez LR: An extragradient-type algorithm for non-smooth variational inequalities. Optimization 2000, 48: 309–332. 10.1080/02331930008844508MathSciNetView 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
- Lions JL, Stampacchia G: Variational inequalities. Commun. Pure Appl. Math. 1967, 20: 493–517. 10.1002/cpa.3160200302MathSciNetView 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
- Bello Cruz JY, Iusem AN: A strongly convergent direct method for monotone variational inequalities in Hilbert space. Numer. Funct. Anal. Optim. 2009, 30(1–2):23–36. 10.1080/01630560902735223MathSciNetView ArticleGoogle Scholar
- Glowinski R: Numerical Methods for Nonlinear Variational Problems. Springer, New York; 1984.View ArticleGoogle Scholar
- Yao Y, Noor MA, Liou YC: Strong convergence of a modified extra-gradient method to the minimum-norm solution of variational inequalities. Abstr. Appl. Anal. 2012., 2012: Article ID 817436. doi:10.1155/2012/817436Google 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: The hybrid steepest descent for the variational inequality problems over the intersection of fixed points sets of nonexpansive mappings. In Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications. Edited by: Butnariu D, Censor Y, Reich S. Elsevier, New York; 2001:473–504.View 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: 801–811. doi:10.1007/s10898–011–9804–0 10.1007/s10898-011-9804-0MathSciNetView ArticleGoogle Scholar
- Yao Y, Chen R, Liou YC: A unified implicit algorithm for solving the triple-hierarchical constrained optimization problem. Math. Comput. Model. 2012, 55: 1506–1515. 10.1016/j.mcm.2011.10.041MathSciNetView ArticleGoogle Scholar
- Yao Y, Noor MA, Noor KI, Liou YC, Yaqoob H: Modified extragradient method for a system of variational inequalities in Banach spaces. Acta Appl. Math. 2010, 110: 1211–1224. 10.1007/s10440-009-9502-9MathSciNetView ArticleGoogle Scholar
- Cho YJ, Yao Y, Zhou H: Strong convergence of an iterative algorithm for accretive operators in Banach spaces. J. Comput. Anal. Appl. 2008, 10: 113–125.MathSciNetGoogle Scholar
- Yao Y, Cho YJ, Liou YC: Algorithms of common solutions for variational inclusions, mixed equilibrium problems and fixed point problems. Eur. J. Oper. Res. 2011, 212: 242–250. 10.1016/j.ejor.2011.01.042MathSciNetView ArticleGoogle Scholar
- Cho YJ, Kang SM, Qin X: On systems of generalized nonlinear variational inequalities in Banach spaces. Appl. Math. Comput. 2008, 206: 214–220. 10.1016/j.amc.2008.09.005MathSciNetView ArticleGoogle Scholar
- Cho YJ, Qin X: Systems of generalized nonlinear variational inequalities and its projection methods. Nonlinear Anal. 2008, 69: 4443–4451. 10.1016/j.na.2007.11.001MathSciNetView ArticleGoogle Scholar
- Ceng LC, Ansari QH, Yao JC: Mann type steepest-descent and modified hybrid steepest-descent methods for variational inequalities in Banach spaces. Numer. Funct. Anal. Optim. 2008, 29: 987–1033. 10.1080/01630560802418391MathSciNetView ArticleGoogle Scholar
- Sahu DR, Wong NC, Yao JC: A unified hybrid iterative method for solving variational inequalities involving generalized pseudo-contractive mappings. SIAM J. Control Optim. 2012, 50: 2335–2354. 10.1137/100798648MathSciNetView ArticleGoogle Scholar
- Yao, Y, Marino, G, Muglia, L: A modified Korpelevich’s method convergent to the minimum norm solution of a variational inequality. Optimization (in press). doi:10.1080/02331934.2013.764522Google Scholar
- Aoyama K, Iiduka H, Takahashi W: Weak convergence of an iterative sequence for accretive operators in Banach spaces. Fixed Point Theory Appl. 2006., 2006: Article ID 35390. doi:10.1155/FPTA/2006/35390Google Scholar
- Yao Y, Maruster S: Strong convergence of an iterative algorithm for variational inequalities in Banach spaces. Math. Comput. Model. 2011, 54: 325–329. 10.1016/j.mcm.2011.02.016MathSciNetView ArticleGoogle Scholar
- Xu HK: Inequalities in Banach spaces with applications. Nonlinear Anal. 1991, 16: 1127–1138. 10.1016/0362-546X(91)90200-KMathSciNetView ArticleGoogle Scholar
- Bruck RE Jr.: Nonexpansive retracts of Banach spaces. Bull. Am. Math. Soc. 1970, 76: 384–386. 10.1090/S0002-9904-1970-12486-7MathSciNetView 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
- Browder FE: Nonlinear operators and nonlinear equations of evolution in Banach spaces. In Nonlinear Functional Analysis. Am. Math. Soc., Rhode Island; 1976:1–308. (Proc. Sympos. Pure Math., vol. XVIII, Part 2, Chicago, Ill, 1968)Google Scholar
- Xu HK: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 2002, 66: 240–256. 10.1112/S0024610702003332View ArticleGoogle Scholar
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