- Open Access
Affine algorithms for the split variational inequality and equilibrium problems
© Yao et al.; licensee Springer. 2013
- Received: 11 October 2012
- Accepted: 14 May 2013
- Published: 29 May 2013
An affine algorithm for the split variational inequality and equilibrium problems is presented. Strong convergence result is given.
- affine algorithm
- split method
- variational inequality
- equilibrium problem
Our main motivations are inspired by the following reasons.
has received much attention due to its applications in signal processing and image reconstruction with particular progress in intensity modulated radiation therapy [1–13]. Note that the involved operator g is a bounded linear operator. However, in the present paper, the involved mapping ψ in (1.1) is a nonlinear mapping.
Reason 2 The variational inequality problem [14–24] and equilibrium problem [23–27], which include the fixed point problems and optimization problems [28–30], have been studied by many authors. It is an interesting topic associated with the analytical and algorithmic approach to the variational inequality and equilibrium problems.
Motivated and inspired by the results in the literature, we present an affine algorithm for solving the split problem (1.1). Strong convergence theorem is given under some mild assumptions.
Let H be a real Hilbert space with the inner product and the norm , respectively. Let C be a nonempty closed convex subset of H.
2.1 Monotonicity and convexity
An operator is said to be monotone if for all . is said to be strongly monotone if there exists a constant such that for all . is called an inverse-strongly-monotone operator if there exists such that for all . Let be a nonlinear operator. is said to be α-inverse strongly g-monotone iff for all and for some . Let B be a mapping of H into . The effective domain of B is denoted by , that is, . A multi-valued mapping B is said to be a monotone operator on H iff for all , , and . A monotone operator B on H is said to be maximal iff its graph is not strictly contained in the graph of any other monotone operator on H.
A function is said to be convex if for any and for any , .
2.2 Nonexpansivity and continuity
A mapping is said to be nonexpansive [31–38] if for all . We use to denote the set of fixed points of T. is called a firmly nonexpansive mapping if, for all , . It is known that T is firmly nonexpansive if and only if a mapping is nonexpansive, where I is the identity mapping on H. is said to be L-Lipschitz continuous if there exists a constant such that for all . In such a case, T is said to be L-Lipschitz continuous. Given a nonempty, closed convex subset C of H, the mapping that assigns every point to its unique nearest point in C is called a metric projection onto C and denoted by , that is, and . The metric projection is a typical firmly nonexpansive mapping. The characteristic inequality of the projection is for all , .
2.3 Equilibrium problem
In this paper, we consider the split problem (1.1). In the sequel, we assume that the solution set S of (1.1) is nonempty.
is an α-inverse strongly ψ-monotone mapping;
is a weakly continuous and γ-strongly monotone mapping such that ;
is a bifunction;
is a β-inverse-strongly monotone mapping.
for all ;
F is monotone, i.e., for all ;
for each , ;
for each , is convex and lower semicontinuous.
In order to solve Problem 2.1, we need the following useful lemmas.
2.4 Useful lemmas
The following three lemmas are important tools for our main results in the next section. Note that these lemmas are used extensively in the literature.
Lemma 2.2 (Combettes and Hirstoaga’s lemma )
is single-valued and is firmly nonexpansive;
is closed and convex and .
Lemma 2.3 (Suzuki’s lemma )
Let and be bounded sequences in a Banach space X and let be a sequence in with . Suppose for all and . Then .
Lemma 2.4 (Xu’s lemma )
Assume that is a sequence of nonnegative real numbers such that , where is a sequence in and is a sequence such that and (or ). Then .
In this section, we first present our algorithm for solving Problem 2.1. Assume that the conditions in Problem 2.1 are all satisfied.
Algorithm 3.1 Let C be a nonempty closed and convex subset of a real Hilbert space H.
where is the metric projection, is a real number sequence, is an L-Lipschitz continuous mapping and is a constant.
where is a real number sequence.
where is a real number sequence.
, and ;
This deduces the contraction because of by the assumption. Therefore, . So, the solution of variational inequality (3.1) is unique.
Next, we prove Theorem 3.2.
where and . It is easily seen that and . We can therefore apply Lemma 2.4 to conclude that and . This completes the proof. □
Yonghong Yao was supported in part by NSFC 11071279 and NSFC 71161001-G0105. Rudong Chen was supported in part by NSFC 11071279. Yeong-Cheng Liou was supported in part by NSC 101-2628-E-230-001-MY3 and NSC 101-2622-E-230-005-CC3.
- Censor Y, Elfving T: A multiprojection algorithm using Bregman projections in a product space. Numer. Algorithms 1994, 8: 221–239. 10.1007/BF02142692MATHMathSciNetView ArticleGoogle Scholar
- Censor Y, Bortfeld T, Martin B, Trofimov A: A unified approach for inversion problems in intensity modulated radiation therapy. Phys. Med. Biol. 2006, 51: 2353–2365. 10.1088/0031-9155/51/10/001View ArticleGoogle Scholar
- Censor Y, Elfving T, Kopf N, Bortfeld T: The multiple-sets split feasibility problem and its applications for inverse problems. Inverse Probl. 2005, 21: 2071–2084. 10.1088/0266-5611/21/6/017MATHMathSciNetView ArticleGoogle Scholar
- Byrne C: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Probl. 2004, 20: 103–120. 10.1088/0266-5611/20/1/006MATHMathSciNetView ArticleGoogle Scholar
- Yang Q: The relaxed CQ algorithm for solving the split feasibility problem. Inverse Probl. 2004, 20: 1261–1266. 10.1088/0266-5611/20/4/014MATHView ArticleGoogle Scholar
- Qu B, Xiu N: A note on the CQ algorithm for the split feasibility problem. Inverse Probl. 2005, 21: 1655–1665. 10.1088/0266-5611/21/5/009MATHMathSciNetView ArticleGoogle Scholar
- Zhao J, Yang Q: Several solution methods for the split feasibility problem. Inverse Probl. 2005, 21: 1791–1799. 10.1088/0266-5611/21/5/017MATHView ArticleGoogle Scholar
- Xu HK: A variable Krasnosel’skii-Mann algorithm and the multiple-set split feasibility problem. Inverse Probl. 2006, 22: 2021–2034. 10.1088/0266-5611/22/6/007MATHView ArticleGoogle Scholar
- Dang Y, Gao Y: The strong convergence of a KM-CQ-like algorithm for a split feasibility problem. Inverse Probl. 2011., 27: Article ID 015007Google Scholar
- Wang F, Xu HK: Approximating curve and strong convergence of the CQ algorithm for the split feasibility problem. J. Inequal. Appl. 2010. doi:10.1155/2010/102085Google Scholar
- Xu HK: Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces. Inverse Probl. 2010., 26: Article ID 105018Google Scholar
- Yao Y, Kim TH, Chebbi S, Xu HK: A modified extragradient method for the split feasibility and fixed point problems. J. Nonlinear Convex Anal. 2012, 13(3):383–396.MATHMathSciNetGoogle Scholar
- Yao Y, Liou YC, Shahzad N: A strongly convergent method for the split feasibility problem. Abstr. Appl. Anal. 2012., 2012: Article ID 125046Google Scholar
- Stampacchia G: Formes bilineaires coercivites sur les ensembles convexes. C. R. Math. Acad. Sci. Paris 1964, 258: 4413–4416.MATHMathSciNetGoogle Scholar
- Korpelevich GM: An extragradient method for finding saddle points and for other problems. Èkon. Mat. Metody 1976, 12: 747–756.MATHGoogle Scholar
- Glowinski R: Numerical Methods for Nonlinear Variational Problems. Springer, New York; 1984.MATHView ArticleGoogle Scholar
- Iusem AN: An iterative algorithm for the variational inequality problem. Comput. Appl. Math. 1994, 13: 103–114.MATHMathSciNetGoogle Scholar
- Noor MA: Some development in general variational inequalities. Appl. Math. Comput. 2004, 152: 199–277. 10.1016/S0096-3003(03)00558-7MATHMathSciNetView ArticleGoogle Scholar
- Facchinei F, Pang JS Springer Series in Operations Research I. In Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, New York; 2003.Google Scholar
- Facchinei F, Pang JS Springer Series in Operations Research II. In Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, New York; 2003.Google Scholar
- Xu HK, Kim TH: Convergence of hybrid steepest-descent methods for variational inequalities. J. Optim. Theory Appl. 2003, 119: 185–201.MATHMathSciNetView ArticleGoogle Scholar
- Yao JC: Variational inequalities with generalized monotone operators. Math. Oper. Res. 1994, 19: 691–705. 10.1287/moor.19.3.691MATHMathSciNetView ArticleGoogle Scholar
- Ceng LC, Yao JC: Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems. Taiwan. J. Math. 2006, 10: 1293–1303.MathSciNetGoogle Scholar
- Ceng LC, Al-Homidan S, Ansari QH, Yao JC: An iterative scheme for equilibrium problems and fixed point problems of strict pseudo-contraction mappings. J. Comput. Appl. Math. 2009, 223: 967–974. 10.1016/j.cam.2008.03.032MATHMathSciNetView ArticleGoogle Scholar
- Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. Math. Stud. 1994, 63: 123–145.MATHMathSciNetGoogle Scholar
- Combettes PL, Hirstoaga A: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 2005, 6: 117–136.MATHMathSciNetGoogle 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.2011.01.042MATHMathSciNetView ArticleGoogle Scholar
- Qin X, Cho SY, Kang SM: Some results on fixed points of asymptotically strict quasi- ϕ -pseudocontractions in the intermediate sense. Fixed Point Theory Appl. 2012., 2012: Article ID 143Google Scholar
- Zegeye H, Shahzad N, Alghamdi MA: Strong convergence theorems for a common point of solution of variational inequality, solutions of equilibrium and fixed point problems. Fixed Point Theory Appl. 2012., 2012: Article ID 119. doi:10.1186/1687–1812–2012–119Google Scholar
- Qin X, Agarwal RP, Cho SY, Kang SM: Convergence of algorithms for fixed points of generalized asymptotically quasi- ψ -nonexpansive mappings with applications. Fixed Point Theory Appl. 2012., 2012: Article ID 58. doi:10.1186/1687–1812–2012–58Google Scholar
- Browder FE: Convergence of approximation to fixed points of nonexpansive nonlinear mappings in Hilbert spaces. Arch. Ration. Mech. Anal. 1967, 24: 82–90.MATHMathSciNetView ArticleGoogle Scholar
- Halpern B: Fixed points of nonexpansive maps. Bull. Am. Math. Soc. 1967, 73: 957–961. 10.1090/S0002-9904-1967-11864-0MATHView ArticleGoogle Scholar
- Geobel K, Kirk WA Cambridge Studies in Advanced Mathematics 28. In Topics in Metric Fixed Point Theory. Cambridge University Press, Cambridge; 1990.View ArticleGoogle Scholar
- Lions PL: Approximation de points fixes de contractions. C. R. Hebd. Séances Acad. Sci., Sér. A, Sci. Math. 1977, 284: 1357–1359.MATHGoogle Scholar
- Opial Z: Weak convergence of the sequence of successive approximations of nonexpansive mappings. Bull. Am. Math. Soc. 1967, 73: 595–597.MathSciNetView ArticleGoogle Scholar
- Wittmann R: Approximation of fixed points of non-expansive mappings. Arch. Math. 1992, 58: 486–491. 10.1007/BF01190119MATHMathSciNetView ArticleGoogle Scholar
- Moudafi A: Viscosity approximation methods for fixed-points problems. J. Math. Anal. Appl. 2000, 241: 46–55. 10.1006/jmaa.1999.6615MATHMathSciNetView ArticleGoogle Scholar
- Xu HK: Viscosity approximation methods for nonexpansive mappings. J. Math. Anal. Appl. 2004, 298: 279–291. 10.1016/j.jmaa.2004.04.059MATHMathSciNetView 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.MATHView ArticleGoogle Scholar
- Xu HK: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 2002, 2: 1–17.Google Scholar
- Zhang LJ, Chen JM, Hou ZB: Viscosity approximation methods for nonexpansive mappings and generalized variational inequalities. Acta Math. Sin. 2010, 53: 691–6988.MATHMathSciNetGoogle 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.