Scalar gap functions and error bounds for generalized mixed vector equilibrium problems with applications
- Wenyan Zhang^{1},
- Jiawei Chen^{2}Email author,
- Shu Xu^{3} and
- Wen Dong^{2}
https://doi.org/10.1186/s13663-015-0422-2
© Zhang et al. 2015
Received: 21 April 2015
Accepted: 8 September 2015
Published: 18 September 2015
Abstract
It is well known that equilibrium problems are very important mathematical models and are closely related with fixed point problems, variational inequalities, and Nash equilibrium problems. Gap functions and error bounds which play a vital role in algorithms design, are two much-addressed topics of vector equilibrium problems. This paper is devoted to studying the scalar-valued gap functions and error bounds for the generalized mixed vector equilibrium problem (GMVE). First, a scalar gap function for (GMVE) is proposed without any scalarization methods, and then error bounds of (GMVE) are established in terms of the gap function. As applications, error bounds for generalized vector variational inequalities and vector variational inequalities are derived, respectively. The main results obtained are new and improve corresponding results of Charitha and Dutta (Pac. J. Optim. 6:497-510, 2010) and Sun and Chai (Optim. Lett. 8:1663-1673, 2014).
Keywords
error bound scalar gap function generalized mixed vector equilibrium1 Introduction
Error bounds, which play a critical role in algorithm design, can be used to measure how much the approximate solution fails to be in the solution set and to analyze the convergence rates of various methods. Recently, kinds of error bounds have been presented for variational inequalities in [11–18]. Results for error bounds have been established for a weak vector variational inequality (WVVI) in [12–15, 19]. Xu and Li [15] obtained error bounds for a weak vector variational inequality with cone constraints by using a method of image space analysis. By using a scalarization approach of Konnov [20], Li and Mastroeni [13] established error bounds for two kinds of (WVVI) with set-valued mappings. By a regularized gap function and a D-gap function, Charitha and Dutta [12] used a projection operator method to obtain error bounds of (WVVI), respectively. Sun and Chai [14] studied some error bounds for generalized vector variational inequalities in virtue of the regularized gap functions. Very recently, a global error bound of a weak vector variational inequality was established by the nonlinear scalarization method in Li [19].
However, to the best of our knowledge, an error bound of the generalized mixed vector equilibrium problem (GMVE) has never been investigated. In this paper, motivated by ideas in Sun and Chai [14] and Yamashita et al. [18], we introduce a scalar gap function for (GMVE). Then an error bound of (GMVE) is presented. As an application of an error bound for (GMVE), we also get error bounds of (GVVI) and (VVI), respectively.
This paper is organized as follows: In Section 2, we first recall some basic definitions. In Section 3, we introduce scalar gap functions for (GMVE), (GVVI), and (VVI). By using these gap functions, we obtain some error bound results for (GMVE), (GVVI), and (VVI), respectively.
2 Mathematical preliminaries
For \(i=1,2,\ldots,m\), we denote the generalized mixed vector equilibrium problems (GMVE) associated with \(F_{i}\), \(A_{i}\) and \(g_{i}\) as \((GMVE)^{i}\), the generalized vector variational inequality problems (GVVI) associated with \(A_{i}\) and \(g_{i} \) as \((GVVI)^{i}\), and the vector variational inequality problems (VVI) associated with \(A_{i}\) as \((VVI)^{i}\), respectively. The solution sets of \((GMVE)^{i}\), \((GVVI)^{i}\), and \((VVI)^{i}\) will be denoted by \(S_{GMVE}^{i}\), \(S_{GVVI}^{i}\), and \(S_{VVI}^{i}\), respectively.
In the paper, we intend to investigate gap functions and error bounds of (GMVE), (GVVI), and (VVI). We shall recall some notations and definitions, which will be used in the sequel.
Definition 2.1
Definition 2.2
Definition 2.3
- (i)
\(\vartheta(x)\geq0\), for any \(x\in K \),
- (ii)
\(\vartheta(x_{0})=0 \) if and only if \(x_{0} \in K\) is a solution of (GMVE) (resp. (GVVI) and (VVI)).
3 Main results
- (P)For all \(x,y \in\mathbb{R}^{n}\),$$ \beta\|x-y\|^{2}\leq\varphi(x,y)\leq(\gamma- \beta) \|x-y\|^{2},\quad \gamma\geq 2\beta>0. $$(10)
- (A1)
\(F_{i}\) is a convex function about the second variable on \(\mathbb{\mathbb{R}}^{n}\times\mathbb{\mathbb{R}}^{n}\).
- (A2)
\(F_{i}(x,y)=0\), \(\forall x,y\in\mathbb{\mathbb{R}}^{n}\), if and only if \(x=y\).
- (A3)
For any \(x,y,z\in K\), \(F_{i}(x,y)+F_{i}(y,z)\leq F_{i}(x,z)\).
Theorem 3.1
If \(F_{i}:\mathbb{R}^{n}\times\mathbb{R}^{n}\rightarrow\mathbb{R}\) is convex about the second variable and \(g_{i}\) is convex over \(\mathbb {R}^{n}\) for any \(i=1,2,\ldots,m\), then the function \(\vartheta_{\alpha}\), with \(\alpha> 0\), defined by (7) is a gap function for (GMVE).
Proof
By a similar method, we conclude the following results for (GVVI) and (VVI), respectively.
Corollary 3.1
The function \(\psi_{\alpha}\), with \(\alpha> 0\), defined by (8) is a gap function for (GVVI).
Corollary 3.2
The function \(\phi_{\alpha}\), with \(\alpha> 0\), defined by (9) is a gap function for (VVI).
Now, by using the gap function \({\vartheta_{\alpha}}(x)\), we obtain an error bound result for (GMVE).
Theorem 3.2
Proof
The following example shows that, in general, the conditions of Theorem 3.2 can be achieved.
Example 3.1
Similarly, by using gap functions \(\psi_{\alpha}\) and \(\phi_{\alpha}\), we can also obtain error bound results for (GVVI) and (VVI), respectively.
Corollary 3.3
Corollary 3.4
Remark 3.1
(i) In [14], there exist some mistakes in the proof of Theorem 3.2, which lead to the requirement of Lipschitz properties of \(g_{i}\), \(i=1,2,\ldots,n\). Hence, we give the modified error bound for (GVVI) in Corollary 3.3 without Lipschitz assumption.
(ii) In [12], Charitha and Dutta established error bounds for (VVI) by using the projection operator method and strongly monotone assumptions, whereas it seems that our method is more simple from the computational view since there are not any scalarization parameters.
Now we ask: How do we establish error bounds for \(S_{GMVE}\) in terms of the gap function \(\vartheta_{\alpha}\), under mild assumptions, such that \(S_{GMVE}\) need not be a singleton set in general? This problem may be interesting and valuable in vector optimization.
Declarations
Acknowledgements
The authors would like to thank the associated editor and the two anonymous referees for their valuable comments and suggestions, which have helped to improve the paper. This work was partially supported by the National Natural Science Foundation of China (Grant number: 11401487), the Fundamental Research Funds for the Central Universities (Grant numbers: SWU113037, XDJK2014C073), and the Scientific Research Fund of Sichuan Provincial Science and Technology Department (Grant number: 2015JY0237).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Blum, E, Oettli, W: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123-145 (1994) MATHMathSciNetGoogle Scholar
- Agarwal, RP, Cho, YJ, Petrot, N: Systems of general nonlinear set-valued mixed variational inequalities problems in Hilbert spaces. Fixed Point Theory Appl. 2011, Article ID 31 (2011). doi:https://doi.org/10.1186/1687-1812-2011-31 MathSciNetView ArticleGoogle Scholar
- Saewan, S, Kumam, P: The shrinking projection method for solving generalized equilibrium problems and common fixed points for asymptotically quasi-ϕ-nonexpansive mappings. Fixed Point Theory Appl. 2011, Article ID 9 (2011). doi:https://doi.org/10.1186/1687-1812-2011-9 View ArticleGoogle Scholar
- Markshoe, P, Wangkeeree, R, Kamraksa, U: The shrinking projection method for generalized mixed equilibrium problems and fixed point problems in Banach spaces. J. Nonlinear Anal. Optim., Theory Appl. 1, 111-129 (2010) MathSciNetGoogle Scholar
- Kumam, W, Jaiboon, C, Kumam, P: A shrinking projection method for generalized mixed equilibrium problems, variational inclusion problems and a finite family quasi-nonexpansive mappings. J. Inequal. Appl. 2010, Article ID 458274 (2010). doi:https://doi.org/10.1155/2010/458274 View ArticleGoogle Scholar
- Zhang, S: Generalized mixed equilibrium problem in Banach spaces. Appl. Math. Mech. 30, 1105-1112 (2009) MATHView ArticleGoogle Scholar
- Chen, J, Cho, YJ, Wan, Z: Shrinking projection algorithms for equilibrium problems with a bifunction defined on the dual space of a Banach space. Fixed Point Theory Appl. 2011, Article ID 91 (2011). doi:https://doi.org/10.1186/1687-1812-2011-91 MathSciNetView ArticleGoogle Scholar
- Cholamjiak, W, Cholamjiak, P, Suantai, S: Convergence of iterative schemes for solving fixed point problems for multi-valued nonself mappings and equilibrium problems. J. Nonlinear Sci. Appl. 8, 1245-1256 (2015) Google Scholar
- Khan, SA, Chen, JW: Gap functions and error bounds for generalized mixed vector equilibrium problems. J. Optim. Theory Appl. 166, 767-776 (2015) MathSciNetView ArticleGoogle Scholar
- Chang, SS, Joseph Lee, HW, Chan, CK: A new hybrid method for solving generalized equilibrium problem variational inequality and common fixed point in Banach spaces with applications. Nonlinear Anal. 73, 2260-2270 (2010) MATHMathSciNetView ArticleGoogle Scholar
- Facchinei, F, Pang, JS: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, Berlin (2003) Google Scholar
- Charitha, C, Dutta, J: Regularized gap functions and error bounds for vector variational inequalities. Pac. J. Optim. 6, 497-510 (2010) MATHMathSciNetGoogle Scholar
- Li, J, Mastroeni, G: Vector variational inequalities involving set-valued mappings via scalarization with applications to error bounds for gap functions. J. Optim. Theory Appl. 145, 355-372 (2010) MATHMathSciNetView ArticleGoogle Scholar
- Sun, XK, Chai, Y: Gap functions and error bounds for generalized vector variational inequalities. Optim. Lett. 8, 1663-1673 (2014) MATHMathSciNetView ArticleGoogle Scholar
- Xu, YD, Li, SJ: Gap functions and error bounds for weak vector variational inequalities. Optimization 63, 1339-1352 (2014) MATHMathSciNetView ArticleGoogle Scholar
- Fukushima, M: Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems. Math. Program. 53, 99-110 (1992) MATHMathSciNetView ArticleGoogle Scholar
- Ng, KF, Tan, LL: Error bounds of regularized gap functions for nonsmooth variational inequality problems. Math. Program. 110, 405-429 (2007) MATHMathSciNetView ArticleGoogle Scholar
- Yamashita, N, Taji, K, Fukushima, M: Unconstrained optimization reformulations of variational inequality problems. J. Optim. Theory Appl. 92, 439-456 (1997) MATHMathSciNetView ArticleGoogle Scholar
- Li, MH: Error bounds of regularized gap functions for weak vector variational inequality problems. J. Inequal. Appl. 2014, Article ID 331 (2014). doi:https://doi.org/10.1186/1029-242X-2014-331 View ArticleGoogle Scholar
- Konnov, IV: A scalarization approach for vector variational inequalities with applications. J. Glob. Optim. 32, 517-527 (2005) MATHMathSciNetView ArticleGoogle Scholar
- Giannessi, F: Theorems of the alternative, quadratic programs and complementarity problems. In: Cottle, RW, Giannessi, F, Lions, JL (eds.) Variational Inequalities and Complementarity Problems, pp. 151-186. Wiley, New York (1980) Google Scholar