Error bounds of regularized gap functions for weak vector variational inequality problems
© Li; licensee Springer 2014
Received: 20 June 2014
Accepted: 18 August 2014
Published: 2 September 2014
In this paper, by the nonlinear scalarization method, a global error bound of a weak vector variational inequality is established via a regularized gap function. The result extends some existing results in the literature.
where is a closed convex and pointed cone with nonempty interior intC. (WVVI) was firstly introduced by Giannessi . It has been shown to have many applications in vector optimization problems and traffic equilibrium problems (e.g., [2, 3]).
Error bounds are to depict the distance from a feasible solution to the solution set, and have played an important role not only in sensitivity analysis but also in convergence analysis of iterative algorithms. Recently, kinds of error bounds have been presented for weak vector variational inequalities in [4–7]. By using a scalarization approach of Konnov , Li and Mastroeni  established the error bounds for two kinds of (WVVIs) with set-valued mappings. By a regularized gap function and a D-gap function for a weak vector variational inequality, Charitha and Dutta  obtained the error bounds of (WVVI), respectively. Recently, in virtue of the regularized gap functions, Sun and Chai  studied some error bounds for generalized (WVVIs). By using the image space analysis, Xu and Li  got a gap function for (WVVI). Then, they established an error bound for (WVVI) without the convexity of the constraint set. These papers have a common characteristic: the solution set of (WVVI) is a singleton [6, 7]. Even though the solution set of (WVVI) is not a singleton [4, 5], the solution set of the corresponding variational inequality (VI) is a singleton, when their results reduce to (VI).
In this paper, by the nonlinear scalarization method, we study a global error bound of (WVVI). This paper is organized as follows. In Section 2, we establish a global error bound of (VI) via the generalized gap functions. In Section 3, we discuss a global error bound of (WVVI) by the nonlinear scalarization method.
2 A global error bound of (VI)
where is a continuously differentiable mapping.
where is a real-valued function with the following properties:
(P1) φ is continuously differentiable on .
(P2) , and the equality holds if and only if .
where denotes the partial derivative of φ with respect to the second variable.
Now we recall some properties of φ in (1).
if and only if .
Remark 2.1 In light of (ii) in Proposition 2.1, it holds true that .
Then we list some basic properties of the generalized regularized gap function .
- (i)For every , there exists a unique vector at which the infimum in (1) is attained, i.e.,
is a gap function of (VI).
if and only if x is a solution of (VI).
- (iv)is continuously differentiable on with
and are both locally Lipschitz on .
If f is coercive on K, then (VI) has a nonempty compact solution set.
Thus, one has . □
Theorem 2.1 Let f be coercive on K and . Assume that φ satisfies
(P5) , .
Hence, (3) follows from (5) and (6). The proof is complete. □
Now we use two examples to show that (2) cannot be dropped and that Theorem 2.1 is applicable, respectively.
Example 2.1 Consider , , and . Then we can easily get that , , and . It is clear that f is coercive on K and . Thus, (2) does not hold. Moreover, it is obvious that does not have a global error bound.
Example 2.2 Consider , , and . Then we can easily get that , , , and . It is clear that f is coercive on K and (2) holds. Thus, it follows from Theorem 2.1 that has a global error bound.
Thus, (2) holds. Moreover, the strong monotonicity of f implies the coerciveness of f (cf. [, Remark 2.1]). Thus, by Theorem 2.1, we get that has a global error bound. □
3 A global error bound of (WVVI)
In this section, by the nonlinear scalarization method and by Theorem 2.1, we discuss a global error bound of (WVVI). The dual cone of C is defined by . For each , , where denotes the value of ξ at z. Let and . It is well known that is a compact convex base of .
where and S is the solution set of (WVVI).
where . When , the generalized regularized gap function reduces to the regularized gap function which was defined in .
Hence, has a global error bound with the modulus . □
Moreover, the strong monotonicity of implies the coerciveness of (cf. [, Remark 2.1]) and that (VI) has a unique solution (cf. [, Theorem 2.3.3]). Thus, by Theorem 3.1, we get that has a global error bound. Hence, our results extend those of [, Theorem 2.9].
This research was supported by the Natural Science Foundation of Shaanxi Province, China (Grant number: 2014JQ1023).
- Giannessi F: Theorems of the alternative, quadratic programs and complementarity problems. In Variational Inequalities and Complementarity Problems. Edited by: Cottle RW, Giannessi F, Lions JL. Wiley, New York; 1980:151–186.Google Scholar
- Chen GY, Goh CJ, Yang XQ: Vector network equilibrium problems and nonlinear scalarization methods. Math. Methods Oper. Res. 1999, 49: 239–253.MathSciNetMATHGoogle Scholar
- Li SJ, Teo KL, Yang XQ: A remark on a standard and linear vector network equilibrium problem with capacity constraints. Eur. J. Oper. Res. 2008, 184: 13–23. 10.1016/j.ejor.2005.11.059MathSciNetView ArticleMATHGoogle Scholar
- Charitha C, Dutta J: Regularized gap functions and error bounds for vector variational inequalities. Pac. J. Optim. 2010, 6: 497–510.MathSciNetMATHGoogle 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. 2010, 145: 355–372. 10.1007/s10957-009-9625-1MathSciNetView ArticleMATHGoogle Scholar
- Sun XK, Chai Y: Gap functions and error bounds for generalized vector variational inequalities. Optim. Lett. 2014, 8: 1663–1673. 10.1007/s11590-013-0685-7MathSciNetView ArticleMATHGoogle Scholar
- Xu YD, Li SJ: Gap functions and error bounds for weak vector variational inequalities. Optimization 2014, 63: 1339–1352. 10.1080/02331934.2012.721115MathSciNetView ArticleMATHGoogle Scholar
- Konnov IV: A scalarization approach for vector variational inequalities with applications. J. Glob. Optim. 2005, 32: 517–527. 10.1007/s10898-003-2688-xMathSciNetView ArticleMATHGoogle Scholar
- Auchmuty G: Variational principles for variational inequalities. Numer. Funct. Anal. Optim. 1989, 10: 863–874. 10.1080/01630568908816335MathSciNetView ArticleMATHGoogle Scholar
- Auslender A: Optimization, Méthodes Numériques. Masson, Paris; 1976.MATHGoogle Scholar
- Facchinei F, Pang JS: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, Berlin; 2003.MATHGoogle Scholar
- Fukushima M: Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems. Math. Program. 1992, 53: 99–110. 10.1007/BF01585696MathSciNetView ArticleMATHGoogle Scholar
- Harker PT, Pang JS: Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms, and applications. Math. Program. 1990, 48: 161–220. 10.1007/BF01582255MathSciNetView ArticleMATHGoogle Scholar
- Huang LR, Ng KF: Equivalent optimization formulations and error bounds for variational inequality problems. J. Optim. Theory Appl. 2005, 125: 299–314. 10.1007/s10957-004-1839-7MathSciNetView ArticleMATHGoogle Scholar
- Ng KF, Tan LL: Error bounds of regularized gap functions for nonsmooth variational inequality problems. Math. Program. 2007, 110: 405–429. 10.1007/s10107-006-0007-2MathSciNetView ArticleMATHGoogle Scholar
- Wu JH, Florian M, Marcotte P: A general descent framework for the monotone variational inequality problem. Math. Program. 1993, 61: 281–300. 10.1007/BF01582152MathSciNetView ArticleMATHGoogle Scholar
- Yamashita N, Taji K, Fukushima M: Unconstrained optimization reformulations of variational inequality problems. J. Optim. Theory Appl. 1997, 92: 439–456. 10.1023/A:1022660704427MathSciNetView ArticleMATHGoogle Scholar
- Li G, Tang C, Wei Z: Error bound results for generalized D-gap functions of nonsmooth variational inequality problems. J. Comput. Appl. Math. 2010, 233: 2795–2806. 10.1016/j.cam.2009.11.025MathSciNetView ArticleMATHGoogle Scholar
- Ng KF, Zheng XY: Error bounds for lower semicontinuous functions in normed spaces. SIAM J. Optim. 2001, 12: 1–17. 10.1137/S1052623499358884MathSciNetView ArticleMATHGoogle Scholar
- Lee GM, Kim DS, Lee BS, Yen ND: Vector variational inequality as a tool for studying vector optimization problems. Nonlinear Anal. 1998, 34: 745–765. 10.1016/S0362-546X(97)00578-6MathSciNetView ArticleMATHGoogle Scholar
- Jiang HY, Qi LQ: Local uniqueness and convergence of iterative methods for nonsmooth variational inequalities. J. Math. Anal. Appl. 1995, 196: 314–331. 10.1006/jmaa.1995.1412MathSciNetView ArticleMATHGoogle Scholar
- Li G, Ng KF: Error bounds of generalized D-gap functions for nonsmooth and nonmonotone variational inequality problems. SIAM J. Optim. 2009, 20: 667–690. 10.1137/070696283MathSciNetView ArticleMATHGoogle 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/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.