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.
Keywordserror bound regularized gap function weak vector variational inequality
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).
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