An iterative method for variational inequality problems
© Guan; licensee Springer. 2013
Received: 30 August 2013
Accepted: 11 November 2013
Published: 5 December 2013
In this paper, we present some properties of generalized proximity operators andpropose an iterative method of approximating solutions for a class ofgeneralized variational inequalities and show its convergence in uniformlyconvex and smooth Banach spaces.
MSC: 47J20, 46B20, 46N10, 47N10, 49J40.
thus defined, is called the proximity operator of f. When is the indicator function of a closed convex setK in H, then becomes the metric projection operator onK.
and they proved that has a subsequence converging to a solution of (1.3)when K is a nonempty compact convex subset of a uniformly convex anduniformly smooth Banach space.
where is a proper convex and lower semicontinuous function, is a norm-to-weak continuous operator. Our iterativemethod is different from that given in . We also prove a convergence result for this iterative method in smoothand uniformly convex Banach spaces. Let K be a nonempty closed convex setof X. If we replace f by in (1.5), where is the indicator function of K, then (1.5)reduces to (1.3).
Let X be a reflexive, smooth and strictly convex Banach space with the dualspace . We denote by and the strong and the weak convergence to x ofa sequence in a Banach space X, respectively. Let denote the class of all lower semi-continuous properconvex functions from X to . Let denote the closed ball of and radius . Let be the unit sphere.
A subset C of X is called boundedly compact if for any the intersection is empty or compact.
X is reflexive if and only if J is surjective;
X is strictly convex if and only if J is injective;
X is smooth if and only if J is single-valued;
if X is smooth, then J is norm-to-weak star continuous;
J is monotone, i.e., , ;
if X is strictly convex and smooth, then , ;
if a Banach space X is reflexive strictly convex and smooth, then the duality mapping from into X is the inverse of J, that is, .
where . Since the function is lower semicontinuous convex, one sees that is lower semicontinuous and convex byProposition 4.4 in .
If, in addition, , then .
Lemma 2.1 ()
LetXbe a smooth, strictly convex and reflexive Banach space,letbe a sequence inX, and. If, then, and.
Lemma 2.2 ()
3 Main results
is bounded on each nonempty bounded subset of;
ifis a sequence insuch that, then, and;
if domfis a nonempty boundedly compact convex subset, thenis weak-to-norm continuous, that is, if, then.
- (ii)Suppose that is not bounded on some nonempty bounded subset of C. Then there exists a bounded sequence such that . Fix . From (i), we obtain the following:
- (iii)Let be a sequence in such that . It follows from (ii) that is bounded. From (i), we have
- (iv)From (2.2), we know that(3.1)
Then, by (2.2), we have . Similar to the above arguments, we know that is the unique limit point of . Hence, . □
With the help of the operator , we can show that the envelope function is Gâteaux differentiable.
Proposition 3.2Let. Thenis Gâteaux differentiable and.
Hence is Gâteaux differentiable and. □
In the following, we propose a modification of the iterative method given in  and prove that the iterative sequence has a subsequence converging to asolution of (1.5) when X is a smooth and uniformly convex Banach space andf is not necessarily positively homogeneous.
By (2.2), we can easily prove the following result.
The following lemma will be used in proving the convergence of the iterative methodfor variational inequality problem (1.5).
- (ii)for any,
for all ;
Then generalized variational inequality (1.5) has a solution, and there exists a subsequenceofsuch thatas.
Now it follows from Proposition 3.3 that is a solution of generalized variationalinequality (1.5). □
Note that if g is convex and Gâteaux differentiable, then∇g is norm-to-weak continuous from X to by Corollary 3.1 in . Therefore, as an application of Proposition 3.4, we have thefollowing result.
- (ii)for any,
for all ;
Then problem (P) has a solutionand there exists a subsequenceofsuch thatas.
5 Concluding remark
This paper has improved the iterative method of Wu and Huang  for solving generalized variational inequality problem (1.5), severalresults regarding the generalized proximity operator and its relations with theenvelope function are presented. In addition, it is shown that under an appropriateassumption some optimization problem can be transformed into (1.5) and then theiterative method can be applied.
The work was supported by the Scientific Technology Program of the EducationalDepartment Heilongjiang Province (No. 12511161) and the National NaturalSciences Grant (No. 11071052).
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