An iterative method for variational inequality problems
Journal of Inequalities and Applications volume 2013, Article number: 574 (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.
Let f be a lower semi-continuous proper convex function from a Hilbert spaceH to . The Moreau envelope of the function f isdefined as
It is well known that is a continuous convex function, and for every, the infimum in (1.1) is achieved at a unique point. The operator from H to H, i.e.,
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.
In 1994, Alber extended the metric projection operator to uniformly convex anduniformly smooth Banach spaces. Let K be a closed convex subset of auniformly convex and uniformly smooth Banach space X, Alber  introduced the generalized projections and ,
where J is the duality mapping from X to , and studied their properties in detail. In , Alber presented some applications of the generalized projections toapproximately solving variational inequalities in Banach spaces. Recently, Li  extended the generalized projection operator from uniformly convex and uniformly smooth Banachspaces to reflexive Banach spaces and studied some properties of the generalizedprojection operator with applications to solving the variational inequality inBanach spaces. By employing the generalized projection operators, Zeng and Yao  established some existence results for the variational inequality problemin uniformly convex and uniformly smooth Banach spaces and convergence results forthe variational inequality. In , Wu and Huang further introduced and studied a class of generalizedf-projection operators in Banach spaces. As applications, they proposedan iterative method of approximating solutions for the variational inequalityproblem: find such that
where K is a nonempty closed convex subset of X, is a mapping and is a proper convex, lower semicontinuous andpositively homogeneous function, via
and the parameter sequence satisfies
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.
Motivated and inspired by the above works, we continue to study some properties ofgeneralized proximity operators and propose an iterative method of approximatingsolutions for the following generalized variational inequality problem: find such that
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 Banach space X is said to be strictly convex if for all and . The Banach space X is said to be smoothprovided
exists for each . We recall that uniform convexity of X meansthat for any given , there exists such that for all with , , and , the inequality
A subset C of X is called boundedly compact if for any the intersection is empty or compact.
The duality mapping is defined by
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, .
Consider the following envelope function:
where . Since the function is lower semicontinuous convex, one sees that is lower semicontinuous and convex byProposition 4.4 in .
For every , the infimum in (2.1) is achieved at a unique point, i.e.,
The operator is called the generalized proximity operator. It canbe characterized by the inclusion
From (2.3), we easily know that is maximal monotone by Theorem 2.6.2 in . Observe that when ,
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 ()
Letbe a fixed real number. Then a Banach spaceXis uniformly convex if and only if there is a continuous, strictlyincreasing and convex functionwithsuch that
3 Main results
Proposition 3.1Let. Then the following hold:
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.
Proof (i) Take . Then (2.2) yields
Adding these two inequalities, we obtain
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:
So, we have . This is a contradiction.
Let be a sequence in such that . It follows from (ii) that is bounded. From (i), we have
Thus, Lemma 2.1 implies that , and .
From (2.2), we know that(3.1)
Since , is bounded. It follows from (ii) that is bounded. Since domf is boundedly compact,there exists a subsequence of such that
Since J is norm-to-weak star continuous and f is lowersemicontinuous, we obtain that
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.
Proof For any , by definitions of and , we have
Since , for any , we get that
On the other hand,
By Proposition 3.1(iii), we have as . Hence, we get that
This implies that
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.
Proposition 3.3Let. Then the pointis a solution of the variational inequality
if and only ifis a solution of the following inclusion:
The following lemma will be used in proving the convergence of the iterative methodfor variational inequality problem (1.5).
Lemma 3.1Let. Iffor alland, then
Proof From (2.2), we know that
Noticing that for all and , it follows that
Proposition 3.4LetXbe a smooth and uniformly convex Banach space. Letbe a norm-to-weak continuous operator. Supposethatand domfis nonempty boundedly compact convex. Suppose that
Letand the sequencebe generated by the following iteration scheme:
wheresatisfies the conditions:
for all ;
Then generalized variational inequality (1.5) has a solution, and there exists a subsequenceofsuch thatas.
Proof By (3.3), we have
By (3.4) and condition (ii), we obtain
Then and are bounded. Hence, by Lemma 2.2, there exists acontinuous, strictly increasing and convex function with such that
It follows from (3.5), (3.4) and condition (ii) that
Taking the sum for in (3.6), we get
Due to the condition , we may assume, without loss of generality, that
Applying the properties of g, we can deduce that
Since domf is boundedly compact, there exists a subsequence of such that
Since A is norm-to-weak continuous and J is norm-to-weak starcontinuous, we get that
Since is weak-to-norm continuous byProposition 3.1(iv),
Hence, (3.8) yields
Now it follows from Proposition 3.3 that is a solution of generalized variationalinequality (1.5). □
Let and let be a convex and Gâteaux differentiable function.Consider the optimization problem
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.
Proposition 4.1LetXbe a smooth and uniformly convex Banach space. Letbe convex and Gâteaux differentiable. Suppose thatand domfis a nonempty boundedly compact convex subset ofX. Suppose that
Letand the sequencebe generated by the following iteration scheme:
wheresatisfies the conditions:
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.
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The work was supported by the Scientific Technology Program of the EducationalDepartment Heilongjiang Province (No. 12511161) and the National NaturalSciences Grant (No. 11071052).
The author declares that they have no competing interests.
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Guan, WB. An iterative method for variational inequality problems. J Inequal Appl 2013, 574 (2013). https://doi.org/10.1186/1029-242X-2013-574