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An iterative method for variational inequality problems
Journal of Inequalities and Applications volume 2013, Article number: 574 (2013)
Abstract
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.
1 Introduction
Let f be a lower semicontinuous proper convex function from a Hilbert spaceH to (\mathrm{\infty},+\mathrm{\infty}]. The Moreau envelope of the function f isdefined as
It is well known that {e}^{f}(x) is a continuous convex function, and for everyx\in H, the infimum in (1.1) is achieved at a unique point{prox}_{f}(x). The operator {prox}_{f} from H to H, i.e.,
thus defined, is called the proximity operator of f. Whenf={\iota}_{K} is the indicator function of a closed convex setK in H, then {prox}_{f}(x)={P}_{K}(x) 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 [1] introduced the generalized projections {\pi}_{K}:{X}^{\ast}\to K and {\mathrm{\Pi}}_{K}:X\to K,
and
where J is the duality mapping from X to {X}^{\ast}, and studied their properties in detail. In [2], Alber presented some applications of the generalized projections toapproximately solving variational inequalities in Banach spaces. Recently, Li [3] extended the generalized projection operator {\pi}_{K} 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 [4] established some existence results for the variational inequality problemin uniformly convex and uniformly smooth Banach spaces and convergence results forthe variational inequality. In [5], Wu and Huang further introduced and studied a class of generalizedfprojection operators in Banach spaces. As applications, they proposedan iterative method of approximating solutions for the variational inequalityproblem: find \overline{x}\in K such that
where K is a nonempty closed convex subset of X,A:K\to {X}^{\ast} is a mapping and f:X\to (\mathrm{\infty},+\mathrm{\infty}] is a proper convex, lower semicontinuous andpositively homogeneous function, via
where
and the parameter sequence \{{\alpha}_{n}\} satisfies
and they proved that \{{x}_{n}\} 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\overline{x}\in domf such that
where f:X\to (\mathrm{\infty},+\mathrm{\infty}] is a proper convex and lower semicontinuous function,A:X\to {X}^{\ast} is a normtoweak continuous operator. Our iterativemethod is different from that given in [5]. 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 f+{I}_{K} in (1.5), where {I}_{K} is the indicator function of K, then (1.5)reduces to (1.3).
2 Preliminaries
Let X be a reflexive, smooth and strictly convex Banach space with the dualspace {X}^{\ast}. We denote by {x}_{n}\to x and {x}_{n}\rightharpoonup x the strong and the weak convergence to x ofa sequence \{{x}_{n}\} in a Banach space X, respectively. Let{\mathrm{\Gamma}}_{0}(X) denote the class of all lower semicontinuous properconvex functions from X to (\mathrm{\infty},+\mathrm{\infty}]. Let B(x,\delta ) denote the closed ball of x\in X and radius \delta >0. Let S(X)=\{x\in X:\parallel x\parallel =1\} be the unit sphere.
A Banach space X is said to be strictly convex if \frac{1}{2}\parallel x+y\parallel <1 for all x,y\in S(X) and x\ne y. The Banach space X is said to be smoothprovided
exists for each x,y\in S(X). We recall that uniform convexity of X meansthat for any given \u03f5>0, there exists \delta >0 such that for all x,y\in X with \parallel x\parallel \le 1, \parallel y\parallel \le 1, and \parallel xy\parallel =\u03f5, the inequality
holds.
A subset C of X is called boundedly compact if for any\delta >0 the intersection C\cap B(0,\delta ) is empty or compact.
The duality mapping J:X\rightrightarrows {X}^{\ast} is defined by
The following basic results concerning the duality mapping are well known [2, 6, 7]:

(1)
X is reflexive if and only if J is surjective;

(2)
X is strictly convex if and only if J is injective;

(3)
X is smooth if and only if J is singlevalued;

(4)
if X is smooth, then J is normtoweak star continuous;

(5)
J is monotone, i.e., \u3008JxJy,xy\u3009\ge 0, \mathrm{\forall}x,y\in X;

(6)
if X is strictly convex and smooth, then \u3008JxJy,xy\u3009=0\Rightarrow x=y, \mathrm{\forall}x,y\in X;

(7)
if a Banach space X is reflexive strictly convex and smooth, then the duality mapping {J}^{\ast} from {X}^{\ast} into X is the inverse of J, that is, {J}^{1}={J}^{\ast}.
Consider the following envelope function:
where V({x}^{\ast},x)={\parallel {x}^{\ast}\parallel}^{2}2\u3008{x}^{\ast},x\u3009+{\parallel x\parallel}^{2}. Since the function (x,{x}^{\ast})\to f(x)+\frac{1}{2}V({x}^{\ast},x) is lower semicontinuous convex, one sees that{e}_{V}^{f}({x}^{\ast}) is lower semicontinuous and convex byProposition 4.4 in [8].
For every {x}^{\ast}\in {X}^{\ast}, the infimum in (2.1) is achieved at a unique point{\pi}_{f}({x}^{\ast}), i.e.,
The operator {\pi}_{f} is called the generalized proximity operator. It canbe characterized by the inclusion
equivalently,
From (2.3), we easily know that {\pi}_{f} is maximal monotone by Theorem 2.6.2 in [6]. Observe that when \rho =1,
If, in addition, domf=K, then {\pi}_{K}^{f}({x}^{\ast})={\pi}_{f}({x}^{\ast}).
Lemma 2.1 ([9])
LetXbe a smooth, strictly convex and reflexive Banach space,let\{{x}_{n}\}be a sequence inX, andx\in X. If\u3008{x}_{n}x,J{x}_{n}Jx\u3009\to 0, then{x}_{n}\rightharpoonup x, J{x}_{n}\rightharpoonup Jxand\parallel {x}_{n}\parallel \to \parallel x\parallel.
Lemma 2.2 ([10])
Letr>0be a fixed real number. Then a Banach spaceXis uniformly convex if and only if there is a continuous, strictlyincreasing and convex functiong:{R}^{+}\to {R}^{+}withg(0)=0such that
where{B}_{r}=\{x\in X:\parallel x\parallel \le r\}.
3 Main results
Proposition 3.1Letf\in {\mathrm{\Gamma}}_{0}(X). Then the following hold:

(i)
\u3008{\pi}_{f}({x}^{\ast}){\pi}_{f}({y}^{\ast}),({x}^{\ast}J{\pi}_{f}({x}^{\ast}))({y}^{\ast}J{\pi}_{f}({y}^{\ast}))\u3009\ge 0, \mathrm{\forall}{x}^{\ast},{y}^{\ast}\in {X}^{\ast};

(ii)
{\pi}_{f}is bounded on each nonempty bounded subset ofC\subset {X}^{\ast};

(iii)
if\{{x}_{n}^{\ast}\}is a sequence in{X}^{\ast}such that{x}_{n}^{\ast}\to {x}^{\ast}, then{\pi}_{f}({x}_{n}^{\ast})\rightharpoonup {\pi}_{f}({x}^{\ast}), J{\pi}_{f}({x}_{n}^{\ast})\rightharpoonup J{\pi}_{f}({x}^{\ast})and\parallel {\pi}_{f}({x}_{n}^{\ast})\parallel \to \parallel {\pi}_{f}({x}^{\ast})\parallel;

(iv)
if domfis a nonempty boundedly compact convex subset, then{\pi}_{f}is weaktonorm continuous, that is, if{x}_{n}^{\ast}\rightharpoonup {x}^{\ast}, then{\pi}_{f}({x}_{n}^{\ast})\to {\pi}_{f}({x}^{\ast}).
Proof (i) Take {x}^{\ast},{y}^{\ast}\in {X}^{\ast}. Then (2.2) yields
and
Adding these two inequalities, we obtain

(ii)
Suppose that {\pi}_{f} is not bounded on some nonempty bounded subset of C. Then there exists a bounded sequence \{{x}_{n}^{\ast}\}\subset C such that \parallel {\pi}_{f}({x}_{n}^{\ast})\parallel \to \mathrm{\infty}. Fix {x}^{\ast}\in {X}^{\ast}. From (i), we obtain the following:
\begin{array}{rcl}\parallel {\pi}_{f}\left({x}_{n}^{\ast}\right){\pi}_{f}\left({x}^{\ast}\right)\parallel \parallel {x}_{n}^{\ast}{x}^{\ast}\parallel & \ge & \u3008{\pi}_{f}\left({x}_{n}^{\ast}\right){\pi}_{f}\left({x}^{\ast}\right),{x}_{n}^{\ast}{x}^{\ast}\u3009\\ \ge & \u3008{\pi}_{f}\left({x}_{n}^{\ast}\right){\pi}_{f}\left({x}^{\ast}\right),J{\pi}_{f}\left({x}_{n}^{\ast}\right)J{\pi}_{f}\left({x}^{\ast}\right)\u3009\\ =& \frac{1}{2}(V(J{\pi}_{f}\left({x}_{n}^{\ast}\right),{\pi}_{f}\left({x}^{\ast}\right))+V(J{\pi}_{f}\left({x}^{\ast}\right),{\pi}_{f}\left({x}_{n}^{\ast}\right)))\\ \ge & {(\parallel {\pi}_{f}\left({x}_{n}^{\ast}\right)\parallel \parallel {\pi}_{f}\left({x}^{\ast}\right)\parallel )}^{2}.\end{array}
So, we have \parallel {x}_{n}^{\ast}\parallel \to \mathrm{\infty}. This is a contradiction.

(iii)
Let \{{x}_{n}^{\ast}\} be a sequence in {X}^{\ast} such that {x}_{n}^{\ast}\to {x}^{\ast}. It follows from (ii) that \{{\pi}_{f}({x}_{n}^{\ast})\} is bounded. From (i), we have
0\le \u3008{\pi}_{f}\left({x}_{n}^{\ast}\right){\pi}_{f}\left({x}^{\ast}\right),J{\pi}_{f}\left({x}_{n}^{\ast}\right)J{\pi}_{f}\left({x}^{\ast}\right)\u3009\le \parallel {\pi}_{f}\left({x}_{n}^{\ast}\right){\pi}_{f}\left({x}^{\ast}\right)\parallel \parallel {x}_{n}^{\ast}{x}^{\ast}\parallel \to 0.
Thus, Lemma 2.1 implies that {\pi}_{f}({x}_{n}^{\ast})\rightharpoonup {\pi}_{f}({x}^{\ast}), J{\pi}_{f}({x}_{n}^{\ast})\rightharpoonup J{\pi}_{f}({x}^{\ast}) and \parallel {\pi}_{f}({x}_{n}^{\ast})\parallel \to \parallel {\pi}_{f}({x}^{\ast})\parallel.

(iv)
From (2.2), we know that
\u3008{x}_{n}^{\ast}J{\pi}_{f}\left({x}_{n}^{\ast}\right),y{\pi}_{f}\left({x}_{n}^{\ast}\right)\u3009\le f(y)f\left({\pi}_{f}\left({x}_{n}^{\ast}\right)\right),\phantom{\rule{1em}{0ex}}\mathrm{\forall}y\in domf.(3.1)
Since {x}_{n}^{\ast}\rightharpoonup {x}^{\ast}, \{{x}_{n}^{\ast}\} is bounded. It follows from (ii) that\{{\pi}_{f}({x}_{n}^{\ast})\} is bounded. Since domf is boundedly compact,there exists a subsequence \{{x}_{{n}_{i}}^{\ast}\} of \{{x}_{n}^{\ast}\} such that
Since J is normtoweak star continuous and f is lowersemicontinuous, we obtain that
Then, by (2.2), we have \overline{x}={\pi}_{f}({x}^{\ast}). Similar to the above arguments, we know that{\pi}_{f}({x}^{\ast}) is the unique limit point of \{{\pi}_{f}({x}_{n}^{\ast})\}. Hence, {\pi}_{f}({x}_{n}^{\ast})\to {\pi}_{f}({x}^{\ast}). □
With the help of the operator {\pi}_{f}, we can show that the envelope function{e}_{V}^{f} is Gâteaux differentiable.
Proposition 3.2Letf\in {\mathrm{\Gamma}}_{0}(X). Then{e}_{V}^{f}is Gâteaux differentiable and\mathrm{\nabla}{e}_{V}^{f}({x}^{\ast})={J}^{\ast}{x}^{\ast}{\pi}_{f}({x}^{\ast}).
Proof For any h\in {X}^{\ast}, by definitions of {e}_{V}^{f} and {\pi}_{f}, we have
Since \mathrm{\nabla}({\parallel z\parallel}^{2})=2{J}^{\ast}(z), for any z\in {X}^{\ast}, we get that
On the other hand,
By Proposition 3.1(iii), we have {\pi}_{f}({x}^{\ast}+th)\rightharpoonup {\pi}_{f}({x}^{\ast}) as t\to 0. Hence, we get that
This implies that
Hence {e}_{V}^{f} is Gâteaux differentiable and\mathrm{\nabla}{e}_{V}^{f}({x}^{\ast})={J}^{\ast}{x}^{\ast}{\pi}_{f}({x}^{\ast}). □
In the following, we propose a modification of the iterative method given in [5] 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 3.3Letf\in {\mathrm{\Gamma}}_{0}(X). Then the point\overline{x}\in domfis a solution of the variational inequality
if and only if\overline{x}\in domfis 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 3.1Letf\in {\mathrm{\Gamma}}_{0}(X). Iff(x)\ge 0for allx\in domfandf(0)=0, then
Proof From (2.2), we know that
Noticing that f(x)\ge 0 for all x\in domf and f(0)=0, it follows that
Hence,
and hence
□
Proposition 3.4LetXbe a smooth and uniformly convex Banach space. LetA:X\to {X}^{\ast}be a normtoweak continuous operator. Supposethatf\in {\mathrm{\Gamma}}_{0}(X)and domfis nonempty boundedly compact convex. Suppose that

(i)
f(x)\ge 0for allx\in domfandf(0)=0;

(ii)
for anyx\in domf,
\parallel JxAx\parallel \le \parallel x\parallel .
Let{x}_{0}\in domfand the sequence\{{x}_{n}\}be generated by the following iteration scheme:
where\{{\alpha}_{n}\}satisfies the conditions:

(a)
0\le {\alpha}_{n}\le 1for alln=0,1,2,\dots ;

(b)
{\sum}_{n=0}^{+\mathrm{\infty}}{\alpha}_{n}(1{\alpha}_{n})=+\mathrm{\infty}.
Then generalized variational inequality (1.5) has a solution\overline{x}\in domf, and there exists a subsequence\{{x}_{{n}_{i}}\}of\{{x}_{n}\}such that{x}_{{n}_{i}}\to \overline{x}asi\to \mathrm{\infty}.
Proof By (3.3), we have
By (3.4) and condition (ii), we obtain
Then \{{x}_{n}\} and \{{\pi}_{f}(J{x}_{n}A{x}_{n})\} are bounded. Hence, by Lemma 2.2, there exists acontinuous, strictly increasing and convex function g:{R}^{+}\to {R}^{+} with g(0)=0 such that
It follows from (3.5), (3.4) and condition (ii) that
That is,
Taking the sum for n=0,1,2,\dots ,m in (3.6), we get
Hence,
Due to the condition {\sum}_{n=0}^{+\mathrm{\infty}}{\alpha}_{n}(1{\alpha}_{n})=+\mathrm{\infty}, 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\{{x}_{{n}_{i}}\} of \{{x}_{n}\} such that
Since A is normtoweak continuous and J is normtoweak starcontinuous, we get that
Since {\pi}_{f} is weaktonorm continuous byProposition 3.1(iv),
Hence, (3.8) yields
Now it follows from Proposition 3.3 that \overline{x} is a solution of generalized variationalinequality (1.5). □
4 Application
Let f\in {\mathrm{\Gamma}}_{0}(X) and let g:X\to \mathbb{R} be a convex and Gâteaux differentiable function.Consider the optimization problem
We denote by Sol(\mathrm{P}) the solution set of problem (P). Despite itssimplicity, problem (P) has been shown to cover a wide range of apparently unrelatedsignal recovery formulations (see [11, 12]).
Notice that
Note that if g is convex and Gâteaux differentiable, then∇g is normtoweak continuous from X to{X}^{\ast} by Corollary 3.1 in [13]. Therefore, as an application of Proposition 3.4, we have thefollowing result.
Proposition 4.1LetXbe a smooth and uniformly convex Banach space. Letg:X\to \mathbb{R}be convex and Gâteaux differentiable. Suppose thatf\in {\mathrm{\Gamma}}_{0}(X)and domfis a nonempty boundedly compact convex subset ofX. Suppose that

(i)
f(x)\ge 0for allx\in domfandf(0)=0;

(ii)
for anyx\in domf,
\parallel Jx\mathrm{\nabla}g(x)\parallel \le \parallel x\parallel .
Let{x}_{0}\in domfand the sequence\{{x}_{n}\}be generated by the following iteration scheme:
where\{{\alpha}_{n}\}satisfies the conditions:

(a)
0\le {\alpha}_{n}\le 1for alln=0,1,2,\dots ;

(b)
{\sum}_{n=0}^{+\mathrm{\infty}}{\alpha}_{n}(1{\alpha}_{n})=+\mathrm{\infty}.
Then problem (P) has a solution\overline{x}and there exists a subsequence\{{x}_{{n}_{i}}\}of\{{x}_{n}\}such that{x}_{{n}_{i}}\to \overline{x}asi\to \mathrm{\infty}.
5 Concluding remark
This paper has improved the iterative method of Wu and Huang [5] 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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Acknowledgements
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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Guan, WB. An iterative method for variational inequality problems. J Inequal Appl 2013, 574 (2013). https://doi.org/10.1186/1029242X2013574
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DOI: https://doi.org/10.1186/1029242X2013574