Skip to main content

An iterative method for variational inequality problems

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 semi-continuous proper convex function from a Hilbert spaceH to (,+]. The Moreau envelope of the function f isdefined as

e f (x)= inf y H { f ( y ) + 1 2 x y 2 } .
(1.1)

It is well known that e f (x) is a continuous convex function, and for everyxH, the infimum in (1.1) is achieved at a unique point prox f (x). The operator prox f from H to H, i.e.,

prox f x= arg  min y H { f ( y ) + 1 2 x y 2 }
(1.2)

thus defined, is called the proximity operator of f. Whenf= ι 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 π K : X K and Π K :XK,

π K ( x ) = arg min x K { x 2 2 x , x + x 2 }

and

Π K (x)= arg min y K { J x 2 2 J x , y + y 2 } ,

where J is the duality mapping from X to X , 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 π 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 generalizedf-projection operators in Banach spaces. As applications, they proposedan iterative method of approximating solutions for the variational inequalityproblem: find x ¯ K such that

A x ¯ ,y x ¯ +f(y)f( x ¯ )0,yK,
(1.3)

where K is a nonempty closed convex subset of X,A:K X is a mapping and f:X(,+] is a proper convex, lower semicontinuous andpositively homogeneous function, via

x n + 1 = π K f ( J x n α n J ( x n π K f ( J x n ρ A x n ) ) ) ,
(1.4)

where

π K f ( x ) = arg min x K { 2 ρ f ( x ) + x 2 2 x , x + x 2 } ,

and the parameter sequence { α n } satisfies

0 α n 1, n = 0 + α n (1 α n )=+,ρ>0,

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 x ¯ domf such that

A x ¯ ,y x ¯ +f(y)f( x ¯ )0,ydomf,
(1.5)

where f:X(,+] is a proper convex and lower semicontinuous function,A:X X is a norm-to-weak 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 . We denote by x n x and x n x the strong and the weak convergence to x ofa sequence { x n } in a Banach space X, respectively. Let Γ 0 (X) denote the class of all lower semi-continuous properconvex functions from X to (,+]. Let B(x,δ) denote the closed ball of xX and radius δ>0. Let S(X)={xX:x=1} be the unit sphere.

A Banach space X is said to be strictly convex if 1 2 x+y<1 for all x,yS(X) and xy. The Banach space X is said to be smoothprovided

lim t 0 x + t y x t

exists for each x,yS(X). We recall that uniform convexity of X meansthat for any given ϵ>0, there exists δ>0 such that for all x,yX with x1, y1, and xy=ϵ, the inequality

x+y2(1δ)

holds.

A subset C of X is called boundedly compact if for anyδ>0 the intersection CB(0,δ) is empty or compact.

The duality mapping J:X X is defined by

J(x)= { x X | x , x = x 2 = x 2 } ,xX.

The following basic results concerning the duality mapping are well known [2, 6, 7]:

  1. (1)

    X is reflexive if and only if J is surjective;

  2. (2)

    X is strictly convex if and only if J is injective;

  3. (3)

    X is smooth if and only if J is single-valued;

  4. (4)

    if X is smooth, then J is norm-to-weak star continuous;

  5. (5)

    J is monotone, i.e., JxJy,xy0, x,yX;

  6. (6)

    if X is strictly convex and smooth, then JxJy,xy=0x=y, x,yX;

  7. (7)

    if a Banach space X is reflexive strictly convex and smooth, then the duality mapping J from X into X is the inverse of J, that is, J 1 = J .

Consider the following envelope function:

e V f ( x ) = inf x X { f ( x ) + 1 2 V ( x , x ) } ,
(2.1)

where V( x ,x)= x 2 2 x ,x+ x 2 . Since the function (x, x )f(x)+ 1 2 V( x ,x) is lower semicontinuous convex, one sees that e V f ( x ) is lower semicontinuous and convex byProposition 4.4 in [8].

For every x X , the infimum in (2.1) is achieved at a unique point π f ( x ), i.e.,

π f ( x ) := arg  min x K { f ( x ) + 1 2 V ( x , x ) } .

The operator π f is called the generalized proximity operator. It canbe characterized by the inclusion

x J π f ( x ) f ( π f ( x ) ) ,
(2.2)

equivalently,

π f = ( J + f ) 1 .
(2.3)

From (2.3), we easily know that π f is maximal monotone by Theorem 2.6.2 in [6]. Observe that when ρ=1,

π K f ( x ) = π f + I K ( x ) .

If, in addition, domf=K, then π K f ( x )= π f ( x ).

Lemma 2.1 ([9])

LetXbe a smooth, strictly convex and reflexive Banach space,let{ x n }be a sequence inX, andxX. If x n x,J x n Jx0, then x n x, J x n Jxand x n x.

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 + R + withg(0)=0such that

λ x + ( 1 λ ) y 2 λ x 2 +(1λ) y 2 λ(1λ)g ( x y ) ,x,y B r ,0λ1,

where B r ={xX:xr}.

3 Main results

Proposition 3.1Letf Γ 0 (X). Then the following hold:

  1. (i)

    π f ( x ) π f ( y ),( x J π f ( x ))( y J π f ( y ))0, x , y X ;

  2. (ii)

    π f is bounded on each nonempty bounded subset ofC X ;

  3. (iii)

    if{ x n }is a sequence in X such that x n x , then π f ( x n ) π f ( x ), J π f ( x n )J π f ( x )and π f ( x n ) π f ( x );

  4. (iv)

    if domfis a nonempty boundedly compact convex subset, then π f is weak-to-norm continuous, that is, if x n x , then π f ( x n ) π f ( x ).

Proof (i) Take x , y X . Then (2.2) yields

x J π f ( x ) , π f ( y ) π f ( x ) f ( π f ( y ) ) f ( π f ( x ) )

and

y J π f ( y ) , π f ( x ) π f ( y ) f ( π f ( x ) ) f ( π f ( y ) ) .

Adding these two inequalities, we obtain

π f ( x ) π f ( y ) , ( x J π f ( x ) ) ( y J π f ( y ) ) 0.
  1. (ii)

    Suppose that π f is not bounded on some nonempty bounded subset of C. Then there exists a bounded sequence { x n }C such that π f ( x n ). Fix x X . From (i), we obtain the following:

    π f ( x n ) π f ( x ) x n x π f ( x n ) π f ( x ) , x n x π f ( x n ) π f ( x ) , J π f ( x n ) J π f ( x ) = 1 2 ( V ( J π f ( x n ) , π f ( x ) ) + V ( J π f ( x ) , π f ( x n ) ) ) ( π f ( x n ) π f ( x ) ) 2 .

So, we have x n . This is a contradiction.

  1. (iii)

    Let { x n } be a sequence in X such that x n x . It follows from (ii) that { π f ( x n )} is bounded. From (i), we have

    0 π f ( x n ) π f ( x ) , J π f ( x n ) J π f ( x ) π f ( x n ) π f ( x ) x n x 0.

Thus, Lemma 2.1 implies that π f ( x n ) π f ( x ), J π f ( x n )J π f ( x ) and π f ( x n ) π f ( x ).

  1. (iv)

    From (2.2), we know that

    x n J π f ( x n ) , y π f ( x n ) f(y)f ( π f ( x n ) ) ,ydomf.
    (3.1)

Since x n x , { x n } is bounded. It follows from (ii) that{ π f ( x n )} is bounded. Since domf is boundedly compact,there exists a subsequence { x n i } of { x n } such that

π f ( x n i ) x ¯ domfas i+.

Since J is norm-to-weak star continuous and f is lowersemicontinuous, we obtain that

x J x ¯ , y x ¯ f(y)f( x ¯ ),ydomf.
(3.2)

Then, by (2.2), we have x ¯ = π f ( x ). Similar to the above arguments, we know that π f ( x ) is the unique limit point of { π f ( x n )}. Hence, π f ( x n ) π f ( x ). □

With the help of the operator π f , we can show that the envelope function e V f is Gâteaux differentiable.

Proposition 3.2Letf Γ 0 (X). Then e V f is Gâteaux differentiable and e V f ( x )= J x π f ( x ).

Proof For any h X , by definitions of e V f and π f , we have

e V f ( x + t h ) e V f ( x ) t = f ( π f ( x + t h ) ) + 1 2 V ( π f ( x + t h ) , x + t h ) f ( π f ( x ) ) 1 2 V ( π f ( x ) , x ) t f ( π f ( x ) ) + 1 2 V ( π f ( x ) , x + t h ) f ( π f ( x ) ) 1 2 V ( π f ( x ) , x ) t = 1 2 x + t h 2 1 2 x 2 π f ( x ) , t h t .

Since ( z 2 )=2 J (z), for any z X , we get that

lim sup t 0 e V f ( x + t h ) e V f ( x ) t lim t 0 1 2 x + t h 2 1 2 x 2 t π f ( x ) , h = J x π f ( x ) , h .

On the other hand,

e V f ( x + t h ) e V f ( x ) t = f ( π f ( x + t h ) ) + 1 2 V ( π f ( x + t h ) , x + t h ) f ( π f ( x ) ) 1 2 V ( π f ( x ) , x ) t f ( π f ( x + t h ) ) + 1 2 V ( π f ( x + t h ) , x + t h ) f ( π f ( x + t h ) ) 1 2 V ( π f ( x + t h ) , x ) t = 1 2 x + t h 2 1 2 x 2 t π f ( x + t h ) , h .

By Proposition 3.1(iii), we have π f ( x +th) π f ( x ) as t0. Hence, we get that

lim inf t 0 e V f ( x + t h ) e V f ( x ) t lim t 0 1 2 x + t h 2 1 2 x 2 t π f ( x + t h ) , h = J x π f ( x ) , h .

This implies that

lim t 0 e V f ( x + t h ) e V f ( x ) t = J x π f ( x ) , h .

Hence e V f is Gâteaux differentiable and e V f ( x )= J x π f ( x ). □

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 Γ 0 (X). Then the point x ¯ domfis a solution of the variational inequality

Ax,yx+f(y)f(x)0,ydomf

if and only if x ¯ domfis a solution of the following inclusion:

x= π f (JxAx).

The following lemma will be used in proving the convergence of the iterative methodfor variational inequality problem (1.5).

Lemma 3.1Letf Γ 0 (X). Iff(x)0for allxdomfandf(0)=0, then

π f ( x ) x .
(3.3)

Proof From (2.2), we know that

x J π f ( x ) , y π f ( x ) f(y)f ( π f ( x ) ) ,ydomf.

Noticing that f(x)0 for all xdomf and f(0)=0, it follows that

x J π f ( x ) , π f ( x ) f ( π f ( x ) ) 0.

Hence,

π f ( x ) 2 x , π f ( x ) ,

and hence

π f ( x ) x .

 □

Proposition 3.4LetXbe a smooth and uniformly convex Banach space. LetA:X X be a norm-to-weak continuous operator. Supposethatf Γ 0 (X)and domfis nonempty boundedly compact convex. Suppose that

  1. (i)

    f(x)0for allxdomfandf(0)=0;

  2. (ii)

    for anyxdomf,

    JxAxx.

Let x 0 domfand the sequence{ x n }be generated by the following iteration scheme:

x n + 1 =(1 α n ) x n + α n π f (J x n A x n ),

where{ α n }satisfies the conditions:

  1. (a)

    0 α n 1for alln=0,1,2, ;

  2. (b)

    n = 0 + α n (1 α n )=+.

Then generalized variational inequality (1.5) has a solution x ¯ domf, and there exists a subsequence{ x n i }of{ x n }such that x n i x ¯ asi.

Proof By (3.3), we have

π f ( J x n A x n ) J x n A x n .
(3.4)

By (3.4) and condition (ii), we obtain

x n + 1 (1 α n ) x n + α n π f ( J x n A x n ) x n .

Then { x n } and { π f (J x n A x n )} are bounded. Hence, by Lemma 2.2, there exists acontinuous, strictly increasing and convex function g: R + R + with g(0)=0 such that

x n + 1 2 = ( 1 α n ) x n + α n π f ( J x n A x n ) 2 ( 1 α n ) x n 2 + α n π f ( J x n A x n ) 2 α n ( 1 α n ) g ( x n π f ( J x n A x n ) ) .
(3.5)

It follows from (3.5), (3.4) and condition (ii) that

x n + 1 2 (1 α n ) x n 2 + α n x n 2 α n (1 α n )g ( x n π f ( J x n A x n ) ) .

That is,

x n + 1 2 x n 2 α n (1 α n )g ( x n π f ( J x n A x n ) ) .
(3.6)

Taking the sum for n=0,1,2,,m in (3.6), we get

n = 0 m α n ( 1 α n ) g ( x n π f ( J x n A x n ) ) x 0 2 x m + 1 2 x 0 2 .

Hence,

n = 0 + α n (1 α n )g ( x n π f ( J x n A x n ) ) <+.
(3.7)

Due to the condition n = 0 + α n (1 α n )=+, we may assume, without loss of generality, that

g ( x n π f ( J x n A x n ) ) 0as n+.

Applying the properties of g, we can deduce that

x n π f ( J x n A x n ) 0as n+.
(3.8)

Since domf is boundedly compact, there exists a subsequence{ x n i } of { x n } such that

x n i x ¯ domfas i+.

Since A is norm-to-weak continuous and J is norm-to-weak starcontinuous, we get that

J x n i A x n i J x ¯ A x ¯ as i+.

Since π f is weak-to-norm continuous byProposition 3.1(iv),

π f (J x n i A x n i ) π f (J x ¯ A x ¯ )as i+.

Hence, (3.8) yields

x ¯ = π f (J x ¯ A x ¯ ).

Now it follows from Proposition 3.3 that x ¯ is a solution of generalized variationalinequality (1.5). □

4 Application

Let f Γ 0 (X) and let g:XR be a convex and Gâteaux differentiable function.Consider the optimization problem

min x X f(x)+g(x).
(P)

We denote by Sol(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

x ¯ Sol ( P ) 0 f ( x ¯ ) + g ( x ¯ ) g ( x ¯ ) f ( x ¯ ) g ( x ¯ ) , y x ¯ + f ( y ) f ( x ¯ ) 0 , y X .

Note that if g is convex and Gâteaux differentiable, theng is norm-to-weak continuous from X to X 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:XRbe convex and Gâteaux differentiable. Suppose thatf Γ 0 (X)and domfis a nonempty boundedly compact convex subset ofX. Suppose that

  1. (i)

    f(x)0for allxdomfandf(0)=0;

  2. (ii)

    for anyxdomf,

    J x g ( x ) x.

Let x 0 domfand the sequence{ x n }be generated by the following iteration scheme:

x n + 1 =(1 α n ) x n + α n π f ( J x n g ( x n ) ) ,

where{ α n }satisfies the conditions:

  1. (a)

    0 α n 1for alln=0,1,2, ;

  2. (b)

    n = 0 + α n (1 α n )=+.

Then problem (P) has a solution x ¯ and there exists a subsequence{ x n i }of{ x n }such that x n i x ¯ asi.

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.

References

  1. 1.

    Alber Y: Generalized projection operators in Banach spaces: properties andapplications. 1.Proceedings of the Israel Seminar Ariel, Israel, Funct. Differ.Equ. 1994, 1–21.

    Google Scholar 

  2. 2.

    Alber Y: Metric and generalized projection operators in Banach spaces: properties andapplications. In Theory and Applications of Nonlinear Operators of Accretive and MonotoneType. Edited by: Kartsatos A. Dekker, New York; 1996:15–50.

    Google Scholar 

  3. 3.

    Li J: The generalized projection operator on reflexive Banach spaces and itsapplications.J. Math. Anal. Appl. 2005, 306: 55–71. 10.1016/j.jmaa.2004.11.007

    MathSciNet  Article  Google Scholar 

  4. 4.

    Zeng LC, Yao JC: Existence theorems for variational inequalities in Banach spaces. J. Optim. Theory Appl. 2006, 36: 483–497.

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Wu K-Q, Huang N-J: Properties of the generalized f-projection operator and itsapplications in Banach spaces.Comput. Math. Appl. 2007, 54: 399–406. 10.1016/j.camwa.2007.01.029

    MathSciNet  Article  Google Scholar 

  6. 6.

    Barbu V, Precupanu T: Convexity and Optimization in Banach Spaces. Springer, New York; 2012.

    Google Scholar 

  7. 7.

    Takahashi W Mathematical Analysis Series 2. In Convex Analysis and Approximation of Fixed Points. Yokohama Publishers, Yokohama; 2000.

    Google Scholar 

  8. 8.

    Bonnans JF, Shapiro A: Perturbation Analysis of Optimization Problems. Springer, New York; 2000.

    Google Scholar 

  9. 9.

    Aoyama K, Kohsaka F, Takahashi W: Three generalizations of firmly nonexpansive mappings: their relations andcontinuity properties.J. Nonlinear Convex Anal. 2009, 10: 131–147.

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Xu H-K: Inequalities in Banach spaces with applications. Nonlinear Anal. 1991, 16: 1127–1138. 10.1016/0362-546X(91)90200-K

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Combettes PL: Inconsistent signal feasibility problems: least-squares solutions in aproduct space.IEEE Trans. Signal Process. 1994, 42: 2955–2966. 10.1109/78.330356

    Article  Google Scholar 

  12. 12.

    Combettes PL, Wajs VR: Signal recovery by proximal forward-backward splitting. Multiscale Model. Simul. 2005, 4: 1168–1200. 10.1137/050626090

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Asplund E, Rockafellar RT: Gradient of convex functions. Trans. Am. Math. Soc. 1969, 139: 443–467.

    MathSciNet  Article  MATH  Google Scholar 

Download references

Acknowledgements

The work was supported by the Scientific Technology Program of the EducationalDepartment Heilongjiang Province (No. 12511161) and the National NaturalSciences Grant (No. 11071052).

Author information

Affiliations

Authors

Corresponding author

Correspondence to Wei-Bo Guan.

Additional information

Competing interests

The author declares that they have no competing interests.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Guan, WB. An iterative method for variational inequality problems. J Inequal Appl 2013, 574 (2013). https://doi.org/10.1186/1029-242X-2013-574

Download citation

Keywords

  • Banach space
  • proximity operator
  • variational inequality
  • iterative method
\