R E S E A R C H
Open Access
An iterative method for variational inequality
problems
Wei-Bo Guan
**Correspondence: [email protected] Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, P.R. China
School of Mathematical Sciences, Harbin Normal University, Harbin, 150025, P.R. China
Abstract
In this paper, we present some properties of generalized proximity operators and propose an iterative method of approximating solutions for a class of generalized variational inequalities and show its convergence in uniformly convex and smooth Banach spaces.
MSC: 47J20; 46B20; 46N10; 47N10; 49J40
Keywords: Banach space; proximity operator; variational inequality; iterative method
1 Introduction
Letf be a lower semi-continuous proper convex function from a Hilbert space H to (–∞, +∞]. The Moreau envelope of the functionf is defined as
ef(x) =inf
y∈H
f(y) + x–y
. (.)
It is well known thatef(x) is a continuous convex function, and for everyx∈H, the
in-fimum in (.) is achieved at a unique pointproxf(x). The operatorproxf fromHto H, i.e.,
proxfx=arg min
y∈H
f(y) + x–y
(.)
thus defined, is called the proximity operator off. Whenf =ιKis the indicator function of
a closed convex setKinH, thenproxf(x) =PK(x) becomes the metric projection operator
onK.
In , Alber extended the metric projection operator to uniformly convex and uni-formly smooth Banach spaces. LetK be a closed convex subset of a uniformly convex and uniformly smooth Banach spaceX, Alber [] introduced the generalized projections
πK:X∗→KandK:X→K,
πK
x∗=arg min
x∈K
x∗– x∗,x +x
and
K(x) =arg min y∈K
Jx– Jx,y+y,
whereJis the duality mapping fromXtoX∗, and studied their properties in detail. In [], Alber presented some applications of the generalized projections to approximately solv-ing variational inequalities in Banach spaces. Recently, Li [] extended the generalized projection operatorπK from uniformly convex and uniformly smooth Banach spaces to
reflexive Banach spaces and studied some properties of the generalized projection opera-tor with applications to solving the variational inequality in Banach spaces. By employing the generalized projection operators, Zeng and Yao [] established some existence results for the variational inequality problem in uniformly convex and uniformly smooth Banach spaces and convergence results for the variational inequality. In [], Wu and Huang further introduced and studied a class of generalizedf-projection operators in Banach spaces. As applications, they proposed an iterative method of approximating solutions for the varia-tional inequality problem: findx¯∈Ksuch that
A¯x,y–x¯ +f(y) –f(¯x)≥, ∀y∈K, (.)
whereKis a nonempty closed convex subset ofX,A:K→X∗is a mapping andf:X→ (–∞, +∞] is a proper convex, lower semicontinuous and positively homogeneous func-tion, via
xn+=πKf
Jxn–αnJ
xn–πKf(Jxn–ρAxn)
, (.)
where
πKfx∗=arg min
x∈K
ρf(x) +x∗– x∗,x +x,
and the parameter sequence{αn}satisfies
≤αn≤,
+∞
n=
αn( –αn) = +∞, ρ> ,
and they proved that{xn}has a subsequence converging to a solution of (.) whenKis
a nonempty compact convex subset of a uniformly convex and uniformly smooth Banach space.
Motivated and inspired by the above works, we continue to study some properties of generalized proximity operators and propose an iterative method of approximating so-lutions for the following generalized variational inequality problem: findx¯∈domf such that
A¯x,y–x¯+f(y) –f(¯x)≥, ∀y∈domf, (.)
2 Preliminaries
LetXbe a reflexive, smooth and strictly convex Banach space with the dual spaceX∗. We denote byxn→xandxnxthe strong and the weak convergence toxof a sequence{xn}
in a Banach spaceX, respectively. Let(X) denote the class of all lower semi-continuous
proper convex functions fromXto (–∞, +∞]. LetB(x,δ) denote the closed ball ofx∈X and radiusδ> . LetS(X) ={x∈X:x= }be the unit sphere.
A Banach spaceXis said to be strictly convex if x+y< for allx,y∈S(X) andx=y. The Banach spaceXis said to be smooth provided
lim
t→
x+ty–x t
exists for eachx,y∈S(X). We recall that uniform convexity ofXmeans that for any given
> , there existsδ> such that for allx,y∈Xwithx ≤,y ≤, andx–y=, the inequality
x+y ≤( –δ)
holds.
A subsetCofXis called boundedly compact if for anyδ> the intersectionC∩B(,δ) is empty or compact.
The duality mappingJ:X⇒X∗is defined by
J(x) =x∗∈X∗|x∗,x =x∗=x, ∀x∈X.
The following basic results concerning the duality mapping are well known [, , ]: () Xis reflexive if and only ifJis surjective;
() Xis strictly convex if and only ifJis injective; () Xis smooth if and only ifJis single-valued;
() ifXis smooth, thenJis norm-to-weak star continuous; () Jis monotone,i.e.,Jx–Jy,x–y ≥,∀x,y∈X;
() ifXis strictly convex and smooth, thenJx–Jy,x–y= ⇒x=y,∀x,y∈X; () if a Banach spaceXis reflexive strictly convex and smooth, then the duality mapping
J∗fromX∗intoXis the inverse ofJ, that is,J–=J∗.
Consider the following envelope function:
efVx∗=inf
x∈X
f(x) + V
x∗,x, (.)
where V(x∗,x) =x∗– x∗,x+x. Since the function (x,x∗)→f(x) +V(x∗,x) is lower semicontinuous convex, one sees thatefV(x∗) is lower semicontinuous and convex by Proposition . in [].
For everyx∗∈X∗, the infimum in (.) is achieved at a unique pointπf(x∗),i.e.,
πf
x∗:=arg min
x∈K
f(x) + V
The operatorπf is called the generalized proximity operator. It can be characterized by
the inclusion
x∗–Jπf
x∗∈∂fπf
x∗, (.)
equivalently,
πf = (J+∂f)–. (.)
From (.), we easily know thatπfis maximal monotone by Theorem .. in []. Observe
that whenρ= ,
πKfx∗=πf+IK
x∗.
If, in addition,domf =K, thenπKf(x∗) =πf(x∗).
Lemma .([]) Let X be a smooth,strictly convex and reflexive Banach space,let{xn}be
a sequence in X,and x∈X.Ifxn–x,Jxn–Jx →,then xnx,JxnJx andxn → x.
Lemma .([]) Let r> be a fixed real number.Then a Banach space X is uniformly
convex if and only if there is a continuous,strictly increasing and convex function g:R+→
R+with g() = such that
λx+ ( –λ)y≤λx+ ( –λ)y–λ( –λ)gx–y, ∀x,y∈Br, ≤λ≤,
where Br={x∈X:x ≤r}.
3 Main results
Proposition . Let f ∈(X).Then the following hold:
(i) πf(x∗) –πf(y∗), (x∗–Jπf(x∗)) – (y∗–Jπf(y∗)) ≥,∀x∗,y∗∈X∗;
(ii) πf is bounded on each nonempty bounded subset ofC⊂X∗;
(iii) if{x∗n}is a sequence inX∗such thatx∗n→x∗,thenπf(x∗n) πf(x∗),
Jπf(x∗n)Jπf(x∗)andπf(x∗n) → πf(x∗);
(iv) ifdomf is a nonempty boundedly compact convex subset,thenπf is weak-to-norm
continuous,that is,ifx∗nx∗,thenπf(x∗n)→πf(x∗).
Proof (i) Takex∗,y∗∈X∗. Then (.) yields
x∗–Jπf
x∗,πf
y∗–πf
x∗ ≤fπf
y∗–fπf
x∗
and
y∗–Jπf
y∗,πf
x∗–πf
y∗ ≤fπf
x∗–fπf
y∗.
Adding these two inequalities, we obtain
πf
x∗–πf
y∗,x∗–Jπf
x∗–y∗–Jπf
(ii) Suppose thatπfis not bounded on some nonempty bounded subset ofC. Then there
exists a bounded sequence{x∗n} ⊂Csuch thatπf(x∗n) → ∞. Fixx∗∈X∗. From (i), we
obtain the following:
πf
x∗n–πf
x∗x∗n–x∗≥πf
x∗n–πf
x∗,x∗n–x∗
≥πf
x∗n–πf
x∗,Jπf
x∗n–Jπf
x∗
=
VJπf
x∗n,πf
x∗+VJπf
x∗,πf
x∗n
≥πf
x∗n–πf
x∗.
So, we havex∗n → ∞. This is a contradiction.
(iii) Let{x∗n}be a sequence inX∗such thatx∗n→x∗. It follows from (ii) that{πf(x∗n)}is
bounded. From (i), we have
≤πf
x∗n–πf
x∗,Jπf
x∗n–Jπf
x∗ ≤πf
x∗n–πf
x∗x∗n–x∗→.
Thus, Lemma . implies that πf(x∗n) πf(x∗), Jπf(xn∗) Jπf(x∗) and πf(x∗n) →
πf(x∗).
(iv) From (.), we know that
x∗n–Jπf
x∗n,y–πf
x∗n ≤f(y) –fπf
x∗n, ∀y∈domf. (.)
Sincex∗nx∗,{x∗n}is bounded. It follows from (ii) that{πf(x∗n)}is bounded. Sincedomf
is boundedly compact, there exists a subsequence{x∗ni}of{x∗n}such that
πf
x∗ni→ ¯x∈domf asi→+∞.
SinceJis norm-to-weak star continuous andf is lower semicontinuous, we obtain that
x∗–Jx,¯ y–x¯ ≤f(y) –f(¯x), ∀y∈domf. (.)
Then, by (.), we havex¯=πf(x∗). Similar to the above arguments, we know thatπf(x∗) is
the unique limit point of{πf(x∗n)}. Hence,πf(x∗n)→πf(x∗).
With the help of the operatorπf, we can show that the envelope functionefVis Gâteaux
differentiable.
Proposition . Let f ∈(X).Then efV is Gâteaux differentiable and∇e
f
V(x∗) =J∗x∗– πf(x∗).
Proof For anyh∈X∗, by definitions ofefVandπf, we have
efV(x∗+th) –efV(x∗) t
=f(πf(x
∗+th)) +
V(πf(x∗+th),x∗+th) –f(πf(x∗)) –
V(πf(x∗),x∗)
≤f(πf(x∗)) +V(πf(x∗),x∗+th) –f(πf(x∗)) –V(πf(x∗),x∗)
t
=
x
∗+th– x
∗–π
f(x∗),th
t .
Since∇(z) = J∗(z), for anyz∈X∗, we get that
lim sup
t→
efV(x∗+th) –efV(x∗) t ≤tlim→
x∗+th –
x∗
t –
πf
x∗,h
=J∗x∗–πf
x∗,h.
On the other hand,
efV(x∗+th) –efV(x∗) t
=f(πf(x
∗+th)) +
V(πf(x∗+th),x∗+th) –f(πf(x∗)) –
V(πf(x∗),x∗)
t
≥f(πf(x∗+th)) +V(πf(x∗+th),x∗+th) –f(πf(x∗+th)) –V(πf(x∗+th),x∗)
t
=
x∗+th –
x∗
t –
πf
x∗+th,h.
By Proposition .(iii), we haveπf(x∗+th) πf(x∗) ast→. Hence, we get that
lim inf
t→
efV(x∗+th) –efV(x∗) t ≥limt→
x∗+th –
x∗
t –
πf
x∗+th,h
=J∗x∗–πf
x∗,h.
This implies that
lim
t→
efV(x∗+th) –efV(x∗)
t =
J∗x∗–πf
x∗,h.
HenceefV is Gâteaux differentiable and∇efV(x∗) =J∗x∗–πf(x∗).
In the following, we propose a modification of the iterative method given in [] and prove that the iterative sequence has a subsequence converging to a solution of (.) when Xis a smooth and uniformly convex Banach space andf is not necessarily positively ho-mogeneous.
By (.), we can easily prove the following result.
Proposition . Let f ∈(X).Then the pointx¯∈domf is a solution of the variational
inequality
Ax,y–x+f(y) –f(x)≥, ∀y∈domf
if and only ifx¯∈domf is a solution of the following inclusion:
The following lemma will be used in proving the convergence of the iterative method for variational inequality problem (.).
Lemma . Let f ∈(X).If f(x)≥for all x∈domf and f() = ,then
πf
x∗≤x∗. (.)
Proof From (.), we know that
x∗–Jπf
x∗,y–πf
x∗ ≤f(y) –fπf
x∗, ∀y∈domf.
Noticing thatf(x)≥ for allx∈domf andf() = , it follows that
x∗–Jπf
x∗, –πf
x∗ ≤–fπf
x∗≤.
Hence,
πf
x∗≤x∗,πf
x∗ ,
and hence
πf
x∗≤x∗.
Proposition . Let X be a smooth and uniformly convex Banach space.Let A:X→X∗
be a norm-to-weak continuous operator.Suppose that f ∈(X)anddomf is nonempty
boundedly compact convex.Suppose that (i) f(x)≥for allx∈domf andf() = ; (ii) for anyx∈domf,
Jx–Ax ≤ x.
Let x∈domf and the sequence{xn}be generated by the following iteration scheme:
xn+= ( –αn)xn+αnπf(Jxn–Axn),
where{αn}satisfies the conditions:
(a) ≤αn≤for alln= , , , . . .;
(b) +n∞=αn( –αn) = +∞.
Then generalized variational inequality(.)has a solutionx¯∈domf,and there exists a subsequence{xni}of{xn}such that xni→ ¯x as i→ ∞.
Proof By (.), we have
πf(Jxn–Axn)≤ Jxn–Axn. (.)
By (.) and condition (ii), we obtain
Then{xn}and{πf(Jxn–Axn)}are bounded. Hence, by Lemma ., there exists a
continu-ous, strictly increasing and convex functiong:R+→R+withg() = such that
xn+=( –αn)xn+αnπf(Jxn–Axn)
≤( –αn)xn+αnπf(Jxn–Axn)
–αn( –αn)gxn–πf(Jxn–Axn). (.)
It follows from (.), (.) and condition (ii) that
xn+≤( –αn)xn+αnxn–αn( –αn)gxn–πf(Jxn–Axn).
That is,
xn+≤ xn–αn( –αn)gxn–πf(Jxn–Axn). (.)
Taking the sum forn= , , , . . . ,min (.), we get
m
n=
αn( –αn)gxn–πf(Jxn–Axn)
≤ x–xm+
≤ x.
Hence,
+∞
n=
αn( –αn)gxn–πf(Jxn–Axn)< +∞. (.)
Due to the condition+n∞=αn( –αn) = +∞, we may assume, without loss of generality,
that
gxn–πf(Jxn–Axn)→ asn→+∞.
Applying the properties ofg, we can deduce that
xn–πf(Jxn–Axn)→ asn→+∞. (.)
Sincedomf is boundedly compact, there exists a subsequence{xni}of{xn}such that
xni→ ¯x∈domf asi→+∞.
SinceAis norm-to-weak continuous andJis norm-to-weak star continuous, we get that
Sinceπf is weak-to-norm continuous by Proposition .(iv),
πf(Jxni–Axni)→πf(J¯x–A¯x) asi→+∞.
Hence, (.) yields
¯
x=πf(Jx¯–Ax).¯
Now it follows from Proposition . thatx¯is a solution of generalized variational
inequal-ity (.).
4 Application
Letf ∈(X) and letg:X→Rbe a convex and Gâteaux differentiable function. Consider
the optimization problem
min
x∈Xf(x) +g(x). (P)
We denote bySol(P) the solution set of problem (P). Despite its simplicity, problem (P) has been shown to cover a wide range of apparently unrelated signal recovery formulations (see [, ]).
Notice that
¯
x∈Sol(P) ⇔ ∈∂f(¯x) +∇g(¯x)
⇔ –∇g(¯x)∈∂f(¯x)
⇔ ∇g(¯x),y–x¯ +f(y) –f(¯x)≥, ∀y∈X.
Note that ifgis convex and Gâteaux differentiable, then∇gis norm-to-weak continuous fromXtoX∗by Corollary . in []. Therefore, as an application of Proposition ., we have the following result.
Proposition . Let X be a smooth and uniformly convex Banach space.Let g:X→R
be convex and Gâteaux differentiable.Suppose that f ∈(X)anddomf is a nonempty
boundedly compact convex subset of X.Suppose that (i) f(x)≥for allx∈domf andf() = ;
(ii) for anyx∈domf,
Jx–∇g(x)≤ x.
Let x∈domf and the sequence{xn}be generated by the following iteration scheme:
xn+= ( –αn)xn+αnπf
Jxn–∇g(xn)
,
where{αn}satisfies the conditions:
(a) ≤αn≤for alln= , , , . . .;
(b) +n∞=αn( –αn) = +∞.
Then problem(P)has a solutionx and there exists a subsequence¯ {xni}of{xn}such that
5 Concluding remark
This paper has improved the iterative method of Wu and Huang [] for solving generalized variational inequality problem (.), several results regarding the generalized proximity operator and its relations with the envelope function are presented. In addition, it is shown that under an appropriate assumption some optimization problem can be transformed into (.) and then the iterative method can be applied.
Competing interests
The author declares that they have no competing interests.
Acknowledgements
The work was supported by the Scientific Technology Program of the Educational Department Heilongjiang Province (No. 12511161) and the National Natural Sciences Grant (No. 11071052).
Received: 30 August 2013 Accepted: 11 November 2013 Published:05 Dec 2013 References
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