A projection method for bilevel variational inequalities

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R E S E A R C H

Open Access

A projection method for bilevel variational

inequalities

Tran TH Anh

1*

_{, Le B Long}

2

_{and Tran V Anh}

2

*_{Correspondence:}

[email protected]

1_{Department of Mathematics,}

Hai Phong University, Haiphong, Vietnam

Full list of author information is available at the end of the article

Abstract

A ﬁxed point iteration algorithm is introduced to solve bilevel monotone variational inequalities. The algorithm uses simple projection sequences. Strong convergence of the iteration sequences generated by the algorithm to the solution is guaranteed under some assumptions in a real Hilbert space.

MSC: 65K10; 90C25

Keywords: bilevel variational inequalities; pseudomonotonicity; strongly monotone; Lipschitz continuous; global convergence

1 Introduction

LetCbe a nonempty closed convex subset of a real Hilbert spaceHwith the inner product

·,·and the norm·. We denote weak convergence and strong convergence by notations

and→, respectively. Thebilevel variational inequalities, shortly (BVI), are formulated as follows:

Findx∗∈Sol(G,C) such thatFx∗,x–x∗≥, ∀x∈Sol(G,C),

whereG:H→H,Sol(G,C) denotes the set of all solutions of the variational inequalities:

Findy∗∈Csuch thatGy∗,y–y∗≥, ∀y∈C,

andF:C→H. We denote the solution set of problem (BVI) by.

Bilevel variational inequalities are special classes ofquasivariational inequalities(see [–]) and of equilibrium with equilibrium constraints considered in []. However, they cover some classes of mathematical programs with equilibrium constraints (see []), bilevel minimization problems (see []), variational inequalities (see [–]), minimum-norm problems of the solution set of variational inequalities (see [, ]), bilevel convex programming models (see []) and bilevel linear programming in [].

Suppose thatf :H→R. It is well known in convex programming that iff is convex and diﬀerentiable onSol(G,C), thenx∗is a solution to

minf(x) :x∈Sol(G,C)

if and only ifx∗ is the solution to the bilevel variational inequalities VI(∇f,Sol(G,C)), where∇f is the gradient off. Then the bilevel variational inequalities (BVI) are written in

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a form of mathematical programs with equilibrium constraints as follows: ⎧

⎨ ⎩

minf(x),

x∈ {y∗:G(y∗),z–y∗ ≥,∀z∈C}.

Iff,gare two convex and diﬀerentiable functions, then problem (BVI) (whereF:=∇f and

G:=∇g) becomes the following bilevel minimization problem (see []): ⎧

⎨ ⎩

minf(x),

x∈argmin{g(x) :x∈C}.

In a special caseF(x) =xfor allx∈C, problem (BVI) becomes the minimum-norm prob-lems of the solution set of variational inequalities as follows:

Findx∗∈Csuch thatx∗=PrSol(G,C)(),

where PrSol(G,C)() is the projection of  ontoSol(G,C). A typical example is the least-squares solution to the constrained linear inverse problem in []. For solving this problem under the assumption that the subsetC⊆His nonempty closed convex,G:C→Hisα -inverse strongly monotone andSol(G,C)=∅, Yaoet al.in [] introduced the following extended extragradient method:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

x_∈_C_,

yk=PrC(xk–λG(xk) –αkxk),

xk+₌_Pr

C[xk–λG(xk) +μ(yk–xk)], ∀k≥.

They showed that under certain conditions over parameters, the sequence{xk_}_converges

strongly toxˆ=PrSol(G,C)().

Recently, Anhet al.in [] introduced an extragradient algorithm for solving problem (BVI) in the Euclidean spaceRn_{. Roughly speaking the algorithm consists of two loops. At}

each iterationkof the outer loop, they applied the extragradient method for the lower vari-ational inequality problem. Then, starting from the obtained iterate in the outer loop, they computed ank-solution of problemVI(G,C). The convergence of the algorithm crucially

depends on the starting pointsx_{and the parameters chosen in advance. More precisely,} they presented the following scheme

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

Computeyk_:=_Pr

C(xk–αkG(xk)) andzk:=PrC(xk–αkG(yk)).

Inner iterationsj= , , . . . . Compute ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

xk,_:=_zk_–_λF₍_zk_),

yk,j_:=_Pr

C(xk,j–δjG(xk,j)),

xk,j+_:=_αjxk,₊_βjxk,j₊_γj_Pr

C(xk,j–δjG(yk,j)).

Ifxk,j+–PrSol(G,C)(xk,) ≤ ¯k, then sethk:=xk,j+and go to Step .

Otherwise, increasejby .

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Under assumptions that F is strongly monotone and Lipschitz continuous, Gis pseu-domonotone and Lipschitz continuous onC, the sequences of parameters were chosen appropriately. They showed that two iterative sequences{xk_}_and_{_zk_}_{converged to the}

same pointx∗which is a solution of problem (BVI). However, at each iteration of the outer loop, the scheme requires computing an approximation solution to a variational inequality problem.

There exist some other solution methods for bilevel variational inequalities when the cost operator has some monotonicity (see [, –]). In all of these methods, solving auxiliary variational inequalities is required. In order to avoid this requirement, we com-bine the projected gradient method in [] for solving variational inequalities and the ﬁxed point property thatx∗is a solution to problemVI(F,C) if and only if it is a ﬁxed point of the mappingPr_C(x–λF(x)), whereλ> . Then, the strong convergence of proposed sequences is considered in a real Hilbert space.

In this paper, we are interested in ﬁnding a solution to bilevel variational inequalities (BVI), where the operatorsFandGsatisfy the following usual conditions:

(A) Gisη-inverse strongly monotone onHandFisβ-strongly monotone onC. (A) FisL-Lipschitz continuous onC.

(A) The solution setof problem (BVI) is nonempty.

The purpose of this paper is to propose an algorithm for directly solving bilevel pseu-domonotone variational inequalities by using the projected gradient method and ﬁxed point techniques.

The rest of this paper is divided into two sections. In Section , we recall some properties for monotonicity, the metric projection onto a closed convex set and introduce in detail a new algorithm for solving problem (BVI). The third section is devoted to the convergence analysis for the algorithm.

2 Preliminaries

We list some well-known deﬁnitions and the projection under the Euclidean norm which will be used in our analysis.

Deﬁnition . LetCbe a nonempty closed convex subset inH. We denote the projection

onCbyPrC(·),i.e.,

Pr_C(x) =argminy–x:y∈C, ∀x∈H.

The operatorϕ:C→His said to be

(i) γ-strongly monotoneonCif for eachx,y∈C,

ϕ(x) –ϕ(y),x–y≥γx–y;

(ii) η-inverse strongly monotoneonCif for eachx,y∈C,

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(iii) Lipschitz continuouswith constantL> (shortlyL-Lipschitz continuous) onCif for eachx,y∈C,

ϕ(x) –ϕ(y)≤Lx–y.

Ifϕ:C→CandL= , thenϕis callednonexpansiveonC.

We know that the projectionPrC(·) has the following well-known basic properties.

Property .

(a) PrC(x) –PrC(y) ≤ x–y,∀x,y∈H.

(b) x–PrC(x),y–PrC(x) ≤,∀y∈C,x∈H.

(c) Pr_C(x) –Pr_C(y)_≤_x_–_y_–_Pr

C(x) –x+y–PrC(y),∀x,y∈H.

To prove the main theorem of this paper, we need the following lemma.

Lemma .(see []) Let A:H→Hbeβ-strongly monotone and L-Lipschitz continuous,

λ∈(, ]andμ∈(,β_L).Then the mapping T(x) :=x–λμA(x)for all x∈Hsatisﬁes the inequality

T(x) –T(y)≤( –λτ)x–y, ∀x,y∈H,

whereτ =  – –μ(β–μL₎_∈_{(, ].}

Lemma .(see []) LetHbe a real Hilbert space,C be a nonempty closed and convex subset ofHand S:C→Hbe a nonexpansive mapping.Then I–S(I is the identity operator onH)is demiclosed at y∈H,i.e.,for any sequence(xk₎_{in C such that x}k_x_¯_∈_{D and}

(I–S)(xk₎_→_y_,_{we have}₍_I_–_S₎₍_x_¯_{) =}_y_.

Lemma .(see []) Let{an}be a sequence of nonnegative real numbers such that

an+≤( –γn)an+δn, ∀n≥,

where{γn} ⊂(, )and{δn}is a sequence inRsuch that

(a) ∞_n_=γn=∞,

(b) lim sup_n_→∞δn γn≤or

∞

n=|δnγn|< +∞. Thenlim_n_→∞an= .

Now we are in a position to describe an algorithm for problem (BVI). The proposed algorithm can be considered as a combination of the projected gradient and ﬁxed point methods. Roughly speaking the algorithm consists of two steps. First, we use the well-known projected gradient method for solving the variational inequalitiesVI(G,C) :xk+₌

PrC(xk–λG(xk)) (k= , , . . .), whereλ>  andx∈C. The method generates a sequence

(xk_{) converging strongly to the unique solution of problem}_VI₍_G_,_C_{) under assumptions}

thatGisL-Lipschitz continuous andα-strongly monotone onCwith the step-sizeλ∈

(,α

L). Next, we use the Banach contraction-mapping ﬁxed-point principle for ﬁnding

the unique ﬁxed point of the contraction-mappingTλ=I–λμF, whereFisβ-strongly monotone andL-Lipschitz continuous,Iis the identity mapping,μ∈(,β_L) andλ∈(, ].

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Algorithm .(Projection algorithm for solving (BVI))

Step . Choosex∈C,k= , a positive sequence{αk},λ,μsuch that ⎧

⎪ ⎪ ⎨ ⎪ ⎪ ⎩

 <αk≤min{,τ}, τ=  –

 –μ(β–μL_),

limk→∞αk= , limk→∞|αk+–

 αk|= ,

∞

k=αk=∞,  <λ≤η,  <μ<β_L.

(.)

Step . Compute ⎧ ⎨ ⎩

yk_:=_Pr

C(xk–λG(xk)),

xk+₌_yk_–_μα kF(yk).

Updatek:=k+ , and go to Step .

Note that in the caseF(x) =  for allx∈C, Algorithm . becomes the projected gradient algorithm as follows:

xk+:=PrC

xk–λGxk.

3 Convergence results

In this section, we state and prove our main results.

Theorem . Let C be a nonempty closed convex subset of a real Hilbert spaceH.Let two mappings F:C→Hand G:H→Hsatisfy assumptions(A)-(A).Then the sequences

(xk₎_and₍_yk₎_{in Algorithm}_._{converge strongly to the same point x}∗_∈_.

Proof For conditions (.), we consider the mappingSk:H→Hdeﬁned by

Sk(x) :=PrC

x–λG(x)–μαkFPrC

x–λG(x), ∀x∈H.

Using Property .(a),Gisη-inverse strongly monotone and conditions (.), for each

x,y∈H, we have

PrC

x–λG(x)–PrC

y–λG(y)≤x–λG(x) –y+λG(y)

=x–y+λG(x) –G(y)

– λx–y,G(x) –G(y)

≤ x–y+λ(λ– η)G(x) –G(y)

≤ x–y. (.)

Combining this and Lemma ., we get

Sk(x) –Sk(y)=PrC

x–λG(x)–μαkF

Pr_Cx–λG(x)–Pr_Cy–λG(y)

+μαkF

Pr_Cy–λG(y)

≤( –αkτ)PrC

x–λG(x)–PrC

y–λG(y)

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whereτ:= – –μ(β–μL_{). Thus,}_Sk_{is a contraction on}H. By the Banach contraction principle, there is the unique ﬁxed pointξksuch thatSk(ξk) =ξk. For eachxˆ∈Sol(G,C), set

ˆ C:=

x∈H:x–x ≤ˆ μF(xˆ)

τ

.

Due to this and Property .(a), we get that the mappingSkPrCˆ is contractive onH. Then

there exists the unique pointzksuch thatSk[PrCˆ(zk)] =zk. Set¯zk=PrCˆ(zk). It follows from

(.) that

_zk_–_x_ˆ₌_Sk_z_¯k_–_x_ˆ

≤Skz¯k–Sk(xˆ)+Sk(xˆ) –xˆ

=Skz¯k–Sk(xˆ)+Sk(xˆ) –PrC

ˆ

x–αkG(xˆ)

≤( –αkτ)¯zk–xˆ+μαkF

Pr_Cxˆ–αkG(xˆ)

≤( –αkτ)μF(xˆ)

τ +μαkF(xˆ)

= μF(xˆ)

τ .

Thus,zk_{∈ ˆ}_C_,_Sk_[_Pr

ˆ

C(zk)] =Sk(zk) =zkand henceξk=zk∈ ˆC. Therefore, there exists a

sub-sequence (ξki_{) of the sequence (}_ξk_{) such that}_ξki_ξ¯_{. Combining this and the assumption}

limk→∞αk= , we get

Pr_Cξki_–_λG_ξki_–_ξki ₌ _Pr

C

ξki_–_λG_ξki_–_S

ki

ξki

= μαkiF

PrC

ξki_–_λG_ξki

→ asi→ ∞. (.)

It follows from (.) that the mapping PrC(· –αkG(·)) is nonexpansive on H. Using

Lemma ., (.) andξki_ξ¯_{, we obtain}_Pr

C(ξ¯–λG(ξ¯)) =ξ¯, which impliesξ¯∈Sol(G,C).

Now we will prove thatlimj→∞ξkj=x∗∈Sol(BVI). Setz¯k₌_Pr

C(ξk–λG(ξk)),v∗= (μF–I)(x∗) andvk= (μF–I)(z¯k), whereIis the identity

mapping. SinceSkj(ξ

kj_{) =}_ξkj_and_x∗₌_Pr

C(x∗–λG(x∗)), we have

( –αkj)

ξkj_–¯_zkj₊_α

kj

ξkj₊_vkj_{= }

and

( –αkj)

I–PrC

·–λG(·)x∗+αkj

x∗+v∗=αkj

x∗+v∗.

Then

–αkj

x∗+v∗,ξkj_–_x∗_{= ( –}_α

kj)

ξkj_–_x∗_–_z¯kj_–_x∗_,_ξkj_–_x∗

+αkj

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By the Schwarz inequality, we have

ξkj_–_x∗_–_z¯kj_–_x∗_,_ξkj_–_x∗≥_ξkj_–_x∗_–_z¯kj_–_x∗_ξkj_–_x∗

≥ξkj_–_x∗_–_ξkj_–_x∗

=  (.)

and

ξkj_–_x∗₊_vkj_–_v∗_,_ξkj_–_x∗≥_ξkj_–_x∗_–_vkj_–_v∗_ξkj_–_x∗

≥ξkj_–_x∗_{– ( –}_τ₎_ξkj_–_x∗

=τξkj_–_x∗_. _(.)

Combining (.), (.) and (.), we get

–τξkj_–_x∗≥_x∗₊_v∗_,_ξkj_–_x∗

=μFx∗,ξkj_–_x∗

=μFx∗,ξkj_–_ξ¯₊_μ_F_x∗_,_ξ¯_–_x∗

≥μFx∗,ξkj_–_ξ¯_.

Then we have

τξkj_–_x∗≤_μ_F_x∗_,_ξ¯_–_ξkj_.

Letj→ ∞, and hence the sequence{ξkj}_{converges strongly to}_x∗_{. This implies that the} sequence{ξk}also converges strongly tox∗.

Otherwise, by using (.), we have

xk–ξk≤xk–ξk–+ξk––ξk

=Sk–

xk––Sk–

ξk–+ξk––ξk

≤( –αk–τ)xk––ξk–+ξk––ξk. (.)

Moreover, by Lemma ., we have

ξk––ξk=Sk–

ξk––Sk

ξk

=( –αk)z¯k–αkvk– ( –αk–)¯zk–+αk–vk–

=( –αk)

¯

zk–z¯k––αk

vk–vk–+ (αk––αk)

¯

zk–+vk–

≤( –αk)z¯k–z¯k–+αkvk–vk–+|αk––αk|μF

¯ zk–

≤( –αk)z¯k–z¯k–+αk

 –μβ–μL_ξk_–_ξk–

+|αk––αk|μF

¯ zk–

≤( –αk)ξk–ξk–+αk

 –μβ–μL_ξk_–_ξk–

+|αk––αk|μF

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This implies that

αkτξk––ξk≤ |αk––αk|μF

¯ zk–

and hence

ξk–ξk–≤μ|αk––αk|F(z¯

k–₎

αkτ .

So, we have

xk–ξk≤( –αk–τ)xk––ξk–+

μ|αk––αk|F(z¯k–)

αkτ

.

Let

δk:=

μ|αk–αk+|F(z¯k)

αkαk+τ

, k≥.

Then

xk–ξk≤( –αk–τ)xk––ξk–+αk–τ δk–, ∀k≥.

Since{F(z¯k₎_}_{is bounded,}_F₍_z_¯k₎_≤_K_{for all}_k_≥_{, we have}

lim

k→∞δk=klim→∞

μ|αk–αk+|F(z¯k)

αkαk+τ ≤

μK τ _klim_→∞

_αk_+ – 

αk

= .

Applying Lemma .,limk→∞xk–ξk= . Combining this and the fact that the sequence {ξk}converges strongly tox∗, the sequence{xk_} _{also converges strongly to the unique}

solution to problem (BVI).

Now we consider the special caseF(x) =xfor allx∈H. It is easy to see thatFis Lipschitz continuous with constantL=  and strongly monotone with constantβ=  onH. Prob-lem (BVI) becomes the minimum-norm probProb-lems of the solution set of the variational inequalities.

Corollary . Let C be a nonempty closed convex subset of a real Hilbert spaceH.Let G:H→Hbeη-inverse strongly monotone.The iteration sequence(xk₎_{is deﬁned by}

⎧ ⎨ ⎩

yk_:=_Pr

C(xk–λG(xk)),

xk+_{= ( –}_μα

k)yk.

The parameters satisfy the following: ⎧

⎪ ⎪ ⎨ ⎪ ⎪ ⎩

 <αk≤min{,τ}, τ=  –| –μ|,

limk→∞αk= , limk→∞|αk+–

 αk|= ,

∞

k=αk=∞,  <λ≤η,  <μ< .

Then the sequences{xk_}_and_{yk_}_{converge strongly to the same point}_x_ˆ₌_Pr

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Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The main idea of this paper was proposed by TTHA. The revision is made by LBL and TVA. All authors read and approved the ﬁnal manuscript.

Author details

1_{Department of Mathematics, Hai Phong University, Haiphong, Vietnam.}2_{Faculty of Basic Science 1, PTIT, Hanoi, Vietnam.}

Acknowledgements

The authors are very grateful to the anonymous referees for their really helpful and constructive comments that helped us very much in improving the paper.

Received: 19 October 2013 Accepted: 8 May 2014 Published:22 May 2014

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