An iterative method for split hierarchical monotone variational inclusions

R E S E A R C H A R T I C L E

Open Access

An iterative method for split hierarchical

monotone variational inclusions

Qamrul Hasan Ansari

_{and Aisha Rehan}

*_{Correspondence:}

[email protected]

1_{Department of Mathematics,}

Aligarh Muslim University, Aligarh, 202002, India

2_{Department of Mathematics &}

Statistics, King Fahd University of Petroleum & Minerals, Dhahran, Saudi Arabia

Abstract

In this paper, we introduce a split hierarchical monotone variational inclusion problem (SHMVIP) which includes split variational inequality problems, split monotone variational inclusion problems, split hierarchical variational inequality problems,etc., as special cases. An iterative algorithm is proposed to compute the approximate solutions of an SHMVIP. The weak convergence of the sequence generated by the proposed algorithm is studied. We present an example to illustrate our algorithm and convergence result.

MSC: Primary 49J40; 47J20; secondary 49J52; 49M05

Keywords: split hierarchical monotone variational inclusions; fixed point problems; iterative method; maximal monotone set-valued mappings; resolvent operators; convergence analysis

1 Introduction

LetHandHbe real Hilbert spaces,C⊆HandQ⊆Hbe nonempty, closed, and convex sets,A:H→H be a bounded linear operator, andf :H→H andg:H→H be two given operators. Recently, Censoret al.[] introduced the followingsplit variational inequality problem(SVIP):

Findx∗∈Csuch thatfx∗,x–x∗≥, for allx∈C, (.)

and such that

y∗:=Ax∗∈Qsolvesgy∗,y–y∗≥, for ally∈Q. (.)

Letdenote the solution set of the SVIP, that is,

=xsolves (.) :Axsolves (.).

Iff andgare convex and differentiable, then the SVIP is equivalent to the followingsplit minimization problem:

minf(x), subject tox∈C, (.)

and such that

y∗:=Ax∗∈Qsolvesming(y), subject toy∈Q. (.)

For further details on the equivalence between a variational inequality and an optimization problem, we refer to []. The SVIP also contains the split feasibility problem (SFP) [] as a special case. For further details of the SFP, we refer to [, ] and the references therein.

If the sets C andQare the set of fixed points of the operatorsT :H→H andS:

H→H, respectively, then the SVIP is called asplit hierarchical variational inequality

problem(SHVIP). It is introduced and studied by Ansariet al.[]. Several special cases of a SHVIP, namely, the split convex minimization problem, the split variational inequality problem defined over the solution set of monotone variational inclusion problem, the split variational inequality problem defined over the solution set of equilibrium problem, are also considered in [].

LetB:H⇒HandB:H⇒Hbe set-valued mappings with nonempty values, and letf :H→Handg:H→Hbe mappings. Then, inspired by the work in [], Moudafi [] introduced the followingsplit monotone variational inclusion problem(SMVIP):

Findx∗∈Hsuch that ∈f

x∗+B

x∗, (.)

and such that

y∗:=Ax∗∈Hsolves ∈g

y∗+B

y∗. (.)

Letdenote the solution set of SMVIP, that is,

=xsolves (.) :Axsolves (.).

To solve the SMVIP, Moudafi [] proposed the following iterative method: Letλ>  and

xbe the initial guess. Compute

xn+=U

xn+γA∗(T–I)Axn

, for alln∈N, (.)

whereγ ∈(, /L) withLbeing the spectral radius of the operatorA∗A,U=JB

λ (I–λf), T=JB

λ (I–λg), andJ

B

λ andJ

B

λ are the resolvents ofBandB, respectively, with parameter

λ(see []). He obtained the following weak convergence result for iterative method (.).

Theorem .([], Theorem .) Given a bounded linear operator A:H →H.Let f :

H→H and g:H→H beα andαinverse strongly monotone operators on Hand

H,respectively,and B,Bbe two maximal monotone operators,and setα:=min{α,α}.

Consider the operator U:=JB

λ (I–λf),T:=J

B

λ (I–λg)withλ∈(, α).Then the sequence

{xn}generated by (.)converges weakly to an element x∗∈,provided that =∅and

γ ∈(, /L).

work in [] and [], in this paper, we introduce the followingsplit hierarchical monotone variational inclusion problem(SHMVIP):

Findx∗∈Fix(T) such that ∈fx∗+B

x∗, (.)

and such that

y∗:=Ax∗∈Fix(S) solves ∈gy∗+B

y∗. (.)

We denote bythe set of solutions of the SHMVIP, that is,

=xsolves (.) :Axsolves (.).

We propose an iterative algorithm to compute the approximate solutions of the SHMVIP. The weak convergence of the sequence generated by the proposed algorithm is studied. An example is presented to illustrate the proposed algorithm and result.

2 Preliminaries

LetH be a real Hilbert space whose inner product and norm are denoted by ·,·and · , respectively. We denote byxn→x(respectively,xnx) the strong (respectively, weak) convergence of the sequence{xn}tox. LetT:H→Hbe an operator whose range is denoted byR(T). The set of all fixed points ofT is denoted byFix(T), that is,Fix(T) = {x∈H:x=Tx}.

Definition . An operatorT:H→His said to be: (a) nonexpansiveifTx–Ty ≤ x–y, for allx,y∈H; (b) strongly nonexpansive[, , ] ifTis nonexpansive and

lim

n→∞(xn–yn) – (Txn–Tyn)= ,

whenever{xn}and{yn}are bounded sequences inHand

limn→∞(xn–yn–Txn–Tyn) = ;

(c) averaged nonexpansive[] if it can be written as

T= ( –α)I+αS,

whereα∈(, ),Iis the identity operator onH, andS:H→His a nonexpansive mapping;

(d) firmly nonexpansiveifTx–Ty_{≤ x–y,Tx–Ty, for allx,y∈H;}

(e) α-inverse strongly monotone(α-ism) if there exists a constantα> such that

Tx–Ty,x–y ≥αTx–Ty, for allx,y∈H.

Example . LetT: [–, ]→Rbe defined byTx= –x, for allx∈[–, ]. ThenTis non-expansive but not strongly nonnon-expansive.

Indeed, letxn=  andyn= , for alln. Then{xn}and{yn}are bounded sequences. Also,

lim

n→∞ (xn–yn) – (Txn–Tyn) =nlim→∞| + |= = . Thus,Tis not strongly nonexpansive.

The following result will be used in the sequel.

Proposition .[] Let T:H→H be an operator. (i) IfT isν-ism,then forγ > ,γTis _γν-ism.

(ii) Tis averaged if and only if the complementI–Tisν-ism for someν>_.Indeed,for α∈(, ),Tisα-averaged if and only ifI–T is 

α-ism.

(iii) The composite of finitely many averaged mappings is averaged.

Letϕ:H→Hbe a given single-valuedα-inverse strongly monotone operator andλ∈

(, α). Then (I–λϕ) is averaged. Indeed, sinceϕisα-inverse strongly monotone,λϕis

α

λ-ism. Thus,I–λϕis averaged as α λ>

 .

Recall that a Banach spaceXis said to satisfyOpial’s condition[] if whenever{xn}is a sequence inXwhich converges weakly toxasn→ ∞, then

lim sup

n→∞

xn–x<lim sup n→∞

xn–y, for ally∈X,y=x.

It is well known that every Hilbert space satisfies Opial’s condition.

Lemma .(Demiclosedness principle) [], Lemma  Let C be a nonempty,closed,and convex subset of a real Hilbert space H and T:C→C be a nonexpansive operator with

Fix(T)=∅.If the sequence{xn} ⊆C converges weakly to x and the sequence{(I–T)xn}

converges strongly to y,then(I–T)x=y.In particular,if y= ,then x∈Fix(T).

LetB:H⇒Hbe a set-valued mapping. The domain, range, and inverse ofBare denoted by

D(B) =x∈H:B(x)=∅, R(B) = x∈D(B)

B(x), and B–(y) =x∈H:y∈B(x),

respectively.

Definition . The set-valued mappingB:H⇒Hwith nonempty values is said to be: (a) monotoneif

u–v,x–y ≥, for allu∈Bx,v∈By;

(b) maximal monotoneif it is monotone and the graph

G(B) =(x,u)∈H×H:u∈Bx

3 Algorithm and convergence result

It is well known that when the set-valued mappingB:H⇒His maximal monotone, then for eachx∈Handλ> , there is a uniquez∈Hsuch thatx∈(I+λB)z[, ]. In this case, the operatorJB

λ:= (I+λB)–is calledresolventofBwith parameterλ. It is well known

thatJB

λ is a single-valued and firmly nonexpansive mapping.

Indeed, for any givenu∈H, letx,y∈JλB(u). Thenx,y∈(I+λB)–(u), and thusu–x∈λBx

andu–y∈λBy. The monotonicity ofλBimplies that

u–x– (u–y),x–y≥.

This implies thatx–y ≤, and thusx=y. Hence,JB

λis single-valued.

Next we show thatJλBis firmly nonexpansive mapping. For anyx,y∈H, letJλB(x) = (I+

λB)–_{(x) andJ}B

λ(y) = (I+λB)–(y). This implies thatx∈(I+λB)(JλB(x)) andy∈(I+λB)(JλB(y)).

It follows that _λ(x– (JB

λ(x)))∈B(JλB(x)) and



λ(y– (J

B

λ(y)))∈B(JλB(y)). The monotonicity of Bimplies that

JλB(x) –JλB(y),



λ

x–JλB(x)

–

λ

y–JλB(y)

≥,

that is,

J_λB(x) –J_λB(y),x–y≥J_λB(x) –J_λB(y).

Thus,JλBis firmly nonexpansive.

Letφ:H→Hbe a given single-valued operator. Then

∈φx∗+Bx∗ ⇔ x∗∈FixJλB(I–λφ)

x∗. (.)

Indeed, letx∗∈Fix(JB

λ(I–λφ)(x∗)). Thenx∗=JλB(I–λφ)(x∗). It follows that

x∗= (I+λB)–(I–λφ)x∗ ⇔ x∗–λφx∗∈(I+λB)x∗ ⇔ ∈φx∗+Bx∗.

SinceJλBis firmly nonexpansive, and therefore, averaged. It is well known that the

compo-sition of averaged mapping is averaged, thusJB

λ(I–λϕ) is averaged. Since every averaged

mapping is strongly nonexpansive [], it follows thatJB

λ(I–λϕ) is also strongly

nonex-pansive.

LetB:H⇒HandB:H⇒Hbe set-valued mappings with nonempty values, and letf :H→Handg:H→Hbe mappings. LetT:H→HandS:H→Hbe opera-tors such thatFix(T)=∅andFix(S)=∅. LetU:=JB

λ (I–λf) andV:=J

B

λ (I–λg). With the

help of (.), (.), and (.) can be rewritten as

findx∗∈Fix(T) such thatx∗∈FixJB

λ (I–λf)

, (.)

and such that

y∗:=Ax∗∈Fix(S) solvesy∗∈FixJB

λ (I–λg)

Now we propose the following algorithm to compute the approximate solutions of the SHMVIP.

Algorithm . Initialization: Letλ>  and take arbitraryx∈H. Iterative step: For a given currentxn∈H, compute

xn+=TU

xn+γA∗(SV–I)Axn

, (.)

whereγ ∈(,_A).

Last step: Updaten:=n+ .

Next we prove the weak convergence of the sequence generated by Algorithm ..

Theorem . Let A:H→Hbe a bounded linear operator,f :H→Hbe anα-inverse

strongly monotone operator,T :H→H be a strongly nonexpansive operator such that Fix(T)=∅,g:H→H be anα-inverse strongly monotone operator,S:H→H be a

nonexpansive operator such thatFix(S)=∅,andα:=min{α,α}.Consider the operator

U:=JB

λ (I–λf)and V :=J

B

λ (I–λg)withλ∈(, α),and B:H⇒Hand B:H⇒H

are two maximal monotone set-valued mappings with nonempty values.Then the sequence

{xn}generated by Algorithm.converges weakly to an element x∗∈,provided=∅.

Proof Letp∈. ThenTp=p,Up=p,SAp=Ap, andVAp=Ap. Letyn:=xn+γA∗(SV–

I)Axnand consider

yn–p=xn+γA∗(SV–I)Axn–p 

=xn–p+γA∗(SV–I)Axn 

+ γxn–p,A∗(SV–I)Axn

≤ xn–p+γA(SV–I)Axn 

+ γxn–p,A∗(SV–I)Axn

. (.)

Consider the third term of inequality (.), we have

xn–p,A∗(SV–I)Axn

=Axn–Ap, (SV–I)Axn

=(SV–I)Axn–Ap+Axn– (SV–I)Axn, (SV–I)Axn

= SVAxn–Ap,SVAxn–Axn–(SV–I)Axn 

=

SVAxn–Ap ₊

SVAxn–Axn _–

Axn–Ap

_{–(SV–I)Ax} n

=

SVAxn–SVAp ₊

SVAxn–Axn _–

Axn–Ap

_{–(SV–I)Ax} n



≤

Axn–Ap ₊

SVAxn–Axn _–

Axn–Ap

_{–(SV–I)Ax} n



= –

(SV–I)Axn 

Combining (.) and (.), we obtain

yn–p≤ xn–p+γA(I–SV)Axn 

–γ(SV–I)Axn 

=xn–p–γ

 –γA(SV–I)Axn. (.)

Sinceγ∈(,_A), we haveγ( –γA) > , and thus,

yn–p ≤ xn–p. (.)

From the above inequality (.), we have

xn+–p=TUyn–TUp ≤ Uyn–Up ≤ yn–p

≤ xn–p–γ

 –γA(SV–I)Axn 

≤ xn–p. (.)

This shows thatxn+–p ≤ xn–pand this implies that{xn–p}∞n= is a monotonic decreasing sequence, also{xn}is a bounded sequence, see [], Theorem .. and, hence,

lim_n→∞xn–p exists. Taking the limit at both sides in (.), and noticing thatγ( –

γA_{) > , we have}

lim

n→∞(SV–I)Axn= , (.)

and sinceyn:=xn+γA∗(SV–I)Axn, we haveyn–xn=γA(SV–I)Axnthus in view of (.) we have

lim

n→∞yn–xn= . (.)

Since{xn}is a bounded sequence and it has a weakly convergent subsequence say,xnix∗, [] or with the help of Opial’s condition [], we can see that xnx∗. Thus, we have

AxnAx∗. SinceSV is nonexpansive, from (.) and the closedness ofSV–Iat  we obtainSVAx∗=Ax∗. Next we show thatVAx∗=Ax∗. We have

SVAxn–Ap–Axn–Ap ≤ SVAxn–Axn.

Taking the limit at both sides and by using (.), we obtain

lim

n→∞ SVAxn–Ap–Axn–Ap = , _lim

n→∞

SVAxn–Ap–Axn–Ap = ,

lim

n→∞

SVAxn–Ap–Axn–Ap

= .

SinceSAp=ApandVAp=Ap, by the nonexpansiveness ofSandV, we have

SVAxn–Ap ≤ VAxn–Ap ≤ Axn–Ap,

and therefore

SVAxn–Ap–Axn–Ap ≤ VAxn–Ap–Axn–Ap ≤.

Thus, we have

lim

n→∞

SVAxn–Ap–Axn–Ap

≤ lim

n→∞

VAxn–Ap–Axn–Ap

≤. (.)

From (.) and (.), we obtain

lim

n→∞

VAxn–Ap–Axn–Ap

= . (.)

SinceVis averaged nonexpansive and every averaged nonexpansive map is strongly non-expansive, we see thatV is strongly nonexpansive. Since {Ap} and{Axn} are bounded sequences, by the definition of strong nonexpansiveness ofV, we have

lim

n→∞VAxn–Axn= .

SinceVis nonexpansive, by the demiclosedness principle, we have

VAx∗=Ax∗.

Now, we show thatTx∗=x∗andUx∗=x∗. By using the nonexpansiveness ofTandU, in view of (.) and (.), we have

≤ Uyn–p–TUyn–p ≤ yn–p–TUyn–p ≤ xn–p–xn+–p.

This implies that

lim

n→∞

Uyn–p–TUyn–p

= . (.)

From (.), we see that{yn}is a bounded sequence and thus{Uyn}is bounded and since {p}is a constant sequence thus{p}is also bounded and sinceTis strongly nonexpansive, we have

lim

n→∞Uyn–TUyn= . (.)

In view of (.) and (.), by using the nonexpansiveness ofTU, we have

≤ yn–p–TUyn–p ≤ xn–p–xn+–p.

It follows that

lim

n→∞

yn–p–TUyn–p

By using the nonexpansiveness ofT andU, we have

TUyn–p ≤ Uyn–p ≤ yn–p,

and therefore

TUyn–p–yn–p ≤ Uyn–p–yn–p ≤.

Thus, from (.), we have

lim

n→∞

Uyn–p–yn–p

= . (.)

From (.) we see that{yn}is a bounded sequence and{p}being a constant sequence, also bounded, by the strong nonexpansiveness ofU, we have

lim

n→∞Uyn–yn= . (.)

Next consider, for allf ∈H,

f(yn) –f

x∗=f(yn) –f(xn) +f(xn) –f

x∗

≤f(yn) –f(xn)+f(xn) –f

x∗

≤ fyn–xn+f(xn) –f

x∗.

Sincexnx∗and from (.), we havelimn→∞f(yn) –f(x∗)= , thusynx∗. Thus, in view of (.) and by applying the demiclosedness principle, we haveUx∗=x∗. Again, sinceynx∗, in view of (.), we haveUynx∗. Thus, again in view of (.) and by applying the demiclosedness principle, we haveTx∗=x∗.

Now, we illustrate Algorithm . and Theorem . by the following example.

Example . LetH=H=H=RandB:H⇒Hbe defined by

B(x) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

{}, ifx> , [, ], ifx= , {}, ifx< .

(.)

Then, as shown in [],Bis a set-valued maximal monotone mapping. We define the map-pingsA,f,h,T,S:H→Hby

Ax=x

, for allx∈H,

fx=hx=x

, for allx∈H,

Tx=x

Table 1 The values of{xn}with initial guessx1= 10,x1= 15, andx1= 20

n 1 2 3 4 5 6 7 8 9 10

xn 10 0.7037 0.0495 0.0035 0.0002 0.0000 0.0000 0.0000 0.0000 0.0000 xn 15 1.0556 0.0743 0.0052 0.0004 0.0000 0.0000 0.0000 0.0000 0.0000 xn 20 1.4074 0.0990 0.0070 0.0005 0.0000 0.0000 0.0000 0.0000 0.0000

and

Sx=x

 , for allx∈H,

respectively. It is easy to show that Ais a bounded linear operator,f andhare _-ism,

Tis firmly nonexpansive, and thusTis strongly nonexpansive [] andSis nonexpansive. LetB(x) =B(x) =Bx. ThenBandBare maximal monotone set-valued mappings. Let

JB

λ (x) =J

B

λ (x) =

x

 be the resolvent operator. The values of{xn}with different values ofn are reported in the Table . All codes were written in Matlab R.

Table  shows that the sequence{xn}converges to , which is the required solution.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Received: 10 March 2015 Accepted: 26 June 2015

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