Generalized proximal-type methods for weak vector variational inequality problems in Banach spaces

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R E S E A R C H

Open Access

Generalized proximal-type methods for

weak vector variational inequality problems

in Banach spaces

Lin-chang Pu

1

_{, Xue-ying Wang}

2

_{and Zhe Chen}

3*

*_{Correspondence:} [email protected] 3_{Business School, Sichuan} University, Chengdu, 610065, China Full list of author information is available at the end of the article

Abstract

In this paper, we propose a class of generalized proximal-type method by the virtue of Bregman functions to solve weak vector variational inequality problems in Banach spaces. We carry out a convergence analysis on the method and prove the weak convergence of the generated sequence to a solution of the weak vector variational inequality problems under some mild conditions. Our results extend some known results to more general cases.

MSC: 90C29; 65K10; 49M05

Keywords: vector variational inequality problems; proximal-type method; Banach spaces; Bregman functions; weak normal mapping; convergence analysis

1 Introduction

The well-known proximal point algorithm (PPA, for short) is a powerful tool for solving optimization problems and variational inequality problems, which was ﬁrst introduced by Martinet [] and its celebrated progress was attained in the work of Rockafellar []. The classical proximal point algorithm generated a sequence{zk_{} ⊂}_H_{with an initial point}_z

through the following iteration:

zk+= (I+ckT)–zk, (.)

where{ck}is a sequence of positive real numbers bounded away from zero. Rockafellar [] proved that for a maximal monotone operatorT, the sequence{zk_}_{weakly converges to} a zero ofT under some mild conditions. From then on, many works have been devoted to the investigation of the proximal point algorithms, its applications, and generalizations (see [–] and the references therein for scalar-valued optimization problems and varia-tional inequality problems; see [–] for vector-valued optimization problems). The no-tion of a Bregman funcno-tion has its origin in [] and the name was first used in optimizano-tion problems and its related topics by Censor and Lent in []. Bregman function algorithms have been extensively used for various optimization problems and variational inequal-ity problems (e.g.[–] in finite-dimensional spaces, [, ] in infinite-dimensional spaces).

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The concept of vector variational inequality was firstly introduced by Giannessi [] in finite-dimensional spaces. The vector variational inequality problems have found a lot of important applications in multiobjective decision making problems, network equilibrium problems, traffic equilibrium problems, and so on. Because of these significant applica-tions, the study of vector variational inequalities has attracted wide attention. Chen and Yang [] investigated general vector variational inequality problems and vector comple-mentary problems in infinite-dimensional spaces. Chen [] considered vector variational inequality problems with a variable ordering structure. Through the last  years of de-velopment, existence results of solutions, duality theorems, and topological properties of solution sets of several kinds of vector variational inequalities have been derived. One can find a fairly complete review of the main results as regards the vector variational inequal-ities in the monograph []. Very recently, Chen [] constructed a class of vector-valued proximal-type method for solving a weak vector variational inequality problem in finite-dimensional spaces. They generalized some the classical results of Rockafellar [] from the scalar-valued case to the vector-valued case.

An important motivation for making an analysis of the convergence properties of both proximal point algorithms and Bregman function algorithms is related to the mesh in-dependence principle [–]. The mesh independence principle relies on infinite di-mensional convergence results for predicting the convergence properties of the discrete finite-dimensional method. Furthermore it can provide a theoretical foundation for the justification of refinement strategies and help to design the refinement process. As we know, many practical problems in economics and engineering are modeled in infinite-dimensional spaces, thus it is necessary to analyze the convergence property of the algo-rithms in abstract spaces. Motivated by [, ], in this paper we propose a class of gener-alized proximal-type methods by virtue of the Bregman function for solving weak vector variational inequality problems in Banach spaces. We carry out a convergence analysis of the method and prove convergence of the generated sequence to a solution of the weak vector variational inequality problems under some mild conditions.

The paper is organized as follows.

In Section , we present some basic concepts, assumptions and preliminary results. In Section , we introduce the proximal-type method and carry out convergence analysis on the method. In Section , we draw a conclusion and make some remarks.

2 Preliminaries

In this section, we present some basic deﬁnitions and propositions for the proof of our main results.

LetXbe a smooth reflexive Banach space with norm · X, denote byX∗the dual space ofXandY is also a smooth reflexive Banach space ordered by a nontrivial, closed and convex coneCwith nonempty interiorintC, which defines a partial order≤CinY,i.e.,

y≤Cyif and only ify–y∈Cfor anyy,y∈Yand for relation≤intC, given byy≤intCy

if and only ify–y∈intCfor anyy,y∈Y. We denote byY∗the dual space ofY, byC∗

the positive polar cone ofC,i.e.,

C∗=z∈Y∗:z,y ≥,∀z∈C.

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Lemma .[] Let e∈intC be ﬁxed and C₌_{z_∈_C∗_{| z}_,_e_{= }_}_,_{then C}_{is a weak}∗ -compact subset of C∗.

Denote byL(X,Y) the set of all continuous linear operators fromXtoY. Denote byϕ,x the valueϕ(x) ofϕatx∈Xifϕ∈L(X,Y). LetX⊂Xbe nonempty, closed, and convex, consider the weak vector variational inequality problem of ﬁndingx¯∈Xsuch that

(WVVI) T(x¯),x–x¯intC ∀x∈X,

where T:X→L(X,Y) be a mapping. Denote byX¯ the solution set of (VVI). Letλ∈

C_and_λ₍_T_{) :}_X_→_X∗_{, consider the corresponding scalar-valued variational inequality}

problem of ﬁndingx∗∈Xsuch that

(VIPλ)

λ(T)x∗,x–x∗≥ ∀x∈X.

Denote byXλ¯ the solution set of (VIPλ).

Deﬁnition .[] LetX⊂Xbe nonempty, closed, and convex, andF:X→X∗ be a single-valued mapping.

(i) Fis said to bemonotoneonXif, for anyx,x∈X, we have

F(x) –F(x),x–x≥.

(ii) Fis said to bepseudomonotoneonXif, for anyx,x∈X, we have

F(x),x–x

≥ ⇒ F(x),x–x

≥.

Deﬁnition .[] LetX⊂Xbe nonempty, closed, and convex.T:X→L(X,Y) is a mapping, which is said to beC-monotoneonXif, for anyx,x∈X, we have

T(x) –T(x),x–x≥C.

Proposition .[] Let Xand T be deﬁned as in Deﬁnition.,we have the following statements:

(i) T isC-monotone if and only if,for anyλ∈C_,_{the mapping}_λ₍_T_{) :}_X

→X∗is monotone;

(ii) ifTisC-monotone,then for anyλ∈C_,_λ₍_T_{) :}_X_→_X∗_{is pseudomonotone}_.

Deﬁnition .[] LetL⊂L(X,Y) be a nonempty set. The weak and strongC-polar cones ofLare deﬁned, respectively, by

Lw

C :=

x∈X:l,xC,∀l∈L (.)

and

Ls

C :=

x∈X:l,x ≤C,∀l∈L

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Deﬁnition . [] Let K⊂X be nonempty, closed, and convex, let F:K →Y be a vector-valued mapping. A linear operatorV is said to be a strong subgradient ofF at

¯ x∈Kif

F(x) –F(x¯) –V,x–x ≥¯ C ∀x∈K.

A linear operatorVis said to be a weak subgradient ofFatx¯∈Kif

F(x) –F(x¯) –V,x–x¯ intC ∀x∈K.

Denote by∂_CwF(x¯) the set of weak subgradients ofFonKatx¯.

LetK⊂Xbe nonempty, closed, and convex. A vector-valued indicator functionδ(x|K) ofKatxis deﬁned by

δ(x|K) =

⎧ ⎨ ⎩

∈Y, ifx∈K;

+∞C, ifx∈/K.

An important and special case in the theory of weak subgradients is the case thatF(x) =

δ(x|K) becomes a vector-valued indicator function of K. By Deﬁnition ., we obtain

V∈∂_Cwδ(x∗|K) if and only if

V,x–x∗intC ∀x∈K. (.)

Deﬁnition . A setVNw

K(x∗)⊂L(X,Y) is said to be a weak normality operator set toK atx∗, if for everyV∈VNw

K(x∗), the inequality (.) holds.

Clearly,VNw

K(x∗) =∂Cwδ(x∗|K). As for the scalar-valued case, from [] we know that

v∗∈∂δK(x∗) =NK(x∗) if and only if

v∗,x–x∗≤ ∀x∈K, (.)

whereδK(x) is the scalar-valued indicator function ofK. The inequality (.) means that

v∗is normal toKatx∗.

Deﬁnition . LetVN_Kw(·) :X⇒L(X,Y) be a set-valued mapping, which is said to be a weak normal mapping forK, if for anyy∈K,V∈VN_Kw(y) such that

V,x–yintC ∀x∈K. (.)

VNs

K(·) is said to be a strong normal mapping forK, if for anyy∈K,V∈VNKs(y) such that

V,x–y ≤C ∀x∈K. (.)

As in [], the normal mapping forKis a set-valued mapping, which is deﬁned, for any

y∈K,v∈NK(y), by

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Next we will introduce the deﬁnition and some basic results as regards themaximal mono-tone mapping.

Deﬁnition .[] Let a set-valued mapG:X⊂X⇒X∗be given, it is said to be mono-toneif

z–¯z,w–w ≥¯ 

for allzand¯zinX, allwinG(z) andw¯ inG(z¯). It is said to bemaximal monotoneif, in addition, the graph

gph(G) =(z,w)∈X×X∗|w∈G(z)

is not properly contained in the graph of any other monotone operator fromXtoX∗.

Deﬁnition .[] An operatorA:X→X∗is said to be regular if for anyu∈R(A) and

for anyy∈dom(A),

sup (z,v∈grh(A))

v–u,y–z<∞. (.)

Proposition .[] The subdiﬀerential of a proper lower semicontinuous convex func-tionφis a regular operator.

Lemma .[] Let K be a nonempty,closed,and convex subset of X.

(a) IfT:X⇒X∗is the normal mapping to K andT:X→X∗is a single-valued monotone operator such thatintK∩dom(T)=∅andTis continuous on K,then T+Tis a maximal monotone operator.

(b) IfT,T:X⇒X∗are maximal monotone operators,andint(domT)∩domT=∅,

thenT+Tis a maximal monotone operator.

Lemma .[] Assume that X is a reﬂexive Banach space and J its normalized duality mapping.Let T,T:X⇒X∗be maximal monotone operators such that

(a) Tis regular.

(b) domT∩domT=∅andR(T) =X∗. (c) T+Tis maximal monotone.

Then we have R(T+T) =X∗.

Proposition .[] Any maximal monotone operator S:X⇒X∗is demiclosed,if the conditions below hold:

• If{xk}converges weakly to x and{wk∈S(xk)}converges strongly to w,thenw∈S(x). • If{xk}converges strongly to x and{wk∈S(xk)}converges weakly to w,thenw∈S(x).

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distance with respect tof is the functionBf(z,x) :D×intD→Rdeﬁned by

Bf(z,x) :=f(z) –f(x) –

∇f(x),z–x, (.)

where∇f(x) is the diﬀerential off deﬁned inintD. Consider the following set of assumptions onf.

A: The right level sets ofBf(y,·),

Sy,α=

z∈intD:Bf(y,z)≤α

, (.)

are bounded for allα≥and for ally∈D.

A: If{xk} ⊂intKand{yk} ⊂intKare bounded sequences such thatlimk→+∞xk–yk=, then

lim

k→+∞

∇f(xk) –∇f(yk)= . (.)

A: Assume{xk} ⊂K is bounded,{yk} ⊂intK is such thatw-limk→+∞yk=y. Further

as-sume thatw-limk→+∞Bf(xk,yk) = , then

w- lim

k→+∞x

k₌_y_.

A: For everyy∈X∗, there existsx∈intDsuch that∇f(x) =y.

We say thatf is said to be a Bregman function if it satisﬁes A-A, andf is said to be

a coercive Bregman function if it satisﬁes A-A. Without loss of generality, we assume X⊂domf.

Definition .[] LetX⊂X be a closed and convex set withintX=∅,f :X→ Ra convex function which is Gâteaux differentiable inintX. We define the modulus of convexity off,vf:intX×R+→R+by

vf(z,t) :=inf

Bf(x,z) :x–z=t

, (.)

whereBf(x,z) is the Bregman distance with respect tof.

(a) The functionf is said to be totally convex inintXif and only ifvf(z,t) > for all

z∈intXandt> .

(b) The functionf is said to be uniformly totally convex if for any bounded set

K⊂intXandt> , we have

inf

x∈Kvf(x,t) > . (.)

Next we will introduce some fundamental deﬁnitions of the asymptotic analysis in inﬁ-nite spaces.

Deﬁnition .[] Let K be a nonempty set inX. Then the asymptotic cone of the set

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the sequence{xk} ⊂K, namely

K∞=

d∈X∃tk→+∞, andxk∈K,w- lim k→+∞

xk

tk =d

. (.)

In the case thatKis convex and closed, then, for anyx∈K,

K∞={d∈X|x+td∈K,∀t> }. (.)

3 Main results

Proposition . Let X⊂X be nonempty,closed, and convex,and VNXw(·)be a weak

normal mapping for X.For any x∗∈X,andϕ∈VN_Xw_(x∗),there exists aλ∈Csuch that

λ(ϕ)∈NX(x∗).

Proof By the deﬁnition of the weak normal mapping, we know that

ϕ,x–x∗intC ∀x∈X.

It follows that

ϕ,x–x∗∈Y\intC ∀x∈X,

and

ϕ,X–x∗⊂Y\intC.

That is,

ϕ,X–x∗∩intC=∅.

By virtue of the convexity ofXand the Hahn-Banach theorem, there exists aλ∗∈C∗\{} such that

λ∗(ϕ),x–x∗≤ ∀x∈X.

Sinceλ∗> , one obtains

λ∗

λ∗(ϕ),x–x

∗_≤_ _∀x_∈_X_.

Clearly, we have λ_λ∗∗∈C. Without loss of generality, letλ= λ ∗

λ∗, one has

λ(ϕ),x–x∗≤ ∀x∈X.

That is,λ(ϕ)∈NX(x∗). The proof is complete.

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Step () takex∈X;

Step () given anyxk∈X, ifxk∈ ¯X, then the algorithm stops; otherwise go to Step (); Step () takexk∈ ¯/X; we deﬁnexk+by the following expression:

∈T(xk+)λk+VNXw(xk+)λk+εk

∇f(xk+) –∇f(xk)

, (.)

where the sequenceλk∈C,εk∈(,ε],ε> , andVNXw(·) is the weak normal mapping to

X,f is a coercive Bregman function; go to Step ().

Remark . The algorithm GPTM is actually a kind of exact proximal point algorithm, where the sequence{λk} ∈Cis called a sequence of scalarization parameters, a bounded exogenous sequence of positive real numbers{εk}is called a sequence of regularization parameters. For everyxk∈ ¯/X, we try to ﬁnd axk+such that ∈X∗belongs to the inclusion

(.).

Next we will show the main results of this paper.

Theorem . Let X⊂X be nonempty,closed,and convex,T:X→L(X,Y)be contin-uous and C-monotone on X,if domT∩intX=∅.The sequence{xk} generated by the

method(GPTM)is well deﬁned.

Proof Letx∈Xbe an initial point and suppose that the method (GPTM) reaches Step

(k). We then show that the next iteratexk+does exist. By the assumptions,T(·) is

con-tinuous andC-monotone onX. By virtue of Proposition ., we see thatλ(T) is mono-tone and continuous onXfor anyλ∈C_{. From Proposition ., there exists a}_λ¯ _∈_C

such that the mapping VNw

X(·)λ¯ is a normal mapping onX. Thus, by the assumption

domT∩intX=∅and the statement (a) of Lemma ., one sees that, for anyx∈X, the

mapping (VNX(x) +T(x))λ¯ is maximal monotone. Without loss of generality, letλk=λ¯.

Once again by the statement (b) of Lemma ., we see thatT(·)λk+VNXw(·)λk+εk∇f(·) is

maximal monotone. Takingεk∇f asTin Lemma ., it is easy to check that all assump-tions are satisﬁed. It follows thatR(T(·)λk+VNXw(·)λk+εk∇f(·)) =X

∗_{. Then we see that,}

for any givenεk∇f(xk)∈X∗, there exists axk+such that

εk∇f(xk)∈T(xk+)λk+VNXw(xk+)λk+εk∇f(xk+). (.)

The proof is complete.

Theorem . Let the same assumptions as those in Theorem.hold.Further suppose that X_∞∩[T(X)]w

C ={}andX is nonempty and bounded¯ .Then the sequence{xk}generated

by the method(GPTM)is bounded and

∞

k=

Bf(xk+,xk) <∞. (.)

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λk∈Candϕk+ ∈VNXw(xk+) such thatλk(ϕk+)∈NX(xk+). From the inclusion (.),

one has

 =T(xk+)λk+λk(ϕk+) +εk

∇f(xk+) –∇f(xk)

.

By the fact thatλk(ϕk+)∈NX(xk+), we obtain

λk(ϕk+),x–xk+

≤ ∀x∈X. (.)

It follows that

T(xk+)λk+εk

∇f(xk+) –∇f(xk)

,x–xk+

≥ ∀x∈X. (.)

On the other hand, from the assumption thatX¯ is nonempty and bounded, without loss of generality, letx∗∈ ¯X. By virtue of Theorem . in [], we know thatx∗is also a solution of the problem (VIPλk), where

(VIPλk)

Tx∗λk,x–x∗≥ ∀x∈X.

It follows that

Tx∗λk,x∗–xk+

≤.

By theC-monotonicity ofT, one has

T(xk+)λk,x∗–xk+

≤. (.)

Combining (.) with (.), we obtain

εk

∇f(xk+) –∇f(xk)

,x∗–xk+

≥.

By Property . of [] (‘three points property’), we have

Bf(x,xk+) =Bf(x,xk) –Bf(xk+,xk) +

∇f(xk) –∇f(xk+),x–xk+

(.)

for anyx∈X. Hence from the inclusion (.), we know there existsλk(ϕk+)∈VNX(xk+)

such that



εk

T(xk+)λk+λk(ϕk+)

=∇f(xk) –∇f(xk+). (.)

Combining (.) with (.), we obtain

Bf(x,xk+) =Bf(x,xk) –Bf(xk+,xk) + 

εk

T(xk+)λk+λk(ϕk+)

. (.)

Sinceλk(ϕk+)∈NX(xk+), we have

λk(ϕk+),x∗–xk+

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It follows that

Bf

x∗,xk+

≤Bf

x∗,xk

–Bf(xk+,xk) –



εk

T(xk+)λk,xk+–x∗

. (.)

Clearly, we have _ε

kT(xk+)λk,xk+–x

∗_≥_{. It follows that}

Bf

x∗,xk+

≤Bf

x∗,xk

–Bf(xk+,xk). (.)

Hence we see that the sequenceBf(x∗,xk) is decreasing ask→ ∞. Takingα=Bf(x∗,x), we obtain the sequence{xk} ⊂Sx∗,α, which is a bounded set, by the assumptionAof the

Bregman function.

As for the sequenceBf(x∗,xk), it converges to a limitl∗. From (.), we have

Bf(xk+,xk)≤Bf

x∗,xk

–Bf

x∗,xk+

. (.)

Summing up the above inequality, we obtain

∞

k=

Bf(xk+,xk)≤

∞ k= Bf x∗,xk

–Bf

x∗,xk+

=Bf

x∗,x–l∗<∞. (.)

Theorem . Let the same assumptions as those in Theorem.hold.If we assume that the Bregman function f is uniformly totally convex,then any weak accumulation point of {xk}is a solution of the problem(WVVI).

Proof If there existsk≥ such thatxk+p=xk, ∀p≥, then it is clear thatxk is the

unique weak cluster point of{xk}and it is also a solution of problem (WVVI). Suppose that the algorithm does not terminate ﬁnitely. Then, by Theorem ., we see that {xk} is bounded and it has some weak cluster points. Next we show that all of weak cluster points are solutions of problem (WVVI). Letxˆbe a weak cluster point of{xk}and{xkj}be a subsequence of{xk}, which weakly converges toxˆ. From the inequality (.), we know that

lim

j→+∞Bf(xkj+,xkj) = . (.)

Hence by the assumptionAoff, we obtain

w- lim

j→+∞xkj+=xˆ. (.)

Next we will show thatlimj→+∞xkj+–xkj= . Suppose the contrary, without loss of generality we assume there exists aβ>  such that

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for allj≥. LetDbe a bounded set which contains the sequence{xkj}. We have

Bf(xkj+,xkj)≥inf

Bf(z,xkj) :xkj+–xkj=z–xkj

=vf

xkj,xkj+–xkj

. (.)

From the assumption (.) and Property . in [], we obtain

vf

xkj,xkj+–xkj

>vf(xkj,β)≥inf

x∈Dvf(x,β). (.)

Since we assumef is uniformly totally convex, it follows thatinf_x_∈_Dvf(x,β) > . One has

Bf(xkj+,xkj)≥inf

x∈Dvf(x,β) > . (.)

Clearly, there is a contradiction between (.) and (.), hence we have

lim

j→+∞xkj+–xkj= .

By the assumptionAof the Bregman functionf, one has

lim

j→+∞

∇f(xkj+) –∇f(xkj)

= . (.)

By the inclusion (.), one sees that there existλkj∈Candϕkj+∈VNXw(xkj+), such that

 =T(xkj+)λkj+λkj(ϕkj+) +εkj

∇f(xkj+) –∇f(xkj)

.

It follows that

lim

j→+∞



εkj

T(xkj+)λkj+λkj(ϕkj+)

= . (.)

{λkj} ⊂C, which is a weak∗-compact set. Without loss of generality, we assume the se-quencew∗–limj→+∞λkj=λ¯. From Proposition ., for everyjwe have

λkj(ϕkj+)∈NX(xkj+).

By the demiclosedness of the graphT+NX(see Proposition .), we conclude that

∈T(x¯)λ¯+VN_Xw_(x¯)λ¯.

Meanwhile, from Proposition ., we know that there exists aω¯∈VNw

X(x¯)λ¯. By the

deﬁ-nition ofNX(x¯), we know that

¯ω,x–x ≤¯  ∀x∈X.

That is,

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Thus

T(x¯),x–x¯∈/–intC ∀x∈X. (.)

We conclude thatx¯is a solution of problem (WVVI). The proof is complete.

Theorem . Let the same assumptions as those in Theorem.hold.If∇f is a weak to weak continuous mapping,then the whole sequence{xk}converges weakly to a solution of

problem(WVVI).

Proof In this part we will showx¯ is the only solution of (WVVI). Suppose the contrary, assume that bothxˆandx¯are weak cluster points of{xn}

w- lim

j→+∞xkj=xˆ, w-i→lim+∞xki=x¯

andxˆandx¯are also solutions of (WVVI). From the proof of Theorem ., we see that the sequencesBf(xˆ,xk) andBf(x¯,xk) are bounded and letlandlbe their limits, then

lim

k→+∞

Bf(xˆ,xk) –Bf(x¯,xk)

=l–l=f(xˆ) –f(x¯) + lim

k→+∞

∇f(xk),x¯–xˆ. (.)

Letlimk→+∞∇f(xk),x¯–xˆ =l. Takingk=kjin (.) and by virtue of the weak to weak continuity of∇f(·) we obtain

l=∇f(xˆ),x¯–xˆ. (.)

Once again takingk=kiin (.) and by virtue of the weak to weak continuity of∇f(·) we obtain

l=∇f(x¯),x¯–xˆ. (.)

Combining the equality (.) with the equality (.), one has

∇f(x¯) –∇f(xˆ),x¯–xˆ= . (.)

From the strict convexity off, we havex¯=xˆ, which meansx¯is the only solution of (WVVI).

4 Conclusions

In this paper, we proposed a class of generalized proximal-type method by virtue of the Bregman distance functions to solve weak vector variational inequality problems in Ba-nach spaces. We carried out a convergence analysis on the method and proved weak con-vergence of the generated sequence to a solution of the weak vector variational inequality problems under some mild conditions. Our results extended some well-known results to more general cases.

Competing interests

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Authors’ contributions

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

Author details

1_{School of Economics and Management, Chongqing Normal University, Chongqing, 401331, China.}2_{School of} Mathematics, Chongqing Normal University, Chongqqing, 401331, China.3_{Business School, Sichuan University, Chengdu,} 610065, China.

Acknowledgements

This work is partially supported by the National Science Foundation of China (No. 11001289, 71471122). The authors thank the anonymous referees for their helpful suggestions, which have contributed to a major improvement of the manuscript.

Received: 21 August 2015 Accepted: 12 October 2015 References

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