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Split hierarchical variational inequality problems and related problems
Fixed Point Theory and Applications volume 2014, Article number: 208 (2014)
The main objective of this paper is to introduce a split hierarchical variational inequality problem. Several related problems are also considered. We propose an iterative method for finding a solution of our problem. The weak convergence of the sequence generated by the proposed method is studied.
MSC:47H09, 47H10, 47J25, 49J40, 65K10.
In 1994, Censor and Elfving  introduced the following split feasibility problem (in short, SFP): find a point
where C and Q are nonempty closed convex subsets of and , respectively, and A is an matrix. They proposed an algorithm to find the solution of SFP. Their algorithm did not become popular, since it concerns the complicated matrix inverse computations and, subsequently, is considered in the case when . Based on these observations, Byrne  applied the forward-backward method, a type of projected gradient method, presenting the so-called CQ-iterative procedure, which is defined by
where an initial , , and and denote the metric projections onto C and Q, respectively. The convergence result of the sequence to a solution of the considered split feasibility problem was presented. Further, Byrne also proposed an application to dynamic emission tomographic image reconstruction. A few years later, Censor et al. [3, 4] proposed an incredible application of the split feasibility problem to the inverse problem of intensity-modulated radiation therapy treatment planning. Recently, Xu  considered SFP in the setting of infinite-dimensional Hilbert spaces and established the following CQ-algorithm: Let and be real Hilbert spaces and be a bounded linear operator. For a given , generated a sequence by the following iterative scheme:
where , and is the adjoint operator of A. He also proved the weak convergence of the sequence produced by the above procedure to a solution of the SFP. The details of the CQ-algorithm for SFP problem are given in . A comprehensive literature, survey, and references on SFP can be found in .
In 2009, Censor and Segal  presented an important form of the split feasibility problem called the split common fixed point problem, which is to find a point
where T and S are some nonlinear operators on and , respectively, and A is a real matrix. Based on the properties of operators T and S, called cutter or directed operators, they presented the following algorithm for solving the split common fixed point problem:
where an initial , . They also presented a convergence result for this algorithm. Moudafi  studied the split common fixed point problem in the context of the demicontractive operators T and S in the setting of infinite-dimensional Hilbert spaces. He established the weak convergence of the sequence generated by his scheme to a solution of the split common fixed point problem.
On the other hand, the theory of variational inequalities is well known and well developed because of its applications in different areas of science, social science, engineering, and management. There are several monographs on variational inequalities, but we mention here a few [10–12]. Let C be a nonempty closed convex subset of a real Hilbert space H and be an operator. The variational inequality problem defined by C and f is to find such that
() , for all .
Another problem closely related to is known as the Minty variational inequality problem: find such that
() , for all .
The trivial unlikeness of two proposed problems is the linearity of variational inequalities. In fact, the Minty variational inequality is linear but the variational inequality is not. However, under the (hemi)continuity and monotonicity of f, the solution sets of these problems are the same (see [, Lemma 1]).
If the constrained set C in variational inequality formulations is a set of fixed points of an operator, then the variational inequality problem is known as hierarchical variational inequality problem.
Let be a nonlinear operator with the set of fixed points and be an operator. The hierarchical variational inequality problem is to find such that
A closely related problem to hierarchical variational inequality problem is the hierarchical Minty variational inequality problem: find such that
The goal of this paper is to introduce a split-type problem, by combining a split fixed point problem and a hierarchical variational inequality problem. The considered problem can be applied to solve many existing problems. We present an iterative procedure for finding a solution of the proposed problem and show that under some suitable assumptions, the sequence generated by our algorithm converges weakly to a solution of the considered problem.
The rest of this paper is divided into four sections. In Section 2, we recall and state preliminaries on numerous nonlinear operators and their useful properties. Section 3 we divide in two subsections, that is, first, we present a split problem, called the split hierarchical Minty variational inequality problem. We subsequently propose an algorithm for solving our problem and establish a convergence result under some assumptions. Second, we present the split hierarchical variational inequality problem. In the last section, we investigate some related problems, where we can apply the considered problem.
Let H be a real Hilbert space whose inner product and norm are denoted by and , respectively. The strong convergence and weak convergence of a sequence to are denoted by and , respectively. Let be an operator. We denote by the range of T, and by the set of all fixed points of T, that is, . The operator T is said to be nonexpansive if for all , ; strongly nonexpansive [20, 21] if T is nonexpansive and for all bounded sequences , in H, the condition implies ; averaged nonexpansive if for all , holds for a nonexpansive operator and ; firmly nonexpansive if is nonexpansive, or equivalently for all , ; cutter  if for all and all ; monotone if , for all ; α-inverse strongly monotone (or α-cocoercive) if there exists a positive real number α such that , for all .
Let be a set-valued operator, we define a graph of B by and an inverse operator of B, denoted , by . A set-valued operator is called monotone if for all , and such that . A monotone operator B is said to be maximal monotone if there exists no other monotone operator such that its graph properly contains the graph of B. For a maximal monotone operator B, we know that for each and a positive real number σ, there is a unique such that . We define the resolvent of B with parameter σ by . It is well known that the resolvent is a single-valued and firmly nonexpansive operator.
Remark 2.1 It can be seen that the class of averaged nonexpansive operators is a proper subclass of the class of strongly nonexpansive operators. Since any firmly nonexpansive operator is an averaged nonexpansive operator, it is clear that a class of firmly nonexpansive operators is contained in the class of strongly nonexpansive operators. Further, a firmly nonexpansive operator with fixed point is cutter. However, a nonexpansive cutter operator need not to be firmly nonexpansive; for further details, see .
The following well-known lemma is due to Opial .
Lemma 2.2 (Demiclosedness principle) [, Lemma 2]
Let C be a nonempty closed convex subset of a real Hilbert space H and be a nonexpansive operator. If the sequence converges weakly to an element and the sequence converges strongly to 0, then x is a fixed point of the operator T.
To prove the main theorem of this paper, we need the following lemma.
Lemma 2.3 [, Section 2.2.1, Lemma 2]
Assume that and are nonnegative real sequences such that . If , then exists.
The following lemma can be immediately obtained by the properties of an inner product.
Lemma 2.4 Let H be a real Hilbert space H. Then, for all ,
3 Split hierarchical variational inequality problems and convergent results
In this section we introduce split hierarchical variational inequality problems and discuss some related problems. Further, we propose an iterative method for finding a solution of the hierarchical variational inequality problem and prove the convergence result for the sequence generated by the proposed iterative method.
3.1 Split hierarchical Minty variational inequality problem
Let and be two real Hilbert spaces, be operators such that , and be operators such that . Let be an operator with . The split hierarchical Minty variational inequality problem (in short, SHMVIP) is to find such that
and such that satisfies
The solution set of SHMVIP (3.1)-(3.2) is denoted by Γ, that is,
We note that the problems (3.1) and (3.2) are nothing but hierarchical Minty variational inequality problems. If f and h are zero operators, that is, , then SHMVIP (3.1) and (3.2) reduces to the split fixed point problem (1.2). Moreover, SHMVIP (3.1) and (3.2) can be applied to several existing split-type problems, which we will discuss in Section 4.
Inspired by the iterative scheme (1.3) for solving split common fixed point problem (1.2) and the existing algorithms, a generalization of the projected gradient method, for solving the hierarchical variational inequality problem of Yamada  and Iiduka , we now present an iterative algorithm for solving HMVIP (3.1)-(3.2).
Algorithm 3.1 Initialization: Choose . Take arbitrary .
Iterative Step: For a given current iterate , compute
where and define by
Rest of the section, unless otherwise specified, we assume that T is a strongly nonexpansive operator on with and S is a strongly nonexpansive cutter operator on with , f (respectively, h) is a monotone and continuous operator on (respectively, ) and is a bounded linear operator with .
Remark 3.2 (i) The Algorithm 3.1 can be applied to the iterative scheme (1.3) by setting the operators .
(ii) The iterative Algorithm 3.1 extends and develops the algorithms in [, Algorithm 6.1] and [, Algorithm (8)] in many aspects. Indeed, the metric projections and in [, Algorithm 6.1] and the resolvent operators and in [, Algorithm (8)] are firmly nonexpansive operators which are special cases of the assumption on the operators T and S. Second, its involves control sequences and , while in [, Algorithm 6.1] and [, Algorithm (8)] they are constant. Furthermore, we assume , while in [, Algorithm 6.1] and [, Algorithm (8)], it was assumed to be in , which clearly is a more restrictive assumption.
(iii) By setting an operator A to be zero operator, the iterative Algorithm 3.1 relates to existing iterative schemes for solving the hierarchical variational inequality problem in, for example, [, Algorithm 3.1] and [, Algorithm 5].
The following theorem provides the weak convergence of the sequence generated by the Algorithm 3.1 to an element of Γ.
Theorem 3.3 Let a sequence be generated by Algorithm 3.1 with , and let and be sequences such that and . If , then the following statements hold.
If there exists a natural number such that
then, for all and , we have
If the sequence is bounded, then exists for all .
If the sequence is bounded, then , , and .
If , and , then the sequence converges weakly to an element of Γ.
Proof (i) Let be given. Then
for all . On the other hand, since S is a cutter operator, we have with
for all . By using (3.4), the inequality (3.3) becomes
for all , as required. Furthermore, since , we observe that , and hence
(ii) Let . For , we have . Since , by using Lemma 2.3, we obtain the result that exists.
(iii) We first show that .
Let for all . Note that is a bounded sequence, since for all we know that and is a bounded sequence. By using the nonexpansiveness of T and (3.5), we have
Subsequently, by the existence of and the fact that , it follows that
By the strong nonexpansiveness of T and the boundedness of , we have
On the other hand, by (i), we observe that
which is equivalent to
for all . Taking the limit as , we get
which implies, by the definition of , that
These together imply that
We now show that .
Observe that as . Using this one together with (3.8), (3.9) and the nonexpansiveness of T, we obtain
Next, we show that .
Set for all . For every , we observe that
Thus, the boundedness of and (3.7) yield
and hence, by the fact that the sequence is bounded and S is a strongly nonexpansive mapping, we have
We note that as . Also, we observe that
for all . By using (3.11) and taking the limit as , we get
Hence, we get
Since is a bounded sequence, there exist a subsequence of and such that . By (iii) and the demiclosed principle of the nonexpansive operator T, we obtain . We claim that solves (3.1).
In fact, for , we have
where . Since , , and , by (3.8), we have for all which means that solves (3.1).
Next, by together with (iii) and the demiclosedness of the nonexpansive operator S, we know that . We show that such solves (3.2).
Since , we may assume that for all . From (3.6), for all , we have
where . This implies that, for ,
where . Subsequently, since , , and , we have
For , we compute
where . Using this together with and (3.14), we obtain for all . Thus, solves (3.2), and therefore, .
Finally, it remains to show that . By the boundedness of , it suffices to show that there is no subsequence of such that and .
Indeed, if this is not true, the well-known Opial theorem would imply
which leads to a contradiction. Therefore, the sequence converges weakly to a point . □
3.2 Split hierarchical variational inequality problems
We consider another kind of split hierarchical variational inequality problem (in short, SHVIP) in which we consider a variational inequality formulation instead of the Minty variational inequality. More precisely, we consider the following split hierarchical variational inequality problem: find a point such that
such that and it satisfies
The solution set of the SHVIP is denoted by Ω. Since and are nonempty closed convex and f and h are monotone and continuous, by the Minty lemma [, Lemma 1], SHVIP (3.16)-(3.17) and SHMVIP (3.1)-(3.2) are equivalent. Hence, Algorithm 3.1 and Theorem 3.3 are also applicable for SHVIP (3.16)-(3.17).
When a closed convex subset of a Hilbert space and a closed convex subset of a Hilbert space , then SHVIP (3.16)-(3.17) is considered and studied by Censor et al. .
Remark 3.4 It is worth to note that, in the context of SHVIP (3.16)-(3.17), if we assume either the finite dimensional settings of and or the compactness of spaces and , then the monotonicity assumptions of operators f and h can be omitted.
In fact, as in the statement and the proof of Theorem 3.3, since we know that , we also have . Recalling the inequality (3.13), we have, for ,
which is equivalent to
where . Since we know that and f is continuous, by approaching j to infinity, we obtain , for all , which means that solves (3.16). Similarly, from the inequality (3.15), we note that, for ,
which is equivalent to
where . Since and h is continuous, we also get , for all , which means that solves (3.17), and subsequently, .
Remark 3.5 The strong nonexpansiveness of operators T and S in Theorem 3.3 can be applied to the cases when the operators T and S are not only firmly nonexpansive, but with relaxation of being firmly nonexpansive and averaged nonexpansive but also strictly nonexpansive, that is, or for all x, y; regarding the compactness of the spaces and , for more details, see [, Remark 2.3.3].
4 Some related problems
In this section, we present some split-type problems which are special cases of SHVIP (3.1) and (3.2) and can be solved by using Algorithm 3.1.
4.1 A split convex minimization problem
Let and be convex continuously differentiable functions and be a bounded linear operator such that . Consider the following split convex minimization problem (in short, SCMP): find
and such that
We know that the SCMP (4.1)-(4.2) can be formulated as the following split hierarchical variational inequality problem: find a point
and such that the point
where ∇ϕ and ∇φ denote the gradient of ϕ and φ, respectively. Since ∇ϕ and ∇φ are monotone [, Proposition 5.3] and continuous, we can apply Algorithm 3.1 to obtain the solution of SCMP (4.1)-(4.2), and Theorem 3.3 will provide the convergence of the sequence to a solution of SCMP (4.1)-(4.2). Furthermore, in the context of finite-dimensional cases, we can remove the convexity of ϕ and φ.
4.2 A split variational inequality problem over the solution set of monotone variational inclusion problem
Let and be two real Hilbert spaces, , be , (respectively, )-inverse strongly monotone operators, , are set-valued maximal monotone operators. Let us consider the following monotone variational inclusion problem (in short, MVIP): find
We denote the solution set of MVIP by .
For , , we know that the operator is an averaged nonexpansive operator and
where represents the resolvent of with parameter σ.
We consider the following split variational inequality problem: find such that
and such that satisfies
By using these facts and adding the assumption that is cutter, we can apply Algorithm 3.1 and Theorem 3.3 to obtain the solution of SVIP (4.3)-(4.4).
4.3 A split variational inequality problem over the solution set of equilibrium problem
Let and be real Hilbert spaces, and be nonempty closed convex sets, and be bifunctions. The equilibrium problem defined by C and ϕ is the problem of finding such that
and we denote the solution set of equilibrium problem of C and ϕ by .
Recall that Blum and Oettli  showed that the bifunction ϕ satisfies the following conditions:
(A1) for all ;
(A2) ϕ is monotone, that is, for all ;
(A3) for all ,
(A4) for all , is a convex and lower semicontinuous function,
and let and , then the set
Moreover, from , we know that
is firmly nonexpansive;
We now consider the following split variational inequality problem: find such that
and such that satisfies
Similarly, by assuming that conditions (A1)-(A4) in the context of the equilibrium problem defined by Q and φ hold, we see that, for all and , the set
is a nonempty set. Moreover, we also see that is single-valued and firmly nonexpansive; and .
By using these facts, we obtain the result that the problem (4.5)-(4.6) forms a special case of the SHVIP (3.1)-(3.2).
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The authors would like to thank the anonymous referees for their careful reading and suggestions, which allowed us to improve the first version of this paper. This research is partially support by the Center of Excellence in Mathematics the Commission on Higher Education, Thailand. QH Ansari was partially supported by a KFUPM Funded Research Project No. IN121037. N Nimana was supported by the Thailand Research Fund through the Royal Golden Jubilee PhD Program (Grant No. PHD/0079/2554).
The authors declare that they have no competing interests.
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
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Cite this article
Ansari, Q.H., Nimana, N. & Petrot, N. Split hierarchical variational inequality problems and related problems. Fixed Point Theory Appl 2014, 208 (2014). https://doi.org/10.1186/1687-1812-2014-208
- split hierarchical variational inequality problem
- split common fixed point problem
- split convex minimization problem
- strongly nonexpansive operators
- cutter operators