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Iterative algorithms with regularization for hierarchical variational inequality problems and convex minimization problems
Fixed Point Theory and Applications volume 2013, Article number: 284 (2013)
In this paper, we consider a variational inequality problem which is defined over the set of intersections of the set of fixed points of a ζ-strictly pseudocontractive mapping, the set of fixed points of a nonexpansive mapping and the set of solutions of a minimization problem. We propose an iterative algorithm with regularization to solve such a variational inequality problem and study the strong convergence of the sequence generated by the proposed algorithm. The results of this paper improve and extend several known results in the literature.
Let H be a real Hilbert space with the inner product and the norm , let C be a nonempty closed convex subset of H, and let be a convex and continuously Fréchet differentiable functional. We consider the following minimization problem (MP):
We denote by Ξ the set of minimizers of problem (1.1), and we assume that . The gradient-projection algorithm (GPA) is one of the most elegant methods to solve the minimization problem (1.1). The convergence of the sequence generated by the GPA depends on the behavior of the gradient ∇f. If ∇f is strongly monotone and Lipschitz continuous, then we get the strong convergence of the sequence generated by the GPA to a unique solution of MP (1.1). However, if the gradient ∇f is assumed to be only Lipschitz continuous, then the sequence generated by the GPA converges weakly if H is infinite-dimensional (a counterexample is given in ). Since the Lipschitz continuity of the gradient ∇f implies that it is actually inverse strongly monotone (ism) , its complement can be an averaged mapping (that is, it can be expressed as a proper convex combination of the identity mapping and a nonexpansive mapping) . Consequently, the GPA can be rewritten as the composite of a projection and an averaged mapping, which is again an averaged mapping. This shows that averaged mappings play an important role in the GPA. Very recently, Xu  used averaged mappings to study the convergence analysis of the GPA, which is an operator-oriented approach. He showed that the sequence generated by the GPA converges in norm to a minimizer of MP (1.1), which is also a unique solution of a particular type of variational inequality problem (VIP). It is worth to emphasize that the regularization, in particular the traditional Tikhonov regularization, is usually used to solve ill-posed optimization problems. The advantage of a regularization method is its possible strong convergence to the minimum-norm solution of the optimization problem. In , Xu introduced a hybrid gradient-projection algorithm with regularization and proved the strong convergence of the sequence to the minimum-norm solution of MP (1.1). Some iterative algorithms with or without regularization for MP (1.1) are proposed and analyzed in [3–5] for finding a common solution of MP (1.1) and the set of solutions of a nonexpansive mapping.
On the other hand, the theory of variational inequalities [6, 7] has emerged as an important tool to study a wide class of problems from science, engineering, social sciences. If the underlying set in the formulation of a variational inequality problem is a set of fixed points of a mapping or, more precisely, of a nonexpansive mapping, then the variational inequality problem is called hierarchical variational problem. For further details on hierarchical variational inequalities, we refer to [8–11] and the references therein.
In this paper, we consider a variational inequality problem which is defined over the set of intersections of the set of fixed points of a ζ-strictly pseudocontractive mapping, the set of fixed points of a nonexpansive mapping and the set of solutions of MP (1.1). We propose an iterative algorithm with regularization to solve such a variational inequality problem and study the strong convergence of the sequence generated by the proposed algorithm. The results of this paper improve and extend several known results in the literature.
2 Preliminaries and formulations
Throughout the paper, unless otherwise specified, we use the following assumptions and notations. Let H be a real Hilbert space whose inner product and norm are denoted by and , respectively. Let C be a nonempty closed convex subset of H. We write (respectively, ) to indicate that the sequence converges strongly (respectively, weakly) to x. Moreover, we use to denote the weak ω-limit set of the sequence , that is,
The metric (or nearest point) projection from H onto C is the mapping which assigns to each point the unique point satisfying
Some important properties of projections are gathered in the following proposition.
Proposition 2.1 For given and , we have
, , which concludes that is nonexpansive and monotone.
Definition 2.1 A mapping is said to be
ζ-strictly pseudocontractive if there exists a constant such that
If , then it is called nonexpansive;
firmly nonexpansive if is nonexpansive, or equivalently,
alternatively, T is firmly nonexpansive if and only if T can be expressed as
where is a nonexpansive mapping.
It can be easily seen that the projection mappings are firmly nonexpansive. It is clear that is ζ-strictly pseudocontractive if and only if
Definition 2.2 Let T be a nonlinear operator with domain and range .
T is said to be monotone if
Given a number , T is said to be β-strongly monotone if
Given a number , T is said to be ν-inverse strongly monotone (ν-ism) if
if T is nonexpansive, then is monotone;
a projection is 1-ism;
if T is a ζ-strictly pseudocontractive mapping, then is -inverse strongly monotone.
Definition 2.3 
A mapping is said to be an averaged mapping if it can be written as the average of the identity I and a nonexpansive mapping, that is,
where and is a nonexpansive mapping. More precisely, when the last equality holds, we say that T is α-averaged. Thus, firmly nonexpansive mappings (in particular, projections) are -averaged maps.
Proposition 2.2 
Let be a given mapping.
T is nonexpansive if and only if the complement is -ism.
If T is ν-ism, then for , γT is -ism.
T is averaged if and only if the complement is ν-ism for some . Indeed, for , T is α-averaged if and only if is -ism.
Let be given operators.
If for some and if S is averaged and V is nonexpansive, then T is averaged.
T is firmly nonexpansive if and only if the complement is firmly nonexpansive.
If for some and if S is firmly nonexpansive and V is nonexpansive, then T is averaged.
The composite of finitely many averaged mappings is averaged, that is, if each of the mappings is averaged, then so is the composite . In particular, if is -averaged and is -averaged, where , then the composite is α-averaged, where .
Lemma 2.1 [, Proposition 2.1]
Let C be a nonempty closed convex subset of a real Hilbert space H, and let be a mapping.
If T is a ζ-strictly pseudocontractive mapping, then T satisfies the Lipschitz condition
If T is a ζ-strictly pseudocontractive mapping, then the mapping is semiclosed at 0, that is, if is a sequence in C such that weakly and strongly, then .
If T is a ζ-(quasi-)strict pseudocontraction, then the fixed point set of T is closed and convex so that the projection is well defined.
The following lemma is an immediate consequence of an inner product.
Lemma 2.2 In a real Hilbert space H, we have
The following elementary result on real sequences is quite well known.
Lemma 2.3 
Let be a sequence of nonnegative real numbers such that
where and satisfy the following conditions:
either or ;
, where , .
Lemma 2.4 
Let C be a nonempty closed convex subset of a real Hilbert space H, and let be a ζ-strictly pseudocontractive mapping. Let γ and δ be two nonnegative real numbers such that . Then
The following lemma appeared implicitly in the paper of Reineermann .
Lemma 2.5 
Let H be a real Hilbert space. Then, for all and ,
Let C be a nonempty closed convex subset of a real Hilbert space H, and let be a monotone mapping. The variational inequality problem (VIP) is to find such that
The solution set of the VIP is denoted by . It is well known that
A set-valued mapping is called monotone if for all , and imply that . A monotone set-valued mapping is called maximal if its graph is not properly contained in the graph of any other monotone set-valued mapping. It is known that a monotone set-valued mapping is maximal if and only if for , for every implies that . Let be a monotone and Lipschitz continuous mapping and be the normal cone to C at , that is,
Lemma 2.6 
Let be a monotone mapping. Then
V is maximal monotone;
Throughout the paper, we denote by and the set of fixed points of T and Γ, respectively. We also assume that the set is nonempty closed and convex.
Let be nonexpansive mappings and be a ζ-strictly pseudocontractive mapping with . In this paper, we consider and study the following hierarchical variational inequality problem which is defined on .
We denote by Ω the solution set of problem (2.1). It is not difficult to verify that solving (2.1) is equivalent to the fixed point problem of finding such that
where stands for the metric projection onto the closed convex set .
By using the definition of the normal cone to , we have the mapping :
and we readily prove that (2.1) is equivalent to the variational inequality
where is a ρ-contraction mapping, , and with , . It is proven that under appropriate assumptions, the above iterative sequence converges strongly to an element .
3 Main results
Let us consider the following assumptions:
the mapping is a ρ-contraction;
the mapping is a ζ-strict pseudocontraction;
are two nonexpansive mappings;
is Lipschitz continuous with ;
is a sequence in with ;
, , are sequences in with ;
, are sequences in with , ;
and , .
Theorem 3.1 Let be a bounded sequence generated from any given by (2.2). Assume that the following conditions hold:
Then the following assertions hold:
Proof First of all, we show that is ξ-averaged for each , where
Indeed, the Lipschitz continuity of ∇f implies that ∇f is -ism , that is,
Therefore, it follows that is -ism. Thus, by Proposition 2.2(b), is -ism. From Proposition 2.2(c), the complement is -averaged. Therefore, noting that is -averaged and utilizing Proposition 2.3(d), we obtain that for each , is ξ-averaged with
This shows that is nonexpansive. For , utilizing the fact that , we may assume that
Consequently, it follows that for each integer , is -averaged with
This implies that is nonexpansive for all .
The rest of the proof is divided into several steps.
Step 1. .
For simplicity, we put and for every . Then and for every .
Taking into account , without loss of generality, we may assume that for some . We write , , where . It follows that for all ,
Since for all , by Lemma 2.4, we have
Now, we estimate . Observe that for every ,
Similarly, for all , we have
From (2.2), we have
which implies that
Also, from (2.2) we have
then simple calculations show that
and thus, from (3.3)-(3.4), we have
where , for some . This together with (3.1)-(3.3) implies that
where , for some .
Further, we observe that
and then by simple calculations, we have
By taking norm and using (3.5)-(3.6), we get
where , for some . Therefore,
where , for some . From (H1)-(H5), it follows that and
Thus, by applying Lemma 2.3 to (3.7), we conclude that
which implies that
Step 2. .
Indeed, let . Then we have
Similarly, we get
By Lemma 2.5 and (3.9), we have
Since for all , utilizing Lemma 2.4, we obtain
Since , we may assume that for some . Therefore, we deduce
Since , , and as , we conclude from the boundedness of , and that as . This together with implies that
Step 3. and .
Let . Then, by Lemmas 2.2 and 2.5, we have
Therefore, we obtain
Since , , , and , from the boundedness of , and , we obtain , and hence
from and , it follows that
Furthermore, from the firm nonexpansiveness of , we obtain
Similarly, we have
Thus, we have
which implies that
Since , and , from the boundedness of , , and , it follows that
In addition, since for all , utilizing Lemma 2.4, we get from (3.12)
which implies that
Since , and , from the boundedness of , and , it follows that
Step 4. .
Let . Then there exists a subsequence of such that . Since
Hence from , and , we get . Since and , we have . By Lemma 2.1(b) (demiclosedness principle), we obtain .
Meanwhile, observe that
This together with yields . Since and , we have . By Lemma 2.1(b) (demiclosedness principle), we have .
Further, let us show . Indeed, from and , we have and . Define
where . Then V is maximal monotone and if and only if (see ). Let . Then we have
Therefore, we have
On the other hand, from
Hence, we obtain
Since V is maximal monotone, we have , and hence , which leads to . Consequently, . This shows that .
Finally, let us show . Indeed, it follows from (2.2) that for every ,
Since for all , by Lemma 2.4, we have
which implies that
Since and as , from the boundedness of , and , we deduce that
So, from , we get
Taking into con