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On the hierarchical variational inclusion problems in Hilbert spaces
Fixed Point Theory and Applications volume 2013, Article number: 179 (2013)
Abstract
The purpose of this paper is by using Maingé’s approach to study the existence and approximation problem of solutions for a class of hierarchical variational inclusion problems in the setting of Hilbert spaces. As applications, we solve the convex programming problems and quadratic minimization problems by using the main theorems. Our results extend and improve the corresponding recent results announced by many authors.
MSC: 47J05, 47H09, 49J25.
1 Introduction
Throughout this paper, we assume that H is a real Hilbert space, C is a nonempty closed and convex subset of H and denote by the set of fixed points of a mapping .
Let be a single-valued nonlinear mapping and let be a multi-valued mapping. The so-called quasi-variational inclusion problem (see [1–3]) is to find a point such that
A number of problems arising in structural analysis, mechanics and economics can be considered in the framework of this kind of variational inclusions (see, for example, [4]).
The set of solutions of the variational inclusion (1.1) is denoted by Ω.
Special cases
(I) If , where is a proper convex and lower semi-continuous function and ∂ϕ is the sub-differential of ϕ, then variational inclusion problem (1.1) is equivalent to finding such that
which is called the mixed quasi-variational inequality.
Especially, if , then (1.2) is equivalent to the minimizing problem of ϕ on H, i.e., to find such that .
(II) If , where C is a nonempty closed and convex subset of H and is the indicator function of C, i.e.,
then variational inclusion problem (1.2) is equivalent to finding such that
This problem is called Hartman-Stampacchia variational inequality problem.
(III) If and where I is an identity mapping and is a nonlinear mapping, then problem (1.1) is equivalent to the fixed point problem of T. That is, find such that
Recently, hierarchical fixed point problems, hierarchical optimization problems and hierarchical minimization problems have attracted many authors’ attention due to their link with convex programming problems, optimization problems and monotone variational inequality problems etc. (see [5–21] and others).
The purpose of this paper is to introduce and study the following bi-level hierarchical variational inclusion problem in the setting of Hilbert spaces:
Find such that for given positive real numbers ρ and η, the following inequalities hold:
where are mappings and are multi-valued mappings, is the set of solutions to variational inclusion problem (1.1) with , for .
Special examples
(I) If , , where is a nonlinear mapping for each , then and bi-level hierarchical variational inclusion problem (1.5) is equivalent to finding such that
This problem, which is called bi-level hierarchical optimization problem, was studied by Maingé [20] and Kraikaew et al. [21].
(II) In (1.6), if for each , where is the metric projection from H onto a nonempty closed convex subset , then it is clear that the and bi-level hierarchical optimization problem (1.6) is equivalent to finding such that
This system forms a more general problem originated from Nash equilibrium points and it was treated from a theoretical viewpoint in [22–24].
(III) If , and both sets and are nonempty closed and convex subsets of H, then bi-level hierarchical variational inclusion problem (1.5) reduces to the following (one-level) hierarchical variational inclusion problem:
Find such that for a given positive real number ρ, the following inequality holds:
(IV) If and , , then (1.7) reduces to the classic variational inequality, i.e., the problem of finding such that
In (1.5), it is worth noting that if , are nonempty closed convex subsets in H, then the metric projections and from H onto and , respectively, are well defined and problem (1.5) is equivalent to the problem of finding such that
However, in practice, both solution sets and (and hence the two projections) are not given explicitly.
To overcome this drawback, inspired by the method studied by Yamada et al. [25, 26], Maingé [20] and Kraikaew et al. [21], we investigate a more general variant of the scheme proposed by Maingé [20], Kraikaew et al. [21] to replace the projection by some suitable mappings with a nice fixed point set. This strategy also suggests an effective approximation process. Our analysis and method allow us to prove the existence and approximation of solutions to problem (1.5). As applications, we utilize the main results to study the quadratic minimization problems and convex programming problems in Hilbert spaces. The results presented in the paper extend and improve the corresponding results in [20, 21, 25, 26] and others.
2 Preliminaries
For the sake of convenience, we first recall some definitions and lemmas for our main results.
Definition 2.1 A mapping is said to be α-inverse-strongly monotone if there exists such that
A multi-valued mapping is called monotone if for all , and imply that
A multi-valued mapping is said to be maximal monotone if it is monotone and for any ,
for every (the graph of mapping M) implies that .
Lemma 2.2 [19]
Let be an α-inverse-strongly monotone mapping. Then
-
(i)
A is an -Lipschitz continuous and monotone mapping;
-
(ii)
For any constant , we have
(2.1) -
(iii)
If , then is a nonexpansive mapping, where I is the identity mapping on H.
Let H be a real Hilbert space, C be a nonempty closed convex subset of H. For each , there exists a unique nearest point in C, denoted by , such that
Such a mapping from H onto C is called the metric projection.
Remark 2.3 It is well known that the metric projection has the following properties:
-
(i)
is nonexpansive;
-
(ii)
is firmly nonexpansive, i.e.,
-
(iii)
For each ,
(2.2)
Definition 2.4 Let be a multi-valued maximal monotone mapping. Then the mapping defined by
is called the resolvent operator associated with M, where λ is any positive number and I is the identity mapping.
Proposition 2.5 [19]
Let be a multi-valued maximal monotone mapping, and let be an α-inverse-strongly monotone mapping. Then the following conclusions hold.
-
(i)
The resolvent operator associated with M is single-valued and nonexpansive for all .
-
(ii)
The resolvent operator is 1-inverse-strongly monotone, i.e.,
-
(iii)
is a solution of the variational inclusion (1.1) if and only if , , i.e., u is a fixed point of the mapping . Therefore we have
(2.3)where Ω is the set of solutions of variational inclusion problem (1.1).
-
(iv)
If , then Ω is a closed convex subset in H.
In the sequel, we denote the strong and weak convergence of a sequence in H to an element by and , respectively.
Lemma 2.6 [27]
For and , the following statements hold:
-
(i)
;
-
(ii)
.
Lemma 2.7 [28]
Let be a sequence of real numbers, and there exists a subsequence of such that for all , where N is the set of all positive integers. Then there exists a nondecreasing sequence of N such that and the following properties are satisfied by all (sufficiently large) number :
In fact, is the largest number n in the set such that holds.
Lemma 2.8 [21]
Let , , , be such that
-
(i)
is a bounded sequence;
-
(ii)
, ;
-
(iii)
whenever is a subsequence of satisfying
it follows that ;
-
(iv)
and .
Then .
Definition 2.9
-
(i)
A mapping is said to be nonexpansive if
-
(ii)
A mapping is said to be quasi-nonexpansive if and
It should be noted that T is quasi-nonexpansive if and only if ,
(2.4) -
(iii)
A mapping is said to be strongly quasi-nonexpansive if T is quasi-nonexpansive and
(2.5)whenever is a bounded sequence in H and for some .
Lemma 2.10 Let be a multi-valued maximal monotone mapping, be an α-inverse-strongly monotone mapping and let Ω be the set of solutions of variational inclusion problem (1.1) and . Then the following statements hold.
-
(i)
If , then the mapping defined by
(2.6)is quasi-nonexpansive, where I is the identity mapping and is the resolvent operator associated with M.
-
(ii)
The mapping is demiclosed at zero, i.e., for any sequence , if and , then .
-
(iii)
For any , the mapping defined by
(2.7)is a strongly quasi-nonexpansive mapping and .
-
(iv)
, is demiclosed at zero.
Proof (i) Since , it follows from Lemma 2.2(iii) and Proposition 2.5 that the mapping K is nonexpansive and . This implies that K is quasi-nonexpansive.
(ii) Since K is a nonexpansive mapping on H, is demiclosed at zero.
(iii) It is obvious that and is quasi-nonexpansive.
Next we prove that , is a strongly quasi-nonexpansive mapping.
In fact, let be any bounded sequence in H and let be a given point such that
Now we prove that .
In fact, it follows from (2.4) that
Hence from (2.8), we have
Since , , and so
(iv) Since and is demi-closed at zero, hence is demi-closed at zero. This completes the proof. □
3 Main results
Throughout this section we always assume that the following conditions are satisfied:
-
(C1)
, , is a multi-valued maximal monotone mapping, is an α-inverse-strongly monotone mapping and is the set of solutions to variational inclusion problem (1.1) with , and ;
-
(C2)
and , , , are the mappings defined by
(3.1)respectively.
We have the following result.
Theorem 3.1 Let , , , , , , satisfy the conditions (C1) and (C2), and let be contractions with a contractive constant . Let and be two sequences defined by
where is a sequence in satisfying and . Then the sequences and converge to and , respectively, where is the unique solution of the following (bi-level) hierarchical optimization problem:
Proof (I) First we prove that (3.3) has a unique solution .
Indeed, it follows from Proposition 2.5 and Lemma 2.10 that both sets , are nonempty closed and convex and for each . Hence the metric projection for each is well defined. It is clear that the mapping
is a contraction. By the Banach contractive mapping principle, there exists a unique element such that
Letting , then it is easy to see that
are the unique solution of (3.3).
(II) Now we prove that and are bounded.
In fact, it follows from Lemma 2.10 that , , is strongly quasi-nonexpansive and . Since f is h-contractive and , , we have
Similarly, we can also prove that
This implies that
By induction, we have
This implies that and are bounded. Consequently, the sequences and both are bounded.
(III) Next we prove that for each the following inequality holds.
In fact, it follows from (3.2) and Lemma 2.6(i) that
Similarly, we have
Adding up the last two inequalities, the inequality (3.4) is proved.
(IV) Next we prove the following fact.
If there exists a subsequence such that
then
In fact, since the norm is convex and , from (3.2) we have that
The above conclusion can be proved as follows.
Indeed, since the sequences and are bounded, and , , is quasi-nonexpansive, we have
The conclusion is proved. Therefore we have that
By Lemma 2.10(iii), the mapping , , is strongly quasi-nonexpansive. Hence from (3.5) we have that
This together with (3.2) shows that
Since is bounded and H is reflexive, there exists a subsequence such that and
On the other hand, by virtue of Lemma 2.10(iv), is demiclosed at zero, and so . Hence from (3.3) we have
Consequently,
Similarly, by using the same argument, we have
The desired inequality is proved.
(V) Finally we prove that the sequences and defined by (3.2) converge to and , respectively.
It is easy to see that
Substituting (3.7) into (3.4), simplifying and putting
then we have the following conclusions:
-
(i)
By (II), is a bounded sequence;
-
(ii)
From (3.4), , ;
-
(iii)
By (IV), for any subsequence satisfying
it follows that .
Hence it follows from Lemma 2.8 that and . This completes the proof of Theorem 3.1. □
Definition 3.2 A mapping is said to be μ-Lipschitzian and r-strongly monotone, if there exist constants and such that
Remark 3.3 It is easy to prove that if is a μ-Lipschitzian and r-strongly monotone mapping and if , then the mapping is a contraction.
Now we are in a position to prove the following main result.
Theorem 3.4 Let , , , , , , be the same as in Theorem 3.1. Let be a μ-Lipschitzian and r-strongly monotone mapping. Let and be the sequences defined by
where , with , and is a sequence in satisfying the following conditions:
Then the sequence converges strongly to the unique solution of bi-level hierarchical variational inclusion problem (1.5).
Proof Indeed, it follows from Remark 3.3 that both mappings are contractive. Therefore all the conditions in Theorem 3.1 are satisfied. By Theorem 3.1, the sequence converges strongly to , which is the unique solution of the following bi-level hierarchical optimization problem:
Since and , we have
This implies that the sequence converges strongly to , which is the unique solution of bi-level hierarchical variational inclusion problem (1.5). This completes the proof of Theorem 3.4. □
4 Some applications
In this section, we shall utilize Theorem 3.1 and Theorem 3.4 to study the convex mathematical programming problem and quadratic minimization problem.
(I) Applications to convex mathematical programming problems.
Let be a convex and lower semi-continuous function with ▽ψ being μ-Lipschitzian and r-strongly monotone, i.e., it satisfies the following conditions:
and
In (1.5) taking , and , then hierarchical variational inclusion problem (1.5) reduces to the following problem:
Find a point such that
By using the subdifferential inequality, this implies that
Therefore we have
Thus problem (4.3) reduces to the convex mathematical programming problem on :
Find a point such that
Hence, we have the following result.
Theorem 4.1 Let , , , , , be the same as in Theorem 3.4. Let be the iterative sequence defined by
where , . Then converges strongly to , which is the unique solution of convex mathematical programming problem (4.5).
(II) Applications to quadratic minimization problems.
Recall that a linear bounded operator is said to be ξ-strongly positive if there exists a positive constant ξ such that
Lemma 4.2 Let be a ξ-strongly positive linear operator and let γ be a positive number with , where is the norm of T defined by
Then we have
-
(1)
The linear operator is μ-Lipschitzian and r-strongly monotone, where and .
-
(2)
If , then the linear operator is contractive with a contractive constant .
Proof (1) In fact, for any , we have
Again, since is a ξ-strongly positive linear operator, we have
Conclusion (1) is proved.
(2) By the definition of the norm of the bounded linear operator , we have
Therefore, is contractive with a contractive constant . This completes the proof. □
From Theorem 3.4 and Lemma 4.2 we have the following result.
Theorem 4.3 Let A, M, K, , Ω and satisfy the same conditions as given in Theorem 3.4. Let the linear mappings T and F satisfy the same conditions as in Lemma 4.2. Then the sequence defined by
where , , converges strongly to , which is the unique solution of the hierarchical variational inclusion problem:
that is,
Letting , then it is easy to know that is a continuous and convex functional and . By the subdifferential inequality of g, we have
This implies that solves the following quadratic minimization problem:
and . This completes the proof.
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Acknowledgements
The second author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2012R1A1A2042138).
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The main idea of this paper was proposed by JKK. JKK and SC prepared the manuscript initially and performed all the steps of the proof in this research. All authors read and approved the final manuscript.
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Chang, Ss., Kim, J.K., Joseph Lee, H. et al. On the hierarchical variational inclusion problems in Hilbert spaces. Fixed Point Theory Appl 2013, 179 (2013). https://doi.org/10.1186/1687-1812-2013-179
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DOI: https://doi.org/10.1186/1687-1812-2013-179