Abstract
By combining Korpelevich’s extragradient method, the viscosity approximation method, the hybrid steepest-descent method, Mann’s iteration method, and the gradient-projection method with regularization, a hybrid multi-step extragradient algorithm with regularization for finding a solution of triple hierarchical variational inequality problem is introduced and analyzed. It is proven that under appropriate assumptions, the proposed algorithm converges strongly to a unique solution of a triple hierarchical variational inequality problem which is defined over the set of solutions of a hierarchical variational inequality problem defined over the set of common solutions of finitely many generalized mixed equilibrium problems (GMEP), finitely many variational inclusions, fixed point problems, and the split feasibility problem (SFP). We also prove the strong convergence of the proposed algorithm to a common solution of the SFP, finitely many GMEPs, finitely many variational inclusions, and the fixed point problem of a strict pseudocontraction. The results presented in this paper improve and extend the corresponding results announced by several others.
MSC:49J30, 47H09, 47J20, 49M05.
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1 Introduction
The following problems have their own importance because of their applications in diverse areas of science, engineering, social sciences, and management:
-
Equilibrium problems including variational inequalities.
-
Variational inclusion problems.
-
Split feasibility problems.
-
Fixed point problems.
One way or the other, these problems are related to each other. They are described as follows.
Equilibrium problem
Let C be a nonempty closed convex subset of a real Hilbert space H and be a real-valued bifunction. The equilibrium problem (EP) is to find an element such that
The set of solutions of EP is denoted by . It includes several problems, namely, variational inequality problems, optimization problems, saddle point problems, fixed point problems, etc., as special cases. For further details on EP, we refer to [1–6] and the references therein.
Let be a nonlinear operator. If , then EP reduces to the variational inequality problem of finding such that
For further details on variational inequalities and their generalizations, we refer to [7–13] and the references therein.
During the last two decades, EP has been extended and generalized in several directions. The generalized mixed equilibrium problem (GMEP), one of the generalizations of EP, is to find such that
where is a real-valued function. The set of solutions of GMEP is denoted by . For different choices of operators/functions Θ, φ, and A, we get different forms of equilibrium problems. For applications of GMEP, we refer to [14, 15] and the references therein.
Variational inclusion problem
Let be a single-valued mapping and be a set-valued mapping with , where denotes the domain of R. The variational inclusion problem is to find such that
We denote by the solution set of the variational inclusion problem (1.2). In particular, if , then . If , then problem (1.2) becomes the inclusion problem introduced by Rockafellar [16]. It is well known that problem (1.2) provides a convenient framework for the unified study of optimal solutions in many optimization related areas including mathematical programming, complementarity problems, variational inequalities, optimal control, mathematical economics, equilibria and game theory, etc. Let a set-valued mapping be maximal monotone. We define the resolvent operator associated with R and as follows:
Huang [17] studied problem (1.2) in the case where R is maximal monotone and B is strongly monotone and Lipschitz continuous with . Zeng et al. [18] further studied problem (1.2) in a more general setting than in [17]. They gave the geometric convergence rate estimate for approximate solutions. Various types of iterative algorithms for solving variational inclusions have been further studied and developed in the literature; see, for example, [19–22] and the references therein.
Split feasibility problem
Let C and Q be nonempty closed convex subsets of real Hilbert spaces and , respectively. The split feasibility problem (SFP) is to find a point x such that
where is a bounded linear operator from to . We denote by Γ the solution set of the SFP. It is a model of an inverse problem which arises in phase retrievals and in medical image reconstruction. A number of image reconstruction problems can be formulated as SFP; see, for example [23] and the references therein. Recently, it is found that the SFP can also be applied to study intensity-modulated radiation therapy (IMRT); see, for example, [24, 25] and the references therein. In the recent past, a wide variety of iterative methods have been proposed to solve SFP; see, for example, [24–28] the references therein.
Fixed point problem
Let C be a nonempty subset of a H and be a mapping. The fixed point problem is to find an element such that .
It is a well-known problem and has tremendous applications in different branches of science, engineering, social sciences, and management.
The following proposition provides some relations among the above mentioned problems.
Proposition 1.1 Given , the following statements are equivalent:
-
(a)
solves the SFP;
-
(b)
solves the fixed point equation
where , , is the projection operator and is the adjoint of A;
-
(c)
solves the variational inequality problem (VIP) of finding such that
A variational inequality problem which is defined over the set of fixed points of a mapping is called hierarchical variational inequality problem; that is, when the set C in variational inequality formulation is equal to the set of fixed points of a mapping. A variational inequality problem which is defined over the set of solutions of a hierarchical variational inequality problem is called a triple hierarchical variational inequality problem. For further details on hierarchical variational inequality problems and triple hierarchical variational inequality problems, we refer to [29], a recent survey on these problems.
Very recently, Kong et al. [30] considered the following triple hierarchical variational inequality problem (THVIP).
Problem 1.1 Let C be a nonempty closed convex subset of a real Hilbert space H and be a κ-Lipschitzian and η-strongly monotone operator, where κ and η are positive constants. Let be a monotone and L-Lipschitzian mapping, be a ρ-contraction with coefficient , be a nonexpansive mapping, and be a ξ-strictly pseudocontractive mapping with , where denotes the set of all fixed points of T. Let and , where . Then the objective is to find such that
where Ξ denotes the solution set of the hierarchical variational inequality problem (HVIP) of finding such that
Kong et al. [30] presented an algorithm for finding a solution of Problem 1.1. Under some conditions, they proved that the sequence generated by the proposed algorithm converges strongly to a point which is a unique solution of Problem 1.1 provided that is bounded and . They also showed under certain conditions that the sequence generated by proposed algorithm converges strongly to a unique solution of the following VIP provided that and the sequence is bounded:
In this paper, we consider the following triple hierarchical variational inequality problem (THVIP).
Problem 1.2 Let M, N be two positive integers. Assume that
-
(i)
is κ-Lipschitzian and η-strongly monotone with positive constants such that and where ;
-
(ii)
for each , satisfies conditions (A1)-(A4) and is a proper lower semicontinuous and convex function with restriction (B1) or (B2) (conditions (A1)-(A4) and (B1)-(B2) are given in the next section);
-
(iii)
for each and , is a maximal monotone mapping, and and are -inverse strongly monotone and -inverse strongly monotone, respectively;
-
(iv)
is a ξ-strict pseudocontraction, is a nonexpansive mapping and is a ρ-contraction with coefficient ;
-
(v)
.
Then the objective is to find such that
where Ξ denotes the solution set of the hierarchical variational inequality problem (HVIP) of finding such that
By combining Korpelevich’s extragradient method, the viscosity approximation method, the hybrid steepest-descent method, Mann’s iteration method, and the gradient-projection method (GPM) with regularization, we introduce and analyze a hybrid multi-step extragradient algorithm with regularization in the setting of Hilbert spaces. It is proven that under appropriate assumptions, the proposed algorithm converges strongly to a unique solution of THVIP (1.6). The algorithm and convergence result of this paper extend and generalize several existing algorithms and results, respectively, in the literature.
2 Preliminaries
Throughout this paper, unless otherwise specified, we assume that H is a real Hilbert space whose inner product and norm are denoted by and , respectively. We write (respectively, ) to indicate that the sequence converges (respectively, weakly) to x. Moreover, we use to denote the weak ω-limit set of the sequence , that is,
Definition 2.1 A mapping is said to be
-
(a)
nonexpansive if
-
(b)
firmly nonexpansive if is nonexpansive, or equivalently, if T is 1-inverse strongly monotone (1-ism),
alternatively, T is firmly nonexpansive if and only if T can be expressed as
where is nonexpansive; projections are firmly nonexpansive.
It can easily be seen that if T is nonexpansive, then is monotone.
Definition 2.2 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 nonexpansive. More precisely, when the last equality holds, we say that T is α-averaged. Thus firmly nonexpansive mappings (in particular, projections) are -averaged mappings.
Proposition 2.1 [31]
Let be a given mapping.
-
(a)
T is nonexpansive if and only if the complement is -ism.
-
(b)
If T is ν-ism, then for , γT is -ism.
-
(c)
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.
-
(a)
If for some and if S is averaged and V is nonexpansive, then T is averaged.
-
(b)
T is firmly nonexpansive if and only if the complement is firmly nonexpansive.
-
(c)
If for some and if S is firmly nonexpansive and V is nonexpansive, then T is averaged.
-
(d)
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 .
-
(e)
If the mappings are averaged and have a common fixed point, then
The notation denotes the set of all fixed points of the mapping T, that is, .
A mapping is said to be ξ-strictly pseudocontractive if there exists such that
In this case, we also say that T is a ξ-strict pseudocontraction. We denote by the set of fixed points of S. In particular, if , T is a nonexpansive mapping.
It is clear that, in a real Hilbert space H, is ξ-strictly pseudocontractive if and only if the following inequality holds:
This immediately implies that if T is a ξ-strictly pseudocontractive mapping, then is -inverse strongly monotone; for further details, we refer to [33] and the references therein. It is well known that the class of strict pseudocontractions strictly includes the class of nonexpansive mappings and that the class of pseudocontractions strictly includes the class of strict pseudocontractions.
Lemma 2.1 [[33], Proposition 2.1]
Let C be a nonempty closed convex subset of a real Hilbert space H and be a mapping.
-
(a)
If T is a ξ-strictly pseudocontractive mapping, then T satisfies the Lipschitzian condition
-
(b)
If T is a ξ-strictly pseudocontractive mapping, then the mapping is semiclosed at 0, that is, if is a sequence in C such that and , then .
-
(c)
If T is ξ-(quasi-)strict pseudocontraction, then the fixed point set of T is closed and convex so that the projection is well defined.
Lemma 2.2 [34]
Let C be a nonempty closed convex subset of a real Hilbert space H. Let be a ξ-strictly pseudocontractive mapping. Let γ and δ be two nonnegative real numbers such that . Then
Lemma 2.3 (Demiclosedness principle)
Let C be a nonempty closed convex subset of a real Hilbert space H. Let S be a nonexpansive self-mapping on C with . Then is demiclosed. That is, whenever is a sequence in C weakly converging to some and the sequence strongly converges to some y, it follows that , where I is the identity operator of H.
Definition 2.3 A nonlinear operator T with the domain and the range is said to be
-
(a)
monotone if
-
(b)
β-strongly monotone if there exists a constant such that
-
(c)
ν-inverse strongly monotone if there exists a constant such that
It is easy to see that the projection is 1-inverse strongly monotone. Inverse strongly monotone (also referred to as co-coercive) operators have been applied widely in solving practical problems in various fields, for instance, in traffic assignment problems; see, for example, [35]. It is obvious that if T is ν-inverse strongly monotone, then T is monotone and -Lipschitz continuous. Moreover, we also have, for all and ,
So, if , then is a nonexpansive mapping.
The metric (or nearest point) projection from H onto C is the mapping which assigns to each point the unique point satisfying the property
Some important properties of projections are gathered in the following proposition.
Proposition 2.3 For given and :
-
(a)
, ;
-
(b)
, ;
-
(c)
, .
Consequently, is nonexpansive and monotone.
Let λ be a number in and let . Associating with a nonexpansive mapping , we define the mapping by
where is an operator such that, for some positive constants , F is κ-Lipschitzian and η-strongly monotone on H, that is, F satisfies the conditions:
for all .
Lemma 2.4 [[36], Lemma 3.1]
is a contraction provided , that is,
where .
Lemma 2.5 Let be a monotone mapping. In the context of the variational inequality problem the characterization of the projection (see Proposition 2.3(a)) implies
Let C be a nonempty closed convex subset of H and satisfy the following conditions.
-
(A1) , ;
-
(A2) T is monotone, that is, , ;
-
(A3) T is upper-hemicontinuous, that is, ,
-
(A4) is convex and lower semicontinuous, for each .
Let be a lower semicontinuous and convex function satisfying either (B1) or (B2), where
-
(B1) for each and , there exist a bounded subset and such that, for any ,
-
(B2) C is a bounded set.
Given a positive number . Let be the solution set of the auxiliary mixed equilibrium problem, that is, for each ,
Next we list some elementary conclusions for the MEP.
Proposition 2.4 [37]
Assume that satisfies (A1)-(A4) and let be a proper lower semicontinuous and convex function. Assume that either (B1) or (B2) holds. For and , define a mapping as follows:
for all . Then
-
(i)
for each , is nonempty and single-valued;
-
(ii)
is firmly nonexpansive, that is, for any ,
-
(iii)
;
-
(iv)
is closed and convex;
-
(v)
, for all and .
We need some facts and tools in a real Hilbert space H which are listed as lemmas below.
Lemma 2.6 Let X be a real inner product space. Then we have the following inequality:
Lemma 2.7 Let H be a real Hilbert space. Then the following hold:
-
(a)
, for all ;
-
(b)
, for all and with ;
-
(c)
if is a sequence in H such that , it follows that
Lemma 2.8 [38]
Let be a sequence of nonnegative real numbers satisfying the property
where and are such that:
-
(i)
;
-
(ii)
either or ;
-
(iii)
where , for all .
Then .
Recall that a set-valued mapping is called monotone if, for all , , and imply . A set-valued mapping T is called maximal monotone if T is monotone and , for each , where I is the identity mapping of H. We denote by the graph of T. It is well known that a monotone mapping T is maximal if and only if, for , for every implies .
Next we provide an example to illustrate the concept of maximal monotone mapping.
Let be a monotone, k-Lipschitz-continuous mapping and let be the normal cone to C at , that is,
Define
Then is maximal monotone (see [16]) such that
Let be a maximal monotone mapping. Let be two positive numbers.
Lemma 2.9 [39]
We have the resolvent identity
Remark 2.1 For , we have the following relation:
The following property for the resolvent operator was considered in [17, 18].
Lemma 2.10 is single-valued and firmly nonexpansive, that is,
Consequently, is nonexpansive and monotone.
Lemma 2.11 [20]
Let R be a maximal monotone mapping with . Then, for any given , is a solution of problem (1.6) if and only if satisfies
Lemma 2.12 [18]
Let R be a maximal monotone mapping with and let be a strongly monotone, continuous and single-valued mapping. Then, for each , the equation has a unique solution for .
Lemma 2.13 [20]
Let R be a maximal monotone mapping with and be a monotone, continuous and single-valued mapping. Then , for each . In this case, is maximal monotone.
3 Algorithms and convergence results
Let H be a real Hilbert space and be a function. Then the minimization problem
is ill-posed. Xu [40] considered the following Tikhonov’s regularization problem:
where is the regularization parameter. It is clear that the gradient
is -Lipschitz continuous.
Throughout the paper, unless otherwise specified, M, N are positive integers and C be a nonempty closed convex subset of a real Hilbert space H.
Algorithm 3.1 The notations and symbols are the same as in Problem 1.2. Start with a given arbitrary , and compute a sequence by
where .
The following result provides the strong convergence of the sequence generated by Algorithm 3.1.
Theorem 3.1 For each , let be a bifunction satisfying conditions (A1)-(A4) and be a proper lower semicontinuous and convex function with restriction (B1) or (B2). For each and , let be a maximal monotone mapping and let and be -inverse strongly monotone and -inverse strongly monotone, respectively. Let be a ξ-strictly pseudocontractive mapping, be a nonexpansive mapping and be a ρ-contraction with coefficient . Let be κ-Lipschitzian and η-strongly monotone with positive constants such that and , where . Assume that the solution set Ξ of HVIP (1.7) is nonempty where . Let , with , with , and , where and . Suppose that
-
(C1) , , and ;
-
(C2) or ;
-
(C3) or ;
-
(C4) or ;
-
(C5) or ;
-
(C6) or ;
-
(C7) , and ;
-
(C8) for each , or ;
-
(C9) for each , or ;
-
(C10) there exist positive constants such that and , for sufficiently large .
If is a sequence generated by Algorithm 3.1 and is bounded, then
-
(a)
;
-
(b)
;
-
(c)
converges strongly to a point provided , which is the unique solution of Problem 1.2.
Proof First of all, taking into account , we know that . Observe that
and
Since and , we deduce that and hence the mapping is -strongly monotone. Moreover, it is clear that the mapping is -Lipschitzian. Thus, there exists a unique solution in Ξ to the VIP
that is, . Now, we put
for all and ,
for all , , and , where I is the identity mapping on H. Then we have and .
Now, we show that is ζ-averaged, for each , where
Indeed, it is easy to see that is -ism, that is,
Observe that
Hence, it follows that is -ism. Thus, by Proposition 2.1(b), is -ism. From Proposition 2.1(c), the complement is -averaged. Therefore, noting that is -averaged and utilizing Proposition 2.2(d), we see that, for each , is ζ-averaged with
This shows that is nonexpansive. Taking into account that and , we get
Without loss of generality, we may assume that , for each . So, is nonexpansive, for each . Similarly, since
it may be confirmed that is nonexpansive, for each .
We divide the rest of the proof into several steps.
Step 1. We prove that is bounded.
Indeed, take a fixed arbitrarily. Utilizing (2.1) and Proposition 2.4(b), we have
Utilizing (2.1) and Lemma 2.10, we have
Combining (3.2) and (3.3), we have
For simplicity, put , for each . Note that for . Hence, from (3.4), it follows that
Since T is a ξ-strictly pseudocontractive mapping and , for all , by Lemma 2.2, we obtain from (3.1) and (3.5) that
Noticing the boundedness of , we get for some . Moreover, utilizing Lemma 2.4 and (3.1), (3.6), we deduce that and that, for all ,
By induction, we get
Thus, is bounded since , and so are the sequences , , , and .
Step 2. We prove that .
Indeed, utilizing (2.1) and (2.3), we obtain
where
for some and for some .
Utilizing Proposition 2.4(b), (e), we deduce that
where is a constant such that, for each ,
Furthermore, we define for all . It follows that
Since T is a ξ-strictly pseudocontractive mapping and , for all , by Lemma 2.2, we obtain
Also, utilizing the nonexpansivity of , we have
Hence, from (3.7)-(3.11), it follows that
In the meantime, simple calculation shows that
So, it follows from (3.12) that
where for some .
On the other hand, we define , for all . Then it is well known that , for all . Simple calculations show that
Since V is a ρ-contraction with coefficient and S is a nonexpansive mapping, we conclude that
which together with (3.13) and implies that
where for some . Consequently,
where for some . Utilizing Lemma 2.8, we conclude from conditions (C1)-(C6) and (C8)-(C9) that and
So, as , it follows that
Step 3. We prove that , , and .
Indeed, utilizing Lemmas 2.2 and 2.7(b), from (3.1), (3.4)-(3.5) and , we deduce that
Observe that
and
for and . Combining (3.5), (3.15)-(3.17), we get
which immediately leads to
Since , , , and , , are bounded sequences, we obtain from ,
for all and .
Furthermore, by Proposition 2.4(b) and Lemma 2.7(a), we have
which implies that
By Lemma 2.7(a) and Lemma 2.10, we obtain
which immediately leads to
Combining (3.15) and (3.21), we conclude that
which yields
So, it follows from , , that
Now we claim that
As a matter of fact, it is easy to see that, for each ,
If , then from (3.22) and (due to (3.9)), we have
That is, . If , then from (3.22), we have
Since (due to (3.19)), it is easy to see that
Thus, from (3.24), it follows that
which leads to a contradiction. This shows that (3.23) holds.
Also, combining (3.3), (3.15), and (3.20), we deduce that
which yields
So, it follows from , , that
Next, we claim that
As a matter of fact, it is easy to see that, for each ,
If , then from (3.25) and (due to (3.19)), we have
That is, . If , then from (3.25), we have
Since (due to (3.19)), it is easy to see that
Consequently, from (3.27), it follows that
which leads to a contradiction. This shows that (3.26) is valid. Therefore, from (3.23) and (3.26), we get
and
respectively. Thus, from (3.28) and (3.29), we obtain
On the other hand, note that . Then, utilizing Lemma 2.6 and the -inverse strong monotonicity of ∇f, we deduce from (2.1) that
Combining (3.4), (3.15), and (3.31), we obtain
which together with and leads to
Since , , and are bounded sequences, we deduce from that
So, it is clear that
Again, utilizing Proposition 2.3(c), from and , we get
which immediately leads to
Combining (3.4), (3.15), and (3.33), we obtain
which immediately yields
Since , , and are bounded sequences, we deduce from that
Observe that
Thus, from (3.32) and (3.34), we have
Taking into account that , from (3.30) and (3.35), we get
Utilizing the relation , we have
which together with (3.36) and , implies that
Since , we obtain
Step 4. We prove that .
Indeed, since H is reflexive and is bounded, there exists at least a weak convergence subsequence of . Hence, . Now, take an arbitrary . Then there exists a subsequence of such that . From (3.23), (3.26), (3.28), (3.30) and (3.36), we have , , , and , where and . Utilizing Lemma 2.1(b), we deduce from and (3.37) that .
Next, we prove that . As a matter of fact, since is -inverse strongly monotone, is a monotone and Lipschitz-continuous mapping. It follows from Lemma 2.13 that is maximal monotone. Let , that is, . Again, since , , , we have
that is,
In terms of the monotonicity of , we get
and hence
In particular,
Since (due to (3.23)) and (due to the Lipschitz continuity of ), we conclude from and that
It follows from the maximal monotonicity of that , that is, . Therefore, .
Next we prove that . Since , , , we have
By (A2), we have
Let , for all and . This implies that . Then we have
By (3.26), we have as . Furthermore, by the monotonicity of , we obtain . Then, by (A4), we obtain
Utilizing (A1), (A4), and (3.39), we obtain
and hence
Letting , we have, for each ,
This implies that , and hence, . Thus, .
Furthermore, let us show that . In fact, define
where . Then is maximal monotone and if and only if ; see [16]. Let . Then we have , and hence, . So, we have , for all .
On the other hand, from and , we get , and hence,
Therefore, from and , we have
Hence, it is easy to see that as . Since is maximal monotone, we have , and hence, . Consequently, . This shows that .
Step 5. We prove that .
Indeed, utilizing Lemmas 2.6 and 2.4, from (3.1) and (3.6), we find that, for all ,
Take an arbitrary . Then there exists a subsequence of such that . Utilizing (3.40), we obtain, for all ,
which implies that
Since the combination of the boundedness of , , , and (due to Step 2) implies that
from (3.41), we conclude that
that is,
Since is -strongly monotone and -Lipschitz continuous, by Minty’s lemma [41] we know that (3.42) is equivalent to the VIP
So, it follows that . This shows that .
Step 6. We prove that where .
Indeed, note that . Since is bounded and H is reflexive, there exists a subsequence of such that and
According to Step 5, we get . So, it follows from that
However, from and condition (C10), we deduce that, for sufficiently large ,
Utilizing Lemma 2.1(a), we have, for sufficiently large ,
where . Hence, for a large enough constant , from (3.44) and (3.45), we have, for sufficiently large ,
Next we prove that . As a matter of fact, putting in (3.40), we obtain from (3.46) that
Since , , , , and , we conclude from (3.36), (3.37), and that , , and
Therefore, applying Lemma 2.8 to (3.47), we infer that . This completes the proof. □
Remark 3.1 Algorithm 3.1 and Theorem 3.1 extend and generalize algorithms and convergence results in [28, 30].
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Acknowledgements
In this research, the first author was partially supported by the National Science Foundation of China (11071169), Innovation Program of Shanghai Municipal Education Commission (09ZZ133), and PhD Program Foundation of Ministry of Education of China (20123127110002). The second author and third author were supported partly by the National Science Council of the Republic of China.
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Ceng, LC., Pang, CT. & Wen, CF. Multi-step extragradient method with regularization for triple hierarchical variational inequalities with variational inclusion and split feasibility constraints. J Inequal Appl 2014, 492 (2014). https://doi.org/10.1186/1029-242X-2014-492
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DOI: https://doi.org/10.1186/1029-242X-2014-492