Abstract
In this paper, we consider a hierarchical variational inequality problem (HVIP) defined over a common set of solutions of finitely many generalized mixed equilibrium problems, finitely many variational inclusions, a general system of variational inequalities, and the fixed point problem of a strictly pseudocontractive mapping. By combining Korpelevich’s extragradient method, the viscosity approximation method, the hybrid steepest-descent method and Mann’s iteration method, we introduce and analyze a multistep hybrid extragradient algorithm for finding a solution of our HVIP. It is proven that under appropriate assumptions, the proposed algorithm converges strongly to a solution of a general system of variational inequalities defined over a common set of solutions of finitely many generalized mixed equilibrium problems (GMEPs), finitely many variational inclusions, and the fixed point problem of a strictly pseudocontractive mapping. In the meantime, we also prove the strong convergence of the proposed algorithm to a unique solution of our HVIP. The results obtained in this paper improve and extend the corresponding results announced by many others.
MSC:49J30, 47H09, 47J20.
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1 Introduction and formulations
1.1 Variational inequalities and equilibrium problems
Let C be a nonempty closed convex subset of a real Hilbert space H and be a nonlinear mapping. A variational inequality problem (VIP) is to find a point such that
The solution set of the VIP (1.1) defined by C and A is denoted by . The theory of variational inequalities is a well established subject in nonlinear analysis and optimization. For different aspects of variational inequalities and their generalizations, we refer to [1–3] and the references therein. Several solution methods for solving different kinds of variational inequality have appeared in literature. Korpelevich’s extragradient method [4] is one of them. It is further studied in [5].
Let be a bifunction. The equilibrium problem (EP) is to find such that
The set of solutions of EP is denoted by . It is a unified model of several problems, namely, variational inequalities, Nash equilibrium problems, optimization problems, saddle point problems, etc. For further details of EP, we refer to [6, 7] and the references therein.
Let be a real-valued function and be a nonlinear mapping. The generalized mixed equilibrium problem (GMEP) [8] is to find such that
We denote the set of solutions of GMEP (1.3) by . The GMEP (1.3) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, Nash equilibrium problems in noncooperative games, and others. The GMEP is further considered and studied in [9, 10] and the references therein.
When , then GMEP (1.3) reduces to the following mixed equilibrium problem (MEP): find such that
The set of solutions of MEP is denoted by .
The common assumptions on the bifunction to study GMEP (1.3) or EP (1.2) are the following:
(A1) for all ;
(A2) Θ is monotone, i.e., for any ;
(A3) Θ is upper-hemicontinuous, i.e., for each ,
(A4) is convex and lower semicontinuous for each ;
We use the assumption that the function is a lower semicontinuous and convex function with restriction (B1) or (B2), where
(B1) for each and , there exists 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 ,
Some elementary conclusions related to MEP are given in the following result.
Proposition 1.1 [11]Let satisfy conditions (A1)-(A4) and be a proper lower semicontinuous and convex function such that either (B1) or (B2) holds. For and , define a mapping by
Then the following conclusions hold:
-
(i)
For each , is nonempty and single-valued;
-
(ii)
is firmly nonexpansive, that is, for any ,
-
(iii)
;
-
(iv)
is closed and convex;
-
(v)
, and .
Recently, Cai and Bu [12] considered a problem of finding a common element of the set of solutions of finitely many generalized mixed equilibrium problems, the set of solutions of finitely many variational inequalities mappings and the set of fixed points of an asymptotically k-strict pseudocontractive mapping in the intermediate sense [13] in a real Hilbert space. They proposed and analyzed an algorithm for finding such a solution. The weak convergence result for the proposed algorithm is also presented.
1.2 General system of variational inequalities
Let be two mappings. We consider the general system of variational inequalities (GSVI) of finding such that
where and are two constants. It was considered and studied in [5, 14, 15]. In particular, if , then the GSVI (1.4) reduces to the problem of finding such that
It is called a new system of variational inequalities (NSVI) [9]. It is worth to mention that the above system of two variational inequalities could be used to solve Nash equilibrium problem. For applications of system of variational inequalities to Nash equilibrium problems, we refer to [16–19] and the references therein. Further, if additionally, then the NSVI reduces to the classical VIP (1.1). Putting and , Ceng et al. [15] transformed the GSVI (1.4) into the following fixed point equation:
1.3 Hierarchical variational inequalities
A variational inequality problem defined over the set of fixed points of a nonexpansive mapping, is called a hierarchical variational inequality problem. Let be nonexpansive mappings. We denote by and the set of fixed points of T and S, respectively. If we replace C by in the formulation of VIP (1.1), then VIP (1.1) is defined by and A and it is called a hierarchical variational inequality problem.
Yao et al. [20] considered the hierarchical variational inequality problem (HVIP) in which the mapping A is replaced by the monotone mapping . They considered the following form of HVIP: find such that
The solution set of HVIP (1.7) is denoted by Λ. It is not hard to check that solving HVIP (1.7) is equivalent to the fixed point problem of the composite mapping , that is, find such that . They proposed and analyzed an iterative method for solving this kind of HTVI. For further details and a comprehensive survey on HVIP, we refer to [21] and the references therein.
1.4 Variational inclusions
Let be a single-valued mapping and be a set-valued mapping with . We consider the variational inclusion problem of finding a point such that
We denote by the solution set of the variational inclusion (1.8). In particular, if , then . If , then problem (1.8) becomes the inclusion problem introduced by Rockafellar [22]. It is well known that problem (1.8) 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.
1.5 Problem to be considered
In this paper, we introduce and study the following hierarchical variational inequality problem (HVIP) defined over a common set of solutions of finitely many GMEPs, finitely many variational inclusions, a general system of variational inequalities, and a fixed point of a strictly pseudocontractive mapping. Throughout the paper, we denote by M and N set of the positive integers.
Problem 1.1 Assume that
-
(i)
for , and are mappings;
-
(ii)
for each , is a bifunction satisfying conditions (A1)-(A4) and is a proper lower semicontinuous and convex function with restriction (B1) or (B2);
-
(iii)
for each and , is a maximal monotone mapping, and and are mappings;
-
(iv)
is a mapping and is a nonexpansive mapping;
-
(v)
.
Then the objective is to find such that
By combining Korpelevich’s extragradient method, the viscosity approximation method, the hybrid steepest-descent method [10], and Mann’s iteration method, we introduce and analyze a multistep hybrid extragradient algorithm for finding a solution of Problem 1.1. It is proven that under appropriate assumptions, the proposed algorithm converges strongly to a solution of GSVI (1.4) defined over a common set of solutions of finitely many generalized mixed equilibrium problems (GMEPs), finitely many variational inclusions and the fixed point problem of a strictly pseudocontractive mapping. In the meantime, we also prove the strong convergence of the proposed algorithm to a unique solution of Problem 1.1. The results obtained in this paper improve and extend the corresponding results announced by many others.
2 Preliminaries
Throughout this paper, we assume that H is 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 to indicate that the sequence converges weakly to x and to indicate that the sequence converges strongly to x. Moreover, we use to denote the weak ω-limit set of the sequence , i.e.,
Definition 2.1 A mapping is called
-
(i)
η-strongly monotone if there exists a constant such that
-
(ii)
ζ-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. It is obvious that if A is ζ-inverse-strongly monotone, then A is monotone and -Lipschitz continuous. Moreover, we also have, for all and ,
So, if , then is a nonexpansive mapping from C to H.
Definition 2.2 A mapping is said to be 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.
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 [23] 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 [[23], Proposition 2.1]
Let C be a nonempty closed convex subset of a real Hilbert space H and be a mapping.
-
(i)
If T is a ξ-strictly pseudocontractive mapping, then T satisfies the Lipschitzian condition
-
(ii)
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 .
-
(iii)
If T is a 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 [24]
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
Let C be a nonempty closed convex subset of a real Hilbert space H. We introduce some notations. Let λ be a number in and let . Associated 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 .
Remark 2.1 Since F is κ-Lipschitzian and η-strongly monotone on H, we get . Hence, whenever , we have
which implies
So, .
Finally, 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 a maximal monotone mapping.
Let be a monotone, k-Lipschitz-continuous mapping and let be the normal cone to C at , i.e.,
Define
Then is maximal monotone (see [22]) such that
Let be a maximal monotone mapping. Let be two positive numbers. Associated with R and λ, we define the resolvent operator by
where λ is a positive number.
The lemma shows that the resolvent operator is nonexpansive.
Lemma 2.3 [25]
is single-valued and firmly nonexpansive, i.e.,
Consequently, is nonexpansive and monotone.
Lemma 2.4 [26]
Let R be a maximal monotone mapping with . Then for any given , is a solution of problem (1.5) if and only if satisfies
Lemma 2.5 [27]
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 2.6 [26]
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.
Lemma 2.7 [28]
We have the resolvent identity
Remark 2.2 For , we have the following relation:
Indeed, whenever , utilizing Lemma 2.7 we deduce that
Similarly, whenever , we get
Combining the above two cases we conclude that (2.3) holds.
We need following fact and lemmas to establish the strong convergence of the sequences generated by the proposed algorithm.
Lemma 2.8 Let X be a real inner product space. Then
Lemma 2.9 Let H be a real Hilbert space. Then:
-
(a)
for all ;
-
(b)
for all and with ;
-
(c)
If is a sequence in H such that , it follows that
Lemma 2.10 (Demiclosedness principle [29])
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.
Lemma 2.11 [[30], Lemma 3.1]
is a contraction provided ; that is,
where .
Remark 2.3 (a) Since F is κ-Lipschitzian and η-strongly monotone on H, we get . Hence, whenever , we have
which implies
Therefore, .
(b) In Lemma 2.11, put and . Then we know that , and
Lemma 2.12 [30]
Let be a sequence of nonnegative numbers satisfying the conditions
where and are sequences of real numbers such that
-
(a)
and , or equivalently,
-
(b)
, or .
Then .
3 Algorithms and convergence results
In this section, we will introduce and analyze a multistep hybrid extragradient algorithm for finding a solution of the HVIP (1.9) (over the fixed point set of a strictly pseudocontractive mapping) with constraints of several problems: GSVI (1.4), finitely many GMEPs, and finitely many variational inclusions in a real Hilbert space. This algorithm is based on Korpelevich’s extragradient method, the viscosity approximation method, the hybrid steepest-descent method [10], and Mann’s iteration method. We prove the strong convergence of the proposed algorithm to a unique solution of HVIP (1.9) under suitable conditions.
We propose the following algorithm to compute the approximate solution of Problem 1.1.
Algorithm 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H. 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 , , let be a maximal monotone mapping, and and be -inverse-strongly monotone and -inverse-strongly monotone, respectively. Let be a ξ-strictly pseudocontractive mapping, is a nonexpansive mapping and be a ρ-contraction with coefficient . For each , let be -inverse-strongly monotone and be κ-Lipschitzian and η-strongly monotone with positive constants such that and where . Assume that . Let , , , and where and . For arbitrarily given , let be a sequence generated by
where with for .
Theorem 3.1 In addition to the assumption in Algorithm 3.1, suppose that
-
(i)
, and ;
-
(ii)
, and ;
-
(iii)
and ;
-
(iv)
and for and ;
-
(v)
and for all ;
-
(vi)
and .
Then
-
(a)
;
-
(b)
;
-
(c)
converges strongly to a point , which is a unique solution of HVIP (1.9), that is,
Proof First of all, observe that
and
Since and , we know that and hence the mapping is -strongly monotone. Moreover, it is clear that the mapping is -Lipschitzian. Thus, there exists a unique solution in Ω 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 .
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 1.1(ii), we have
Utilizing (2.1) and Lemma 2.3, we have
Combining (3.2) and (3.3), we have
Since , is -inverse-strongly monotone for , and for , we deduce that, for any ,
(This shows that is a nonexpansive mapping.) Since for all and T is ξ-strictly pseudocontractive, utilizing Lemma 2.2, we obtain from (3.1), (3.4), and (3.5)
Utilizing Lemma 2.11, we deduce from (3.1), (3.6), and that for all
By induction, we get
Thus, is bounded 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 1.1(ii), (v), we deduce that
where is a constant such that for each
Furthermore, we define for all . It follows that
Since for all , utilizing Lemma 2.2, we have
Hence it follows from (3.7)-(3.10) that
In the meantime, simple calculation shows that
So, it follows from (3.11) 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 (3.12) and implies that
where for some . Consequently,
where for some . From conditions (i)-(iv) it follows that and
Thus, utilizing Lemma 2.12, we immediately conclude that
So, from it follows that
Step 3. We prove that , , and .
Indeed, utilizing Lemmas 2.8 and 2.9(b), from (3.1), (3.4)-(3.5), and we deduce that
and hence
which together with , immediately yields
Since , , , and is bounded, we have
Observe that
and
for and . Combining (3.15), (3.18), and (3.19), we get
which immediately leads to
Since , , , , , , and , are bounded sequences, we have
for all and .
Furthermore, by Proposition 1.1(ii) and Lemma 2.9(a), we have
which implies that
By Lemma 2.9(a) and Lemma 2.3, we obtain
which immediately leads to
Combining (3.15) and (3.22), we conclude
which yields
Since , , , and , , and are bounded sequences, we deduce from (3.20) and that
Also, combining (3.3), (3.15), and (3.21), we deduce
which yields
Since , for , and , are bounded sequences, we deduce from (3.20) and that
Hence from (3.23) and (3.24), we get
and
respectively. Thus, from (3.25) and (3.26), we obtain
On the other hand, for simplicity, we write , , and for all . Then
We now show that , i.e., . As a matter of fact, for , it follows from (3.4), (3.5), and (3.15) that
which immediately yields
Since , , , , and , are bounded sequences, we have
Also, in terms of the firm nonexpansivity of and the -inverse-strong monotonicity of for , we obtain from , ,
and
Thus, we have
and
Consequently, from (3.4), (3.28), and (3.30), it follows that
which hence leads to
Since , , , and , , , are bounded sequences, we obtain from (3.29)
Furthermore, from (3.4), (3.28), and (3.31), it follows that
which hence yields
Since , , , and , , , are bounded sequences, we obtain from (3.29)
Note that
Hence from (3.32) and (3.33), we get
Also, observe that
Hence we find that
So, from , (3.17), (3.27), and (3.34), it follows that
In addition, noticing that
we know from (3.34) and (3.35) that
Step 4. We prove that .
Indeed, since H is reflexive and is bounded, there exists at least a weak convergence subsequence of . Hence it is well known that . Now, take an arbitrary . Then there exists a subsequence of such that . From (3.23)-(3.25), and (3.27), we have , , , and , where and . Utilizing Lemma 2.1(ii), we deduce from and (3.36) that . In the meantime, utilizing Lemma 2.10, we obtain from and (3.34) . 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 2.6 that is maximal monotone. Let , i.e., . 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 , i.e., . Therefore, . Next we prove that . Since , , , we have
By (A2), we have
Let for all and . This implies that . Then we have
By (3.24), we have as . Furthermore, by the monotonicity of , we obtain . Then by (A4) we obtain
Utilizing (A1), (A4), and (3.26), we obtain
and hence
Letting , we have, for each ,
This implies that , and hence, . Thus, . Consequently, . This shows that .
Step 5. We prove that .
Indeed, take an arbitrary . Then there exists a subsequence of such that . Utilizing (3.16), we obtain, for all ,
which implies that
Since , and
from (3.39), we conclude that
that is,
Since is -strongly monotone and -Lipschitz continuous, by Minty’s lemma [29] we know that (3.40) is equivalent to the VIP
This shows that . Taking into account , we know that . Thus, ; that is, .
Next we prove that . As a matter of fact, utilizing (3.16) with , we get
Since and (due to ), we deduce that , and
Therefore, applying Lemma 2.12 to (3.42) we infer that . This completes the proof. □
Putting , , and an inverse-strongly monotone mapping on C in Algorithm 3.1, we have the following algorithm.
Algorithm 3.2 Let C be a nonempty closed convex subset of a real Hilbert space H, be a bifunction from satisfying conditions (A1)-(A4), be a proper lower semicontinuous and convex function with restriction (B1) or (B2), and be -inverse-strongly monotone. Let be a maximal monotone mapping, be -inverse-strongly monotone, be a ξ-strictly pseudocontractive mapping, is a nonexpansive mapping and be a ρ-contraction with coefficient . Let be ζ-inverse-strongly monotone, and be κ-Lipschitzian and η-strongly monotone with positive constants such that and where . Assume that . Let , , , and . For arbitrarily given , let be a sequence generated by
where .
From Theorem 3.1, we have following result.
Corollary 3.1 In addition to assumption of Algorithm 3.2, suppose that
-
(i)
, and ;
-
(ii)
, and ;
-
(iii)
and ;
-
(iv)
and ;
-
(v)
and for all ;
-
(vi)
and .
Then
-
(a)
;
-
(b)
;
-
(c)
converges strongly to a point , which is a unique solution of HVIP (1.9), i.e.,
Proof Since and a ζ-inverse-strongly monotone mapping on C, it is easy to see that . Thus, in terms of Theorem 3.1, we derive the desired result. □
Putting , where is a -strictly pseudocontractive mapping on C, in Algorithm 3.2, we obtain the following algorithm.
Algorithm 3.3 Let C be a nonempty closed convex subset of a real Hilbert space H, be a bifunction satisfying conditions (A1)-(A4), be a proper lower semicontinuous and convex function with restriction (B1) or (B2), and be -inverse-strongly monotone. Let be a maximal monotone mapping, be -inverse-strongly monotone, be a ξ-strictly pseudocontractive mapping, is a nonexpansive mapping and be a ρ-contraction with coefficient . Let be a -strictly pseudocontractive mapping, and be κ-Lipschitzian and η-strongly monotone with positive constants such that and where . Assume that . Let , , , and . For arbitrarily given , let be a sequence generated by
where .
Corollary 3.2 In addition to assumption of Algorithm 3.3, suppose that
-
(i)
, and ;
-
(ii)
, and ;
-
(iii)
and ;
-
(iv)
and ;
-
(v)
and for all ;
-
(vi)
and .
Then
-
(a)
;
-
(b)
;
-
(c)
converges strongly to a point , which is a unique solution of HVIP (1.9), i.e.,
Proof Since is a -strictly pseudocontractive mapping on C, it is well known that for constant ,
It is clear that in this case the mapping is -inverse-strongly monotone. Moreover, we have, for ,
Now let us show . In fact, we have, for ,
Consequently,
Therefore, by Corollary 3.1, we derive the desired result. □
Remark 3.1 Our results generalize and improve results in [12, 20] and the references therein.
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This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. Therefore, the authors acknowledge with thanks DSR, for technical and financial support.
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Ceng, LC., Latif, A., Ansari, Q.H. et al. Hybrid extragradient method for hierarchical variational inequalities. Fixed Point Theory Appl 2014, 222 (2014). https://doi.org/10.1186/1687-1812-2014-222
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DOI: https://doi.org/10.1186/1687-1812-2014-222