Relaxed and hybrid viscosity methods for general system of variational inequalities with split feasibility problem constraint
© Ceng and Yao; licensee Springer 2013
Received: 21 November 2012
Accepted: 31 January 2013
Published: 27 February 2013
In this paper, we propose and analyze some relaxed and hybrid viscosity iterative algorithms for finding a common element of the solution set Ξ of a general system of variational inequalities, the solution set Γ of a split feasibility problem and the fixed point set of a strictly pseudocontractive mapping S in the setting of infinite-dimensional Hilbert spaces. We prove that the sequences generated by the proposed algorithms converge strongly to an element of under mild conditions.
AMS Subject Classification:49J40, 47H05, 47H19.
The solution set of VIP (1.1) is denoted by . Variational inequality theory has been studied quite extensively and has emerged as an important tool in the study of a wide class of obstacle, unilateral, free, moving and equilibrium problems. It is now well known that variational inequalities are equivalent to fixed point problems, the origin of which can be traced back to Lions and Stampacchia . This alternative formulation has been used to suggest and analyze a projection iterative method for solving variational inequalities under the conditions that the involved operator must be strongly monotone and Lipschitz continuous. Related to the variational inequalities, we have the problem of finding fixed points of nonexpansive mappings or strict pseudo-contraction mappings, which is the current interest in functional analysis. Several people considered a unified approach to solve variational inequality problems and fixed point problems; see, for example, [2–11] and the references therein.
where is the metric projection of ℋ onto C, , is a sequence in and is a sequence in . They showed that if , then the sequence converges weakly to some . Nadezhkina and Takahashi  and Zeng and Yao  proposed extragradient methods motivated by Korpelevich  for finding a common element of the fixed point set of a nonexpansive mapping and the solution set of a variational inequality problem. Further, these iterative methods were extended in  to develop a new iterative method for finding elements in .
which is called a general system of variational inequalities (GSVI), where and are two constants. The set of solutions of problem (1.3) is denoted by . In particular, if , then problem (1.3) reduces to the new system of variational inequalities (NSVI) introduced and studied by Verma . Further, if additionally, then the NSVI reduces to VIP (1.1).
Recently, Ceng, Wang and Yao  transformed problem (1.3) into a fixed point problem in the following way.
Lemma 1.1 (see )
In particular, if the mapping is -inverse strongly monotone for , then the mapping G is nonexpansive provided for .
Utilizing Lemma 1.1, they introduced and studied a relaxed extragradient method for solving GSVI (1.3). Throughout this paper, the set of fixed points of the mapping G is denoted by Ξ. Based on the extragradient method and viscosity approximation method, Yao et al.  proposed and analyzed a relaxed extragradient method for finding a common solution of GSVI (1.3) and a fixed point problem of a strictly pseudo-contractive mapping .
Theorem YLK (see [, Theorem 3.2])
and for all ;
Then the sequence generated by (1.4) converges strongly to and is a solution of GSVI (1.3), where .
Subsequently, Ceng, Guu and Yao  further presented and analyzed an iterative scheme for finding a common element of the solution set of VIP (1.1), the solution set of GSVI (1.3) and the fixed point set of a strictly pseudo-contractive mapping .
Theorem CGY (see [, Theorem 3.1])
and for all ;
Then the sequence generated by (1.5) converges strongly to and is a solution of GSVI (1.3), where .
where and denotes the family of all bounded linear operators from to .
It has been extensively investigated in the literature using the projected Landweber iterative method . Comparatively, the SFP has received much less attention so far due to the complexity resulting from the set Q. Therefore, whether various versions of the projected Landweber iterative method  can be extended to solve the SFP remains an interesting open topic. For example, it is not clear whether the dual approach to (1.7) of  can be extended to the SFP. The original algorithm given in  involves the computation of the inverse (assuming the existence of the inverse of A) and thus has not become popular. A seemingly more popular algorithm that solves the SFP is the CQ algorithm of Byrne [19, 24] which is found to be a gradient-projection method (GPM) in convex minimization. It is also a special case of the proximal forward-backward splitting method . The CQ algorithm only involves the computation of the projections and onto the sets C and Q, respectively, and is therefore implementable in the case where and have closed-form expressions, for example, C and Q are closed balls or half-spaces. However, it remains a challenge how to implement the CQ algorithm in the case where the projections and/or fail to have closed-form expressions, though theoretically we can prove the (weak) convergence of the algorithm.
Very recently, Xu  gave a continuation of the study on the CQ algorithm and its convergence. He applied Mann’s algorithm to the SFP and proposed an averaged CQ algorithm which was proved to be weakly convergent to a solution of the SFP. He also established the strong convergence result, which shows that the minimum-norm solution can be obtained.
Furthermore, Korpelevich  introduced the so-called extragradient method for finding a solution of a saddle point problem. She proved that the sequences generated by the proposed iterative algorithm converge to a solution of the saddle point problem.
where is the regularization parameter. The regularized minimization (1.9) has a unique solution which is denoted by . The following results are easy to prove.
Proposition 1.1 (see [, Proposition 3.1])
solves the SFP;
- (ii)solves the fixed point equation
where , and is the adjoint of A;
- (iii)solves the variational inequality problem (VIP) of finding such that(1.10)
for all , where and denote the set of fixed points of and the solution set of VIP (1.10), respectively.
Proposition 1.2 (see )
- (i)the gradient
is -Lipschitz continuous and α-strongly monotone;
- (ii)the mapping is a contraction with coefficient
if the SFP is consistent, then the strong exists and is the minimum-norm solution of the SFP.
Very recently, by combining the regularization method and extragradient method due to Nadezhkina and Takahashi , Ceng, Ansari and Yao  proposed an extragradient algorithm with regularization and proved that the sequences generated by the proposed algorithm converge weakly to an element of , where is a nonexpansive mapping.
Theorem CAY (see [, Theorem 3.1])
where , for some and for some . Then both the sequences and converge weakly to an element .
Motivated and inspired by the research going on in this area, we propose and analyze some relaxed and hybrid viscosity iterative algorithms for finding a common element of the solution set Ξ of GSVI (1.3), the solution set Γ of SFP (1.6) and the fixed point set of a strictly pseudocontractive mapping . These iterative algorithms are based on the regularization method, the viscosity approximation method, the relaxed method in  and the hybrid method in . Furthermore, it is proven that the sequences generated by the proposed algorithms converge strongly to an element of under mild conditions.
Observe that both [, Theorem 5.7] and [, Theorem 3.1] are weak convergence results for solving the SFP and that our problem of finding an element of is more general than the corresponding ones in [, Theorem 5.7] and [, Theorem 3.1], respectively. Hence there is no doubt that our strong convergence results are very interesting and quite valuable. It is worth emphasizing that our relaxed and hybrid viscosity iterative algorithms involve a ρ-contractive self-mapping Q, a k-strictly pseudo-contractive self-mapping S and several parameter sequences, they are more flexible, more advantageous and more subtle than the corresponding ones in [, Theorem 5.7] and [, Theorem 3.1], respectively. Furthermore, relaxed extragradient iterative scheme (1.4) and hybrid extragradient iterative scheme (1.5) are extended to develop our relaxed viscosity iterative algorithms and hybrid viscosity iterative algorithms, respectively. In our strong convergence results, the relaxed and hybrid viscosity iterative algorithms drop the requirement of boundedness for the domain in which various mappings are defined; see, e.g., Yao et al. [, Theorem 3.2]. Therefore, our results represent the modification, supplementation, extension and improvement of [, Theorem 5.7], [, Theorem 3.1], [, Theorem 3.1] and [, Theorem 3.2] to a great extent.
Let ℋ be a real Hilbert space, whose inner product and norm are denoted by and , respectively. Let K be a nonempty, closed and convex subset of ℋ. Now, we present some known results and definitions which will be used in the sequel.
The following properties of projections are useful and pertinent to our purpose.
Proposition 2.1 (see )
, , which hence implies that is nonexpansive and monotone.
- (a)nonexpansive if
- (b)firmly nonexpansive if is nonexpansive, or equivalently,
where is nonexpansive; projections are firmly nonexpansive.
- (a)T is said to be monotone if
- (b)Given a number , T is said to be β-strongly monotone if
- (c)Given a number , T is said to be ν-inverse strongly monotone (ν-ism) if
It can be easily seen that if S is nonexpansive, then is monotone. It is also easy to see that a projection is 1-ism.
Inverse strongly monotone (also referred to as co-coercive) operators have been applied widely to solving practical problems in various fields, for instance, in traffic assignment problems; see, e.g., [36, 37].
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 maps.
Proposition 2.2 (see )
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.
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 .
- (v)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, .
This immediately implies that if S is a k-strictly pseudo-contractive mapping, then is -inverse strongly monotone; for further details, we refer to  and the references therein. It is well known that the class of strict pseudo-contractions strictly includes the class of nonexpansive mappings.
In order to prove the main results of this paper, the following lemmas will be required.
Lemma 2.1 (see )
Let and be bounded sequences in a Banach space X and let be a sequence in with . Suppose for all integers and . Then .
Lemma 2.2 (see [, Proposition 2.1])
- (i)If S is a k-strict pseudo-contractive mapping, then S satisfies the Lipschitz condition
If S is a k-strict pseudo-contractive mapping, then the mapping is semiclosed at 0, that is, if is a sequence in C such that weakly and strongly, then .
If S is k-(quasi-)strict pseudo-contraction, then the fixed point set of S is closed and convex so that the projection is well defined.
The following lemma plays a key role in proving strong convergence of the sequences generated by our algorithms.
Lemma 2.3 (see )
either or ;
, where , .
Lemma 2.4 (see )
The following lemma is an immediate consequence of an inner product.
It is known that in this case the mapping T is maximal monotone and if and only if ; for further details, we refer to  and the references therein.
3 Relaxed viscosity methods and their convergence criteria
In this section, we propose and analyze the following relaxed viscosity iterative algorithms for finding a common element of the solution set of GSVI (1.3), the solution set of SFP (1.6) and the fixed point set of a strictly pseudo-contractive mapping .
Next, we first give the strong convergence criteria of the sequences generated by Algorithm 3.1.
and for all ;
Then the sequences , , converge strongly to the same point if and only if . Furthermore, is a solution of GSVI (1.3), where .
Proof First, taking into account , without loss of generality, we may assume that for some .
This immediately implies that is nonexpansive for all .
Next, we divide the remainder of the proof into several steps.
Step 1. is bounded.
By induction, we conclude that (3.6) is valid. Hence, is bounded. Since , , and are Lipschitz continuous, it is easy to see that , , and are bounded, where for all .
Step 2. .
Step 3. and , where .
Step 4. .
Step 5. , where .
Step 6. .