# Convergence Theorems for the Variational Inequality Problems and Split Feasibility Problems in Hilbert Spaces

Authors: Panisa Lohawech; Anchalee Kaewcharoen; Ali Farajzadeh
Journal: International Journal of Mathematics and Mathematical Sciences
 (2021)

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## Abstract

In this paper, we establish an iterative algorithm by combining Yamada’s hybrid steepest descent method and Wang’s algorithm for finding the common solutions of variational inequality problems and split feasibility problems. The strong convergence of the sequence generated by our suggested iterative algorithm to such a common solution is proved in the setting of Hilbert spaces under some suitable assumptions imposed on the parameters. Moreover, we propose iterative algorithms for finding the common solutions of variational inequality problems and multiple-sets split feasibility problems. Finally, we also give numerical examples for illustrating our algorithms.


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## Body

## 1. Introduction

In 2005, Censor et al. [1] introduced the multiple-sets split feasibility problem (MSSFP), which is formulated as follows:(1)findx∈∩i=1NCisuch thatAx∈∩j=1MQj,where Cii=1,2,…,N and Qjj=1,2,…,M are nonempty closed convex subsets of Hilbert spaces H1 and H2, respectively, and A:H1⟶H2 is a bounded linear mapping. Denote by Ω the set of solutions of MSSFP (1). Many iterative algorithms have been developed to solve the MSSFP (see [1–3]). Moreover, it arises in many fields in the real world, such as inverse problem of intensity-modulated radiation therapy, image reconstruction, and signal processing (see [1, 4, 5] and the references therein).WhenN=M=1, the MSSFP is known as the split feasibility problem (SFP); it was first introduced by Censor and Elfving [5], which is formulated as follows:(2)findx∈Csuch thatAx∈Q.Denote byΓ the set of solutions of SFP (2).Assume that the SFP is consistent (i.e., (2) has a solution). It is well known that x∈C solves (2) if and only if it solves the fixed point equation(3)x=Tx,T=PCI−γA∗I−PQA,x∈C,where γ is a positive constant, A∗ is the adjoint operator of A, and PC and PQ are the metric projections of H1 and H2 onto C and Q, respectively (for more details, see [6]).The variational inequality problem (VIP) was introduced by Stampacchia [7], which is finding a point(4)x∗∈Csuch thatFx∗,x−x∗≥0,for allx∈C,where C is a nonempty closed convex subset of a Hilbert space H and F:C⟶H is a mapping. The ideas of the VIP are being applied in many fields including mechanics, nonlinear programming, game theory, and economic equilibrium (see [8–12]).In [13], we see that x∈C solves (4) if and only if it solves the fixed point equation(5)x=Sx,S=PCI−μF,x∈C.Moreover, it is well known that ifF is k-Lipschitz continuous and η-strongly monotone, then VIP (4) has a unique solution (see, e.g., [14]).Since SFP and VIP include some special cases (see [15, 16]), indeed, convex linear inverse problem and split equality problem are special cases of SFP, and zero point problem and minimization problem are special cases of VIP. Jung [17] studied the common solution of variational inequality problem and split feasibility problem: find a point(6)x∗∈Γsuch thatFx∗,x−x∗≥0,for allx∈Γ,where Γ is the solution set of SFP (2) and F:H1⟶H1 is an η-strongly monotone and k-Lipschitz continuous mapping. After that, for solving problem (6), Buong [2] considered the following algorithms, which were proposed in [14, 18], respectively:(7)xn+1=I−tnμFTxn,n≥0,(8)xn+1=αnxn+1−αnI−tnμFTxn,n≥0,where T=PCI−γA∗I−PQA, and under the following conditions:(C1)tn∈0,1,tn⟶0 as n⟶∞ and ∑n=1∞tn=∞.(C2)0<liminfn⟶∞αn≤limsupn⟶∞αn<1.Moreover, Buong [2] considered the sequence xn that is generated by the following algorithm, which is weakly convergent to a solution of MSSFP (1):(9)xn+1=P1I−γA∗I−P2Axn,where P1=PC1,…,PCN and P2=PQ1,…,PQM or P1=∑i=1NαiPCi and P2=∑j=1MβjPQj in which αi and βj, for 1≤i≤N and 1≤j≤M, are positive real numbers such that ∑i=1Nαi=∑j=1Mβj=1.Motivated by the aforementioned works, we establish an iterative algorithm by combining algorithms (7) and (8) for finding the solution of problem (6) and prove the strong convergence of the sequence generated by our iterative algorithm to the solution of problem (6) in the setting of Hilbert spaces. Moreover, we propose iterative algorithms for solving the common solutions of variational inequality problems and multiple-sets split feasibility problems. Finally, we also give numerical examples for illustrating our algorithms.


## 2. Preliminaries

In order to solve our results, we now recall the following definitions and preliminary results that will be used in the sequel. Throughout this section, letC be a nonempty closed convex subset of a real Hilbert space H with inner product ⋅,⋅ and norm ⋅.Definition 1.
A mappingT:H⟶H is called(i)
k-Lipschitz continuous, if Tx−Ty≤kx−y for all x,y∈H, where k is a positive number.(ii)
Nonexpansive, if (i) holds withk=1.(iii)
η-strongly monotone, if ηx−y2≤Tx−Ty,x−y for all x,y∈H, where η is a positive number.(iv)
Firmly nonexpansive, ifTx−Ty2≤Tx−Ty,x−y for all x,y∈H.(v)
α-Averaged, if T=1−αI+αN for some fixed α∈0,1 and a nonexpansive mapping N.
In [5], we know that the metric projection PC:H⟶C is firmly nonexpansive and 1/2-averaged.
We collect some basic properties of averaged mappings in the following results.Lemma 1 (see [16]).
We have(i)
The composite of finitely many averaged mappings is averaged. In particular, ifTi is αi-averaged, where αi∈0,1 for i=1,2, then the composite T1T2 is α-averaged, where α=α1+α2−α1α2.(ii)
If the mappingsTii=1N are averaged and have a common fixed point, then(10)FixT1,T2,…,TN=∩i=1NFixTi.Proposition 1 (see [19]).
LetD be a nonempty subset of H, m≥2 be an integer, and ϕ;0,1m⟶0,1 be defined by(11)ϕα1,…,αm=11+1/∑i=1mαi/1−αi.For everyi∈1,…,m, let αi∈0,1 and Ti:D⟶D be αi-averaged. Then, T=T1,…,Tm is α-averaged, where α=ϕα1,…,αm.The following properties of the nonexpansive mappings are very convenient and helpful to use.Lemma 2 (see [20]).
Assume thatH1 and H2 are Hilbert spaces. Let A:H1⟶H2 be a linear bounded mapping such that A≠0 and let T:H2⟶H2 be a nonexpansive mapping. Then, for 0≤γ<1/A2,I−γA∗I−TA is γA2-averaged.Proposition 2 (see [19]).
LetC be a nonempty subset of H, and let Tii∈I be a finite family of nonexpansive mappings from C to H. Assume that α˜ii∈I⊂0,1 and δii∈I⊂0,1 such that ∑i∈Iδi=1. Suppose that, for every i∈I, Ti is α˜i-averaged; then, T=∑i∈IδiTi is α-averaged, where α=∑i∈Iδiα˜i.
The following results play a crucial role in the next section.Lemma 3 (see [14]).
Lett be a real number in 0,1. Let F:H⟶H be an η-strongly monotone and k-Lipschitz continuous mapping. The mapping I−tμF, for each fixed point μ∈0,2η/k2, is contractive with constant 1−tτ, i.e.,(12)I−tμFx−I−tμFy≤1−tτx−y,where τ=1−1−μ2η−μk2∈0,1.Theorem 1 (see [21]).
LetF be a k-Lipschitz continuous and η-strongly monotone self-mapping of H. Assume that Tii=1N is a finite family of nonexpansive mappings from H to H such that C=∩i=1NFixTi≠∅. Then, the sequence xn defined by the following algorithm converges strongly to the unique solution x∗ of the variational inequality (4):(13)xn+1=1−βn0xn+βn0I−tnμFTNn,TN−1n,…,T1nxn,n≥0,where μ∈0,2η/k2, Tin:=1−βniI+βniTi, for i=1,…,N, and under the following conditions:(i)
tn∈0,1,tn⟶0 as n⟶∞ and ∑n=0∞tn=∞.(ii)
βni∈α,β, for some α,β∈0,1, and βn+1i−βni⟶0 as n⟶∞,i=0,…,N.Theorem 2 (see [22]).
LetF, C, μ,βnii=1N, tn, and Tii=1N be as in Theorem 1. Then, the sequence xn defined by the following algorithm:(14)xn+1=I−tnμFTNn,TN−1n,…,T1nxn,n≥1,converges strongly to the unique solution x∗ of variational inequality (4).


## 3. Main Results

In this section, we consider the following iterative algorithm by combining Yamada’s hybrid steepest descent method [14] and Wang’s algorithm [18] for solving problem (6):(15)yn=1−αnxn+αnI−tnμFTxn,xn+1=I−tnμFTyn,∀n≥1,where T=PCI−γA∗I−PQA. If we set αn=0 for n∈ℕ, then (15) is reduced to (7) studied by Buong [2]. On the other hand, in the Numerical Example section, we present the example illustrating that the two-step method (15) is more efficient that the one-step method (8) studied by Buong [2] and in terms of the two-step method (15) the generated sequence has the less number of iterations and converges faster than the sequence generated by the one-step method (8).Throughout our results, unless otherwise stated, we assume thatH1 and H2 are two real Hilbert spaces and A:H1⟶H2 is a linear bounded mapping. Let F be an η-strongly monotone and k-Lipschitz continuous mapping on H1 with some positive constants η and k. Assume that μ∈0,2η/k2 is a fixed number.Theorem 3.
LetC and Q be two closed convex subsets in H1 and H2, respectively. Then, as n⟶∞, the sequence xn defined by (15), where the sequences tn and αn satisfy conditions (C1) and (C2), respectively, converges strongly to the solution of (6).Proof.
From Lemma2, we have that I−γA∗I−PQA is γA2-averaged. Since T=PCI−γA∗I−PQA, by Lemma 1 (i), we get that T is λ-averaged where λ=1+γA2/2. Moreover, we obtain that z∈Γ if and only if z∈FixT. It follows from Definition 1 (iv) that T=1−λI+λS, where S is nonexpansive. Then, iterative algorithm (15) can be rewritten as follows:(16)xn+1=I−tnμFTT˜xn,where T˜=1−αnI+αnI−tnμFT and T=1−λI+λS. Since 1−λI+λS and I−tnμF are nonexpansive, then I−tnμFT is also nonexpansive. Therefore, the strong convergence of (15) to the element x∗ in the solution set of (6) follows by Theorem 2.
In [23], Miao and Li showed the weak convergence results of the sequence xn converging to the element of FixT where xn is generated by the following algorithm:(17)yn=1−βnxn+βnI−tnμFTxn,xn+1=1−αnxn+αnI−tnμFTyn,∀n≥1,which tn satisfies condition (C3) ∑n=1∞tn<+∞. Next, we will show the strong convergence for (17) where tn satisfies condition C1.Theorem 4.
LetC and Q be two closed convex subsets in H1 and H2, respectively. Then, as n⟶∞, the sequence xn defined by (17), where the sequence tn satisfies condition (C1) and βn and αn satisfy condition (C2), converges strongly to the solution of (6).Proof.
In the proof of Theorem3, one can rewrite iterative algorithm (17) as follows:(18)xn+1=1−αnxn+αnI−tnμFTT˜xn,where T˜=1−βnI+βnI−tnμFT and T=1−λI+λS. Since I−tnμFT is nonexpansive, then the strong convergence of (17) to the element x∗ in the solution set of (6) follows by Theorem 1.
Moreover, we obtain the following results which are solving the common solution of variational inequality problem and multiple-sets split feasibility problem, i.e., find a point(19)x∗∈Ωsuch thatFx∗,x−x∗≥0,for allx∈Ω,where Ω is a solution set of (1), and F:H1⟶H1 is an η-strongly monotone and k-Lipschitz continuous mapping. This problem has been studied in [2].Theorem 5.
LetCii=1N and Qjj=1M be two finite families of closed convex subsets in H1 and H2, respectively. Assume that γ∈0,1/A2, tn and αn satisfy conditions (C1) and (C2), respectively, and the parameters δn and ζn satisfy the following conditions:(a)
δi>0 for 1≤i≤N such that ∑i=1Nδi=1.(b)
ζj>0 for 1≤j≤M such that ∑i=1Nζj=1.Then, asn⟶∞, the sequence xn, defined by(20)yn=1−αnxn+αnI−tnμFP1I−γAI−P2Axn,xn+1=I−tnμFP1I−γAI−P2Ayn,∀n≥1,with one of the following cases:(A1)P1=PC1,…,PCN and P2=PQ1,…,PQM(A2)P1=∑i=1NδiPCi and P2=∑j=1MζjPQj(A3)P1=PC1,…,PCN and P2=∑j=1MζjPQj(A4)P1=∑i=1NδiPCi and P2=PQ1,…,PQM,converges to the element x∗ in the solution set of (19).Proof.
LetT=P1I−γA∗I−P2A. We will show that T is averaged.
In the case of (A1),P1=PC1,…,PCN and P2=PQ1,…,PQM. Since PCi is 1/2-averaged for all i=1,…,N, by Proposition 1, we get that P1 is λ1-averaged, where λ1=N/N+1. Similarly, we have that P2 is also averaged and so P2 is nonexpansive. By using Lemma 2, we deduce that I−γA∗I−P2A is λ2-averaged, where λ2=γA2. It follows from Lemma 1 (i) that T is λ-averaged with λ=N/N+1+γA2−N/N+1γA2.
IfP1=∑i=1NδiPCi and P2=∑j=1MζjPQj, then by using Proposition 2 and condition (a), we obtain that P1 is 1/2-averaged. From condition (b) and taking into account that PQj is nonexpansive, for all j=1,…,M, we have that P2 is also nonexpansive. It follows from Lemma 2 that I−γA∗I−P2A is γA2-averaged. Thus, T is λ-averaged with λ=1+γA2/2.
Cases (A3) and (A4) are similar. This implies thatT:=1−λI+λS, where S is nonexpansive. Moreover, by Lemma 1, we get that(21)FixT=FixP1∩FixI−γA∗I−P2A=FixP1∩A−1FixP2=∩i=1NCi∩A−1∩j=1MQj=Ω.
Then, iterative algorithm (20) can be rewritten as follows:(22)xn+1=I−tnμFTT˜xn,where T˜=1−αnI+αnI−tnμFT and T=1−λI+λS. Since 1−λI+λS and I−tnμF are nonexpansive, then I−tnμFT is nonexpansive. Thus, the strong convergence of (20) to the element x∗ in the solution set of (19) follows by Theorem 2.Theorem 6.
LetCii=1N, Qjj=1M,γ,tn,δn, and ζn be as in Theorem 5. Then, as n⟶∞, the sequence xn, defined by(23)yn=1−αnxn+αnI−tnμFP1I−γAI−P2Axn,xn+1=1−βnxn+βnI−tnμFP1I−γAI−P2Ayn,∀n≥1,with one of the cases (A1)–(A4), converges strongly to an element in the solution set of (19).Proof.
In the proof of Theorem5, one can rewrite iterative algorithm (23) as follows:(24)xn+1=1−αnxn+αnI−tnμFTT˜xn,where T˜=1−βnI+βnI−tnμFT and T=1−λI+λS. Since I−tnμFT is nonexpansive, the strong convergence of (23) to the element x∗ in the solution set of (19) follows by Theorem 1.


## 4. Numerical Example

In this section, we present the numerical example comparing algorithm (8) which is given by Buong [2] and algorithm (15) (new method) to solve the following test problem in [2]: find an element x∗∈Ω such that(25)φx∗=minx∈Ωφx,Ω=Ci∩A−1Qj≠∅,where φ is a convex function, having a strongly monotone and Lipschitz continuous derivative φ′x on the Euclidian space En, C=∩i=1NCiandQ=∩j=1MQj where(26)Ci=x∈En:∑k=1nakixk≤bi,aki,bi∈−∞,+∞, for 1≤k≤n and 1≤i≤N,(27)Qj=y∈Em:∑l=1myl−alj2≤Rj2,Rj>0,alj∈−∞,+∞, for 1≤l≤m and 1≤j≤M, and A is an n×m-matrix.Example 1.
We consider test problem (25), where N=M=1, n=m=2, φx=1−ax2/2 for some fixed a∈0,1, and(28)A=1−102.
So, we have thatF:=φ′=1−aI is a k-Lipschitz continuous and η-strongly monotone mapping with k=η=1−a. For each algorithm, we set ai=1/i,−1,bi=0, for all i=1,…,N, and aj=1/j,0,Rj=1, for all j=1,…,M. Taking a=0.5,γ=0.3, the stopping criterion is defined by En=xn+1−xn<ε where ε=10−4 and 10−6. The numerical results are listed in Table 1 with different initial points x1, where n is the number of iterations and s is the CPU time in seconds. In Figures 1 and 2, we present the graphs illustrating the number of iterations for both methods using the stopping criterion defined as above with the different initial points shown in Table 1.Table 1
Computational results for Example1 with different methods.
10−410−6Initial pointnsns−2,1TBuong method294610.364595294620431.362283New method117840.241371117848123.4116791,3TBuong method306320.565431306334333.468210New method122520.324808122533625.570356Figure 1
The convergence behavior ofEn with the initial point −2,1T.Figure 2
The convergence behavior ofEn with the initial point 1,3T.Remark 1.
From the numerical analysis of our results in Table1 and Figures 1 and 2, we get that algorithm (15) (new method) has less number of iterations and faster convergence than algorithm (8) (Buong method).Example 2.
In this example, we consider algorithm (23) for solving test problem (25), where N=5 and M=4. Let Cii=1N, Qjj=1M, φ, a, and A be as in Example 1. In the numerical experiment, we take the stopping criterion En<10−4. The numerical results are listed in Table 2 with different cases of P1 and P2. In Figures 3 and 4, we present the graphs illustrating the number of iterations for all cases of P1 and P2 using the stopping criterion as above with the different initial points appeared in Table 2. Moreover, Table 3 shows the effect of different choices of γ.Table 2
Computational results for Example2 with different methods.
Initial pointA1A2A3A4−2,1Tn28577242642857724264s1.4912251.3550741.5344141.2825281,3Tn33407314383340731438s1.7468681.6930691.8168971.690618Figure 3
The convergence behavior ofEn with the initial point −2,1T.Figure 4
The convergence behavior ofEn with the initial point 1,3T.Table 3
Computational results for Example2 with different γ.
γ0.10.20.3−2,1Tn96751920028577s0.6695081.2451361.6667021,3Tn113112244733407s0.7645361.3726001.958486Remark 2.
We observe from the numerical analysis of Table2 that algorithm (23) has the fastest convergence when P1 and P2 satisfy (A4) and the slowest convergence when P1 and P2 satisfy (A3). Moreover, we require less iteration steps and CPU times for convergence when γ is chosen very small and close to zero.



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