# Solving a Split Feasibility Problem by the Strong Convergence of Two Projection Algorithms in Hilbert Spaces

Authors: Hasanen A. Hammad; Habib ur Rehman; Yaé Ulrich Gaba
Journal: Journal of Function Spaces
 (2021)

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## Abstract

The goal of this manuscript is to establish strong convergence theorems for inertial shrinking projection and CQ algorithms to solve a split convex feasibility problem in real Hilbert spaces. Finally, numerical examples were obtained to discuss the performance and effectiveness of our algorithms and compare the proposed algorithms with the previous shrinking projection, hybrid projection, and inertial forward-backward methods.


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## Body

## 1. Introduction

Assume thatY is a real HS defined on the induced norm . and the inner product .,.. Let C be a NCC subset of Y.The mappingT:C⟶C is called NE, if for all κ,ω∈C, the following inequality holds:
(1)Tκ−Tω≤κ−ω.For the mappingT,FT=κ∈C:Tκ=κ refers to the set of all FPs of it.Here, we study the following inclusion problem:(2)Findκ~∈Ysothat0∈Aκ~+Bκ~,where A:Y⟶Y and B:Y⟶2Y are single-valued and set-valued operators, respectively.There are many important applications enjoyed by approximating FP problems for NEMs, such as monotone variational inequalities, image restoration problems, convex optimization problems, and SCFPs, for example, see [1–3]. For more accuracy, these problems can be expressed as mathematical models such a machine learning and the linear inverse problem.In the past, the solution of problem (2) was described by A+B−10 and it relied on the forward-backward splitting method [4–10]. This technique is described as follows: κ1∈Y and
(3)κn+1=I+τB−1κn−τAκn,n≥1,τ>0.In this scene, we do not mean the sum ofA and B in the iterates, but each step of iterates includes only A as the forward term and B as the backward term. As special cases, this technique gets involved heavily in a study of the proximal point algorithm [11–13] and the gradient method [14–17].In 1979, a good splitting iterative scheme in a real HS was introduced by Lions and Mercier [18]. It is described as follows:
(4)κn+1=2JτA−I2JτB−Iκn,n≥1,(5)κn+1=JτA2JτB−Iκn+I−JτBκn,n≥1,where JτS=I+τS−1. In the previous literature, the algorithm (4) is called Peaceman-Rachford [7] and the scheme (5) is called Douglas-Rachford [19]. Generally, the convergence of both procedures is weak [7].In 2001, a heavy ball method involved for studying maximal monotone operators is introduced by Alvarez and Attouch [20]; this idea was developed in [21, 22], where an inertial term was added. This procedure is called the inertial proximal point algorithm and it takes the shape
(6)ωn=κn+θnκn−κn−1,κn+1=I+τnBn−1ωn,n≥1.They got the weak convergence for the mappingB, if τn is nondecreasing and θn⊂0,1 with
(7)∑n=1∞θnκn−κn−12<∞.In particular, condition (7) is true for θn<1/3. Here, θn is an extrapolation factor and the inertia is represented by the term θnκn−κn−1.It should be noted that the inertial term improves and increases the convergence speed of the algorithm [23–25].An inertial proximal point algorithm improved by Moudafi and Oliny [26], where the single-valued, cocoercive, and Lipschitz continuous operator was added, is as follows:
(8)ωn=κn+αnκn−κn−1,κn+1=I+τnBn−1ωn−τnAωn,n≥1.The problem of weak convergence still persists for the algorithm (8) via stipulation (7) and τn<2/L where L is the Lipschitz constant of A.Besides that, the strong convergence is of interest to many researchers, but the study of convergence via norm convergence in infinite-dimensional spaces is often much more desirable than weak convergence [27].The first contribution of researchers to the strong convergence is the algorithm presented by Nakajo and Takahashi [28]. They added the CQ terms to the Mann algorithm as follows: for an arbitrary point κ0∈C, define the sequence κn iteratively by
(9)ωn=αnκn+1−αnTκn,Cn=p∈C:ωn−p≤κn−p,Qn=p∈C:κ0−κn,κn−p≥0,κn+1=PQn∩Cnκ0,n≥0.They showed that the sequenceκn converges strongly to PFixTκ0, whenever the sequence αn is bounded above by 1. We highly recommend seeing [24, 29], for more details on the CQ algorithms for NEMs.Based on the algorithm (9), Dong et al. [30] introduced a strong convergence result by implicating an inertial forward-backward algorithm for monotone inclusions as the following: assume that A:Y⟶Y is an α-ISM operator and B:Y⟶2Y is a MM operator so that A+B−10≠∅. Suppose αn∈ℝ and κn∈Y is a sequence maked iteratively by κ∘,κ1∈Y,(10)ωn=κn+αnκn−κn−1,υn=I+τnB−1ωn−τnAωn,Cn=p∈H:υn−p2≤κn−p2−2αnκn−p,κn−1−κn+αn2κn−1−κn2,Qn=p∈H:κ0−κn,κn−p≤0,κn+1=PQn∩Cnκ0,n≥1.Recently, nice convergence analysis for NEMs via suitable stipulations has been discussed by Dong et al. [31]. They extended the inertial Mann algorithm as follows:
(11)ωn=κn+αnκn−κn−1,υn=κn+βnκn−κn−1,κn+1=1−λnωn+λnTυn,n≥1,where αn, βn, and λn are real sequences that justify the stipulations D1 and D2 [31].Believing in the idea of strong convergence of algorithms in this manuscript, the two-step inertial shrinking projection algorithm is introduced which analyzes the strong convergence. As an application to our main results, the SCFP is solved. Finally, to see the behavior and performance of our algorithms in terms of convergence, numerical results are presented and discussed.


## 2. Preliminaries

This section is devoted to collect some important preliminaries, which we need in the sequel. LetC be a NCC subset of a real HS Y and κn be a sequence in Y. Here, the strong convergence of κn to a point κ is written as κn⟶κ. The metric projection of Y onto C is described by PC, that is, κ−PCκ≤κ−ω for all κ∈Y and ω∈C.Lemma 1 (see [32]).
LetC be a NCC subset of a real HS Y, the metric projection PC is firmly NE, i.e.,
(12)PCκ−PCω2≤PCκ−PCω,κ−ω,for all κ,ω∈Y. Furthermore, for all κ∈Y and ω∈C, κ−PCκ,ω−PCω≤0 is satisfied.Lemma 2 (see [32]).
Assume thatY is a real HS. Then, we get
(i)
κ+ω2≤κ2+2ω,κ+ω(ii)
ρκ+1−ρω2=ρκ2+1−ρω2−ρ1−ρκ−ω2for each κ,ω∈Y and for a real number ρ.Lemma 3 (see [33]).
Suppose thatY is a real HS and κn is a sequence in Y. Then, the following hypotheses hold:
(i)
Ifκn⇀κ and κn⟶κ as n⟶∞, then κn⟶κ as n⟶∞; that is, the HS Y has the Kadec-Klee property(ii)
Ifκn⇀κ as n⟶∞, then κ≤liminfn→∞κnLemma 4 (see [34]).
LetC be a NCC subset of a real HS Y. For each κ,ω,υ∈Y and b∈ℝ, the following set is closed and convex:
(13)δ∈C:ω−δ2≤κ−δ2+υ,δ+b.Lemma 5 (see [28]).
LetC be a NCC subset of a real HS Y and PC:Y⟶C be the metric projection. Then, for all κ∈Y and ω∈C, the following inequality holds:
(14)ω−PCκ2+κ−PCκ2≤κ−ω2.Lemma 6 (see [35]).
LetT be a NE self-mapping of a NCC subset C of a real HS Y. The mapping I−T is demiclosed, i.e., the sequence κn in C weakly converges to some κ∈C and the sequence I−Tκn strongly converges to some ω; it follows that I−Tκ=ω.Definition 7.
Assume thatDA⊂Y is the domain of the mapping A, then for all κ,ω∈DA, the mapping A is called
(i)
monotone ifκ−ω,Aκ−Aω≥0(ii)
σ-strongly monotone if there is β>0 so that κ−ω,Aκ−Aω≥σκ−ω2(iii)
α-ISM if there is α>0 so that κ−ω,Aκ−Aω≥αAκ−Aω2Lemma 8 (see [5]).
LetY be a real HS, A:Y⟶Y be an α-ISM operator, and B:Y⟶2Y be a MM operator. For each ℓ>0, we consider
(15)Tℓ=JℓBI−ℓA=I+ℓB−1I−ℓA,then the following statements hold:
(i)
forℓ>0,FTℓ=A+B−10(ii)
for0<s≤ℓ and κ∈Y, κ−Tsκ≤2κ−TℓκLemma 9 (see [36]).
LetH be a real HS, A:Y⟶Y be an α-ISM operator and B:Y⟶2Y be a MM operator, then for all ℓ>0 and all κ,ω∈Y, we have
(16)Tℓκ−Tℓω2≤κ−ω2−ℓ2α−ℓAκ−Aω2.


## 3. Strong Convergence Results

From now on, we assume thatC be a NCC subset of a real HS Y,A:Y⟶Y is α-ISM operator, B:Y⟶2Y is MM operator, T:Y⟶Y is quasi-NEM so that I−T is demiclosed at zero and Ω=A+B−10.Now, we build our algorithms to finding an element inΩ as follows:Algorithm 1: Shrinking projection algorithm.
Letαn,βn, and γn be sequences of real numbers. Select initial κ0,κ1∈C1=Y.Step (1). Computeωn=κn+αnκn−κn−1,υn=κn+βnκn−κn−1.Step (2). Computeμn=1−γnωn+γnJτnBυn−τnAυn.Step (3). ComputeCn+1=p∈Cn:μn−p2≤κn−p2−2αn1−γn+γnβnκn−p,κn−1−κn+αn21−γn+γnβn2κn−1−κn2,κn+1=PCn+1κ1,n≥1.Algorithm 2: CQ algorithm.
Letαn,βn, and γn be sequences of real numbers. Select initial κ0,κ1∈Y.Step (1). Computeωn=κn+αnκn−κn−1,υn=κn+βnκn−κn−1.Step (2). Computeμn=1−γnωn+γnJτnBυn−τnAυn.Step (3). ComputeCn=p∈Y:μn−p2≤κn−p2−2αn1−γn+γnβnκn−p,κn−1−κn+αn21−γn+γnβn2κn−1−κn2,Qn=p∈Y:p−κn,κ1−κn≤0,κn+1=PCn∩Qnκ1,n≥1.Now, we shall discuss the strong convergence of Algorithm1: by introducing the following theorem.Theorem 10.
Let the sequenceαn, βn be bounded and γn be a sequence in 0,1 and τn be a sequence of positive real numbers so that the following two stipulations hold:
(i)
infnγn≥γ>0(ii)
0<infnτn≤supnτn<2αIfΩ≠∅, then the sequence κn created by Algorithm 1: converges strongly to Θ=PΩκ1.Proof.
The proof will be divided into the following steps:
Step (i). For each κ1∈H, Ω⊂Cn+1, and for n≥0. Prove that PCn+1κ1 is well-defined.
From the stipulation (ii) and Lemma9, we get Tτn=I+τnB−1I−τnA is NEM. Thus, it follows from Lemma 8 that the set Ω is closed and convex. Moreover, Lemma 4 leads that for all n≥1,Cn+1 is closed and convex. Considering p∈Ω, we get
(17)ωn−p2=κn−p−αnκn−1−κn2=κn−p2−2αnκn−p,κn−1−κn+αn2κn−1−κn2.
By the same manner, one can write(18)υn−p2=κn−p2−2βnκn−p,κn−1−κn+βn2κn−1−κn2.
Furthermore, by Lemma2 (ii) and Lemma 9, we obtain that
(19)μn−p2=1−γnωn+γnJτnBυn−τnAυn−p2=1−γnωn−p+γnTτnυn−p2=γnTτnυn−p2+1−γnωn−p2−γn1−γnTτnυn−ωn2≤1−γnωn−p2+γnTτnυn−p2=1−γnωn−p2+γnTτnυn−Tτnp2≤1−γnωn−p2+γnυn−p2−τn2α−τnAυn−Ap2≤1−γnωn−p2+γnυn−p2.
Applying (17) and (18) in (19) and by stipulation (ii) of Theorem 10, we can write
(20)μn−p2≤1−γnκn−p2−2αnκn−p,κn−1−κn+αn2κn−1−κn2+γnκn−p2−2βnκn−p,κn−1−κn+βn2κn−1−κn2=κn−p2−2αn1−γn+γnβnκn−p,κn−1−κn+1−γnαn2+γnβn2κn−1−κn2.
It can be easily obtainedΩ⊂C1=Y. For some n≥1, assume that Ω⊂Cn, then p∈Cn, and by (20), we conclude that p∈Cn+1. Therefore, Ω⊂Cn+1 for all n≥1, and this finishes the requirement of Claim 1.
Step (ii). Prove that the boundedness of κn. Because Ω is a NCC subset of Y, there is a unique u∈Ω so that u=PΩκ1.
Fromκn=PCnκ1,Cn+1⊂Cn, and κn+1∈Cn for all n≥1, we get
(21)κn−κ1≤κn+1−κ1,foralln≥1.
Also, sinceΩ⊂Cn, we get
(22)κn−κ1≤u−κ1,foralln≥1.
By (21) and (22), we obtain that limn→∞κn−κ1 exists; this leads to κn being bounded.
Step (iii). Prove that κn⟶Θ as n⟶∞, for some Θ∈Y. By the structure of Cn, for m>n, one sees that κm=PCmκ1∈Cm⊂Cn. From Lemma 5, we have
(23)κm−κn2≤κm−κ12−κn−κ12.
From Step (ii), we obtain thatκm−κn2⟶0 as m,n⟶∞. This proves that the sequence κn is a Cauchy. Therefore, κn⟶Θ as n⟶∞. In particular, we can obtain
(24)limn→∞κn+1−κn=0.
Step (iv). Prove that Θ∈Ω. Because αn and βn are bounded, then by (24), we have
(25)ωn−κn=αnκn−κn−1⟶0asn⟶∞,(26)υn−κn=βnκn−κn−1⟶0asn⟶∞.
From (24), (25), and (26), we have
(27)κn+1−ωn≤κn+1−κn+ωn−κn⟶0asn⟶∞,(28)κn+1−υn≤κn+1−κn+υn−κn⟶0asn⟶∞.
Sinceκn+1∈Cn+1, we obtain that
(29)μn−κn+1≤κn−κn+12−2αn1−γn+γnβnκn−κn+1,κn−1−κn+1−γnαn2+γnβn2κn−1−κn2≤κn−κn+12+2αn1−γn+γnβnκn−κn+1κn−1−κn+1−γnαn2+γnβn2κn−1−κn2.
Thus, from the boundedness ofαn, βn, and γn and (24), (29), we obtain that
(30)μn−κn+1⟶0.
By (24), (30), (27), and (28) and the following inequalities
(31)μn−κn≤μn−κn+1+κn−κn+1,μn−ωn≤μn−κn+1+ωn−κn+1,μn−υn≤μn−κn+1+υn−κn+1,we get that
(32)μn−κn⟶0,μn−ωn⟶0,μn−υn⟶0.
We now have(33)Tτnυn−υn=1γnμn−1−γnωn−υn=1γnγnυn+1−γnωn−μn=1γnγnυn−μn+1−γnωn−μn≤1γnγnυn−μn+1−γnωn−μn.
Again using the stipulation (i) and (32), we get
(34)limn→∞Tτnυn−υn=0.
Asliminfn→∞τn>0, there is ε>0 so that τn≥ε and ε∈0,2α for all n≥1. Hence, from Lemma 8 (ii) and (34), one can write
(35)Tευn−υn≤2Tτnυn−υn⟶0asn⟶∞.
Based on (32), as κn⟶Θ, we get υn⟶Θ. Because Tε is NE, Tε is continuous mapping. Hence, using (35), we have Θ∈Ω.
Step (v). Prove that Θ=PΩκ1. Since κn=PCnκ1, and Ω⊂Cn, we obtain that
(36)κ1−κn,κn−p≥0,∀p∈Ω.
By taking the limit in (36), we have
(37)κ1−Θ,Θ−p≥0,∀p∈Ω.
This shows thatΘ=PΩκ1 and this finishes the requirement.Next, we shall discuss the strong convergence of Algorithm2: by presenting the following theorem.Theorem 11.
Let the sequenceαn, βn be bounded and γn be a sequence in 0,1 and τn be a sequence of positive real numbers so that the following two stipulations hold:
(i)
infnγn≥γ>0(ii)
0<infnτn≤supnτn<2αIfΩ≠∅, then the sequence κn marked by Algorithm 2: converges strongly to Θ=PΩκ1.Proof.
In the same way as proving Theorem10, we will discuss the following steps:
Step (i). Prove that for all κ1∈Y, Ω⊂Qn∩Cn and for each n≥0.κnn=0∞ is well-defined.
It is clear thatCn is closed and a convex subset of Y (by Lemma 4). So, we can rewrite the set Qn in the shape
(38)Qn=p∈Y:κ1−κn,p≤κ1−κn,κn.It follows that Qn is closed and convex subset of Y too. Thus, Qn∩Cn is also closed and convex, for each n≥0.
Letp∈Ω. One can obtain by similar way of Theorem 10 that
(39)μn−p2≤κn−p2−2αn1−γn+γnβnκn−p,κn−1−κn+1−γnαn2+γnβn2κn−1−κn2.
Hence,p∈Cn for all n≥1; this implies that Ω⊂Cn.
Whenn=1, we get Q1=Y, and hence, Ω⊂C1∩Q1. Suppose that Ω⊂Cl∩Ql for some l≥1. It follows from κl+1=PCl∩Qlκ1 that
(40)κl+1−a,κ1−κl+1≥0,for each a∈Cl∩Ql. Since Ω⊆Cl∩Ql, and p∈Ω, we have
(41)κl+1−p,κ1−κl+1≥0.
This yieldsp∈Ql+1, and hence, Ω⊆Ql+1. This implies that Ω⊆Cl+1∩Ql+1, and hence, κn is well-defined as well as Ω⊂Cn∩Qn.
Step (ii). Prove that the boundedness of κn. Based on Algorithm 2:, one gets
(42)ξ−κn,κ1−κn≤0,∀ξ∈Qn,n≥1.
This implies thatκn=PQnκ1. Since Ω⊂Qn, we have
(43)κn−κ1≤κ1−ξ,∀ξ∈Ω.
Also, sinceκn+1∈Qn, we can write
(44)κn−κ1≤κn+1−κ1.
By (43) and (44), we obtain limn→∞κn−κ1 exists, and hence, κn is bounded.
Step (iii). Prove that κn+1−κn⟶0 as n⟶∞.
Becauseκn+1∈Qn and κn=PQnκ1, it follows from Lemma 5 that
(45)κn+1−κn2≤κn+1−κ12−κn−κ12⟶0.
This implies that∥κn+1−κn∥⟶0 as n⟶∞.
Step (iv). Prove that Θ=PΩκ1. In the same manner as the proof of Step (iv) of Theorem 10, we can write
(46)Tεvn−vn⟶0,vn−κn⟶0,where ε∈0,2α. It follows from the nonexpansivity of Tε that
(47)Tεκn−κn=Tεκn−Tεvn+Tεvn−vn+vn−κn≤2vn−κn+Tεvn−vn.
From (46) and (47), we can get
(48)Tεκn−κn⟶0.
By the boundedness ofκn, there exists a subsequence κnk of κn so that κnk⇀κ∗. This combines with (48), and from Lemma 6, we have κ∗∈FTε; this means that κ∗∈Ω.
SinceΘ=PΩκ1 and κ∗∈Ω, (43) and Lemma 3 (ii) imply that
(49)κ1−Θ≤κ1−κ∗≤liminfk→∞κnk−κ1≤limsupk→∞κnk−κ1≤κ1−Θ.
Since the nearest pointΘ is unique, then we have Θ=κ∗. Also, we get κnk−κ1⟶κ1−Θ. Applying Lemma 3 (i), we have κnk⟶Θ as k⟶∞. Again, the uniqueness of Θ leads to κn⟶Θ as n⟶∞.
This finishes the requirement.


## 4. Application to Solve Split Convex Feasibility Problem

This part is devoted to applying our methods to find a solution to the SCFP. Assume thatT:Y1⟶Y2 is a bounded linear operator and T∗ its adjoint defined on real HSs Y1 and Y2. Let C⊂Y1 and Q⊂Y2 be a NCC sets. Censor and Elfving [37] formulated the SCFP as follows:
(50)Findκ¯∈CsothatTκ¯∈Q.Censor and Elfving in [37] have introduced the SCFP in HSs while using a multidistance approach to find an adaptive approach for resolving it. Many of the problems that emerge from state retrieval and restoration of medical image can be formulated as split variational feasibility problems [38, 39]. This problem is also used in a variety of disciplines like image restoration, dynamic emission tomographic image reconstruction, and radiation therapy treatment planning [40–42]. Let consider
(51)Aκ=∇12Tκ−PQTκ2=T∗I−PQTκ,where PQ is the metric projection on to Q, ∇ is the gradient, and B=∂iC. Due to the above construction, the problem (50) has an inclusion format as described in (2). It can be seen thatA is Lipschitz which continues with constant L=∥T∥2 and B is MM, see, for example, [43].For any NCC subsetC of a real HS Y, the indicator function iC of C is defined by
(52)iCκ=0,ifκ∈C,∞,otherwise.Now, on the basis of the main results, we can deduce the following results for a SCFP.Theorem 12.
Letκn be a sequence iterated as follows: choose initial points κ0,κ1∈C1=Y and let αn, βn be bounded sequences and γn be a sequence in 0,1. Consider that τn is a sequence of positive real numbers so that the following assumptions are fulfilled:
(i)
infnγn≥γ>0(ii)
0<infnτn≤supnτn<2α
Step (1). Compute
(53)ωn=κn+αnκn−κn−1,υn=κn+βnκn−κn−1.
Step (2). Compute
(54)μn=1−γnωn+γnPCυn−τnT∗I−PQTυn.
Step (3). Compute
(55)Cn+1=p∈Cn:μn−p2≤κn−p2−2αn1−γn+γnβnκn−p,κn−1−κn+αn21−γn+γnβn2κn−1−κn2,κn+1=PCn+1κ1,n≥1.If the solution setΓSCFP is nonempty, then the sequence κn converges weakly to an element of solution set ΓSCFP.Theorem 13.
Letκn be a sequence iterated as follows: choose initial points κ0,κ1∈C1=Y and let αn, βn be bounded sequences and γn be a sequence in 0,1. Consider that τn is a sequence of positive real numbers so that the following assumptions are fulfilled:
(i)
infnγn≥γ>0(ii)
0<infnτn≤supnτn<2α
Step (1). Compute
(56)ωn=κn+αnκn−κn−1,υn=κn+βnκn−κn−1.
Step (2). Compute
(57)μn=1−γnωn+γnPCυn−τnT∗I−PQTυn.
Step (3). Compute
(58)Cn=p∈Y:μn−p2≤κn−p2−2αn1−γn+γnβnκn−p,κn−1−κn+αn21−γn+γnβn2κn−1−κn2,Qn=p∈Y:p−κn,κ1−κn≤0,κn+1=PCn∩Qnκ1,n≥1.If the solution setΓSCFP is nonempty, then the sequence κn converges weakly to an element of solution set ΓSCFP.


## 5. Supportive Numerical Examples

This section is the mainstay of the paper as it studies the behavior and performance of our algorithms numerically and graphically. The program used here is MATLAB R2014a running on an HP Compaq 510, Core™ 2 Duo CPU T5870 with 2.0 GHz and 2 GB RAM.Example 14.
LetY1=Y2=L20,2π be two HSs with an inner product
(59)x,y≔∫02πxtytdt,∀x,y∈L20,2π,and the induced norm defined by
(60)x≔∫02πxt2dt,∀x∈L20,2π.Next, consider the feasible setC⊂Y1 as
(61)C=x∈Y1:∫02πxtdt≤1,and Q⊂Y2 is
(62)Q=x∈Y2:∫02πxt−sint2dt≤16.Consider the mappingT:Y1⟶Y2 so that Txs=xs,x∈Y1. Then, T∗xs=xs and T=1. So, we wish to solve the following problem:
(63)Findκ¯∈CsothatTκ¯∈Q.We can also observe that sinceTxs=xs,x∈Y1, the above problem is actually a SCFP in the form of
(64)Findκ¯∈C∩Q.Figures1–5 and Table 1 show the numerical computational results of Algorithm (4.2) by Dong et al. in [30] (St-Alg1), Algorithm 1: (St-Alg2), and Algorithm 2: (St-Alg3) by assuming Dn=κn−κn−1≤10−4.Remark 15.
It is important to note that the different choices of initial points have substantial effect on the CPU (time) and a number of iterations on the proposed algorithms and also for those existing algorithms that are used for comparison. These facts can be seen from Figures1–5 and Table 1. We conclude from these numerical results that our algorithms are faster in convergence than their counterpart presented by Dong et al. in [30].Figure 1
Numerical computational behavior of Algorithm1:, Algorithm 2:, and Algorithm (4.2) while κ0=κ1=2t+3.Figure 2
Numerical computational behavior of Algorithm1:, Algorithm 2:, and Algorithm (4.2) while κ0=κ1=5+t2/5.Figure 3
Numerical computational behavior of Algorithm1:, Algorithm 2:, Algorithm (4.2) while κ0=κ1=3e2tt4.Figure 4
Numerical computational behavior of Algorithm1:, Algorithm 2:, and Algorithm (4.2) while κ0=κ1=3etsint.Figure 5
Numerical computational behavior of Algorithm1:, Algorithm 2:, and Algorithm (4.2) while κ0=κ1=2t3−3etcost.Table 1
Numerical computational results of Algorithm1:, Algorithm 2:, and Algorithm (4.2).
Number of iterationsExecution time in secondsInitial point (κ0=κ1)St-Alg1St-Alg2St-Alg3St-Alg1St-Alg2St-Alg32t+32725200.04970.01840.01845+t2/52528160.04970.01840.01843e2tt43932210.04970.01840.01843etsint4743260.04970.01840.01842t3−3etcost5540300.04970.01840.0184


## 6. Conclusion

The quality of the algorithm is measured by two main factors: velocity in convergence and time. When the convergence is faster in a short time, the results are faster and more accurate. Given the importance of algorithms in many applications in real society, many researchers have studied this logic and try to obtain a strong convergence, which has a prominent role in studying the efficiency and effectiveness of these algorithms. On the basis of this principle, in this paper was studied the effect of shrinking projection and CQ term on two inertial terms to get the strong convergence of new algorithms called two inertial shrinking projection and CQ algorithms. These results have been implicated to obtain a solution to SCFP in HSs. Finally, some numerical results were formulated to illustrate the efficiency and effectiveness of algorithms.



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