# Shrinking Projection Methods for Accelerating Relaxed Inertial Tseng-Type Algorithm with Applications

Authors: Hasanen A. Hammad; Habib ur Rehman; Manuel De la Sen
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

Our main goal in this manuscript is to accelerate the relaxed inertial Tseng-type (RITT) algorithm by adding a shrinking projection (SP) term to the algorithm. Hence, strong convergence results were obtained in a real Hilbert space (RHS). A novel structure was used to solve an inclusion and a minimization problem under proper hypotheses. Finally, numerical experiments to elucidate the applications, performance, quickness, and effectiveness of our procedure are discussed.


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## Body

## 1. Introduction

The standard form of the variational inclusion problem (VIP) on a RHSℸ is(1)0∈¥+ϒϑ∗,where ϑ∗ is the unknown point that we need to find, for an operator ¥:ℸ⟶ℸ and a set-valued operator ϒ:ℸ⟶2ℸ. VIP is a frequent problem in the optimization field, which has a lot of applications in many areas, including equilibrium, machine learning, economics, engineering, image processing, and transportation problems [1–16].The vintage technique to solve problem (1) which is denoted by ¥+ϒ−10 is the forward-backward splitting method [17–22] which is defined as follows: ϑ1∈ℸ and(2)ϑn+1=I+ℓϒ−1I−ℓ¥ϑn,n≥1,where ℓ>0. In (2), each step of iterates includes only the forward step ¥ and the backward step ϒ, but not ¥+ϒ. This technique involves the proximal point algorithm [23–25] and the gradient method [26–28] as special cases.In a RHS, nice splitting iterative procedures presented by Lions and Mercier [29] are shown as follows:(3)ϑn+1=2Jℓ¥−I2JℓΥ−Iϑn,n≥1,and(4)ϑn+1=Jτ¥2JℓΥ−Iϑn+I−JℓΥϑn,n≥1,where Jℓℜ=I+ℓℜ−1. Permanently, two algorithms are weakly convergent [30], knowing that algorithm (3) is called Peaceman–Rachford algorithm [19] and scheme (4) is called Douglas–Rachford algorithm [31].A lot of works are concerned with problem (1) for accretive operators and two monotone operators, for instance, a stationary solution to the initial-valued problem of the evolution equation(5)0∈∂ϖ∂t−Ξϖ,ϖ0=ϖ°can be adjusted as (1) when the governing maximal monotone Ξ=¥+ϒ [29].[1] is used to solve a minimization problem as follows:(6)minϑ∈ℸℶϑ+σϑ,where ℶ,σ:ℸ⟶−∞,∞ are proper and lower semicontinuous convex functions such that ℶ is differentiable with L-Lipschitz gradient, and the proximal mapping of σ is(7)ϑ⟼argminω∈ℸσω+ϑ−ω22ℓ.In particular, if¥=∇ℶ and ϒ=∂σ, where ∇ℶ is the gradient of ℶ and ∂σ is the subdifferential of σ which takes the form ∂σϑ=λ∈ℸ:σω≥σϑ+λ,ω−ϑ∀ω∈ℸ, problem (1) becomes (6), and (3) becomes(8)ϑn+1=proxℓσϑn−ℓ∇ℶϑn,n≥1,where ℓ>0 is the stepsize and proxℓσ=I+ℓ∂σ−1 is the proximity operator of σ.The concept of merging the inertial term with the backward step was initiated by Alvarez and Attouch [32] and studied extensively in [33, 34]. For maximal monotone operators, it was called the inertial proximal point (IPP) algorithm, and they defined it by(9)ℑn=ϑn+Λnϑn−ϑn−1,ϑn+1=I+ℓnϒ−1ℑn,n≥1.It was proved that ifℓn is nondecreasing and Λn⊂0,1 with(10)∑n=1∞Λnϑn−ϑn−12<∞,then algorithm (9) converges weakly to zero of ϒ. In particular, condition (10) is true for Λn<1/3. Here, Λn is an extrapolation factor, and the inertia is represented by the term Λnϑn−ϑn−1. Note that the inertial term improves the performance of the procedure and has good convergence results [35–37].Inertial term was merged with forward-backward algorithm by authors [38]. They added Lipschitz-continuous, a single-valued, cocoercive operator ¥ into the IPP algorithm:(11)ℑn=ϑn+Λnϑn−ϑn−1,ϑn+1=I+ℓnϒn−1ℑn−ℓn¥ℑn,n≥1.Via assumption (10), provided ℓn<2/L with L, the Lipschitz constant of ¥, they obtained a weak convergence result. Note that, for Λn>0, algorithm (11) does not take the form of (2), in spite of ¥ is still evaluated at the points ¥n.Relaxation techniques and inertial effects have many advantages in solving monotone inclusion and convex optimization problems; this effect appeared in several names such as relaxed inertial proximal method, relaxed inertial forward-backward method, and relaxed inertial Douglas–Rachford algorithm; for more details, refer to [22, 24, 39–44].Abubakar et al. [45] introduced the RITT method as follows:(12)ℑn=ϑn+Λϑn−ϑn−1,ψn=1+ℓnϒ−11−ℓn¥ℑn,ϕn+1=1−βℑn+βψn+βℓn¥ℑn−¥ψn,n≥1,where Λ and β are extrapolation and relaxation parameters, respectively. Under this algorithm, they discussed the weak convergence to the solution point of VIP (1) and the problem of image recovery. Note that the extrapolation step works to accelerate but not for the desired acceleration.The concept of the SP method was discussed by Takahashi et al. [46] as in the following algorithm:(13)ϑ0∈ℸbe arbitrarily fixed,C1=C,ϑ1=PC1ϑ0,ωn=Λnϑn+1−Λnℏnϑn,Cn=η∈C:ωn−η≤ϑn−η,ϑn+1=PCn+1ϑ0.They just selected one closed convex (CC) set for a family of nonexpansive mappingsℏn to modify Mann’s iteration method [47] and proved that the sequence ϑn converges strongly to PFixℏϑ0, provided Λn≤e for all n≥1 and for some 0<e<1.In 2019, Yang and Liu [48] selected the stepsize sequence for the iterative algorithm for monotone variational inequalities, which are based on Tseng’s extragradient method and Moudafi viscosity scheme that does not require either the knowledge of the Lipchitz constant of the operator or additional projections.With the incorporation of results of [45, 46, 48], we accelerate RITT algorithm by adding the SP method to algorithm (12). In a RHS, strong convergence results are given under a proposed algorithm. As applications, our algorithm was used to find the solution to a VIP and minimization problem under certain conditions. Eventually, numerical experiments to illustrate the applications, performance, acceleration, and effectiveness of the proposed algorithm are presented.


## 2. Preparatory Lemmas and Definitions

Suppose thatC is a nonempty closed convex subset (CCS) of a RHS ℸ; we shall refer to ”⟶” as the strong convergence, and PC:ℸ⟶C is the nearest point projection, that is, for all ϑ∈ℸ and ω∈C, ϑ−PCϑ≤ϑ−ω. PC is called the metric projection. It is obvious that PC verifies the following inequality:(14)PCϑ−PCω2≤PCϑ−PCω,ϑ−ω,for all ϑ,ω∈ℸ. In other words, the metric projection PC is firmly nonexpansive. Hence, ϑ−PCϑ,ω−PCω≤0 holds for all ϑ∈ℸ and ω∈C, see [49, 50].The following inequality holds in a HS [51]:(15)l±m2=l2+m2±2l,m,for all l,m∈ℸ.Lemma 1.
(see [52]). Let C be a nonempty CCS of a RHS ℸ. For each ϑ,ω,υ∈ℸ and ð∈ℝ, the following set is closed and convex:(16)η∈C:ω−η2≤ϑ−η2+υ,η+δ.Lemma 2.
(see [38]). Let C be a nonempty CCS of a RHS ℸ and PC:ℸ⟶C be the metric projection. Then,(17)ω−PCϑ2+ϑ−PCϑ2≤ϑ−ω2,for all ϑ∈ℸ and ω∈C.Definition 1.
Suppose thatD¥⊂ℸ and R¥⊂ℸ are the domain and the range of an operator ¥, respectively. For all ϑ,ω∈D¥, an operator ¥ is called(1)
Monotone if(18)ϑ−ω,¥ϑ−¥ω≥0.(2)
L−Lipschitz if(19)¥ϑ−¥ω≤Lϑ−ω.(3)
β−Strongly monotone if there exists β>0 such that(20)ϑ−ω,¥ϑ−¥ω≥βϑ−ω2.(4)
Λ−Inverse strongly monotone (Λ−ism) if there exists Λ>0 such that(21)ϑ−ω,¥ϑ−¥ω≥Λ¥ϑ−¥ω2.Lemma 3.
(see [44]). Let ℸ be a RHS, ¥:ℷ⟶ℷ be an Λ−ism operator, and Υ:ℸ⟶2ℸ be a maximal monotone operator. For each ℓ>0, we define(22)℧ℓ=JℓΥI−ℓ¥=I+ℓΥ−1I−ℓ¥.
Then, we get(i)
Forℓ>0, fix℧ℓ=¥+ϒ−10(ii)
For0<s≤ℓ and ϑ∈ℸ, ϑ−℧sϑ≤2ϑ−℧ℓϑLemma 4.
Letℸ be a RHS, ¥:ℸ⟶ℸ be an Λ−ism operator, and ϒ:ℸ⟶2ℸ be a maximal monotone operator. For each ℓ>0, we have(23)℧ℓϑ−℧ℓω2≤ϑ−ω2−ℓ2Λ−ℓ¥ϑ−¥ω2,for all ϑ,ω∈ℸ.Proof.
For allϑ,ω∈ℸ, we get(24)℧ℓϑ−℧ℓω2=JrΥI−ℓ¥ϑ−JrΥI−ℓ¥ω2≤I−ℓ¥ϑ−I−ℓ¥ω2=ϑ−ω−ℓ¥ϑ−¥ω2=ϑ−ω2−2ℓϑ−ω,¥ϑ−¥ω+ℓ2¥ϑ−¥ω2≤ϑ−ω2−2ℓΛ¥ϑ−¥ω2+ℓ2¥ϑ−¥ω2=ϑ−ω2−ℓ2Λ−ℓ¥ϑ−¥ω2.
The proof is ended.


## 3. Shrinking Projection Relaxed Inertial Tseng-Type Algorithm

We provide a method consisting of the forward-backward splitting method with an inertial factor and an explicit stepsize formula, which are being used to ameliorate the convergence average of the iterative scheme and to make the manner independent of the Lipschitz constants. The detailed method is provided in Algorithm1.Algorithm 1: Splitting method for the VIP.
Initialization: select initial ϑ0,ϑ1∈ℸ, ρ∈0,1, Λ≥0, ℓ0>0, and 0<β<1.St. (i). Put ℑn as:ℑn=ϑn+Λϑn−ϑn−1,St. (ii). Calculate:ψn=1+ℓnϒ−11−ℓn¥ℑn.,Ifℑn=ψn, discontinue. ℑn is a solution of (1), otherwise, continue to St. (iii)St. (iii). Calculate:ϕn=1−βℑn+βψn+βℓn¥ℑn−¥ψn,whereℓn+1 is stepsize sequence revised as follows:ℓn+1=minℓn,ρℑn−ψn/¥ℑn−¥ψn,if¥ℑn≠¥ψn,ℓn,else,St. (iv). Calculate:Cn+1=η∈Cn:ϕn−η2≤ϑn−η2+Λ2ϑn−1−ϑn2−2Λϑn−η,ϑn−1−ϑn−βΔℑn−ψn2,whereΔ=2−β−2ρ1−βℓn/ℓn+1−βρ2ℓn2/ℓn+12.St. (v). Computeϑn+1=PCn+1ϑ1,n≥1,putn=n+1, and return to St. (i).Note that(i)
Since¥ is an Λ−ism operator, it is a Lipschitz function with a constant L, ¥ℑn≠¥ψn, and we get(25)ρℑn−ψn¥ℑn−¥ψn≥ρL.It is obvious for¥ℑn=¥ψn that inequality (25) is satisfied. Hence, it follows that ℓn≥minρ/L,ℓ0. This implies that the generated sequence ℓn is bounded below by minρ/L,ℓ0, i.e., ℓn is monotonically decreasing.(ii)
By (i) and (25), we have(26)ℓn+1¥ℑn−¥ψn≤ρℑn−ψn,i.e., the update (28) is well defined.(iii)
If we delete the shrinking projection term from our algorithm, we get the algorithms of the papers [22, 45, 53].Theorem 1.
Letℸ be a RHS and the operators ¥:ℸ⟶ℸ be Λ−ism on ℸ, and ϒ:ℸ⟶2ℸ is maximally monotone. If feasible set Ω=¥+ϒ−10 of (1) is a nonempty CCS of a RHS ℸ, then the sequence ϑn generated by Algorithm 1 converges strongly to a point τ=PΩϑ1, provided that(i)
0<liminfn⟶∞ℓℓn≤limsupn⟶∞ℓn<2Λ.(ii)
limn⟶∞ψn−ℑn=0.Proof.
The proof will be divided as follows:Part 1.
Demonstrate thatPCn+1ϑ1 is well-defined, for each ϑ1∈ℸ, n≥1, and Ω⊂Cn+1. It follows from condition (i) and Lemma 4 that ℧ℓn=I+ℓnϒ−1I−ℓn¥ is a nonexpansive mapping. Lemma 3 implies that Ω is a closed and convex set, and Lemma 1 clarifies that Cn+1 is closed and convex, for all n≥1.
Letη∈Ω; we have(27)ℑn−η2=ϑn−η−Λϑn−1−ϑn2=ϑn−η2−2Λϑn−η,ϑn−1−ϑn+Λ2ϑn−1−ϑn2.
Since the resolvent℧ℓn is firmly a nonexpansive mapping and by Lemma 3, we have(28)ψn−η,ℑn−ψn−ℓn¥ℑn=JℓΥI−ℓn¥ℑn−JℓΥI−ℓn¥η,I−ℓn¥ℑn−I−ℓn¥η+I−ℓn¥η−ψn≥ψn−η2+ψn−η,η−ψn−ψn−η,ℓn¥ψn=−ψn−η,ℓn¥ψn.
Hence, by (28), we get(29)ψn−η,ℑn−ψn−ℓn¥ℑn+¥ψn≥0,which leads to(30)2ℑn−ψn,ψn−η−2ℓn¥ℑn+¥ψn,ψn−η≥0.
It is obvious that(31)2ℑn−ψn,ψn−η=ℑn−η2−ℑn−ψn2−ψn−η2.
Applying (31) in (30), we can write(32)ψn−η2≤ℑn−η2−ℑn−ψn2−2ℓn¥ℑn−¥ψn,ψn−η.
Now, from definitionϕn, we have(33)ϕn−η2=1−βℑn+βψn+βℓn¥ℑn−¥ψn−η2=1−βℑn−η+βψn−η+βℓn¥ℑn−¥ψn2=1−β2ℑn−η2+β2ψn−η+β2ℓn2¥ℑn−¥ψn2+2β1−βℑn−η,ψn−η+2βℓn1−βℑn−η,¥ℑn−¥ψn+2β2ℓnψn−η,¥ℑn−¥ψn.
From equation (15), one can write(34)2ℑn−η,ψn−η=ℑn−η2−ℑn−ψn2+ψn−η2.
Applying (34) in (33), we get(35)ϕn−η2=1−βℑn−η2+βψn−η2−β1−βψn−ℑn2+β2ℓn2¥ℑn−¥ψn2+2βℓn1−βℑn−η,¥ℑn−¥ψn+2β2ℓnψn−η,¥ℑn−¥ψn.
It follows from (32), (35), and (26) that(36)ϕn−η2≤1−βℑn−η2+βℑn−η2−ℑn−ψn2−2ℓn¥ℑn−¥ψn,ψn−η−β1−βψn−ℑn2+β2ℓn2¥ℑn−¥ψn2+2βℓn1−βℑn−η,¥ℑn−¥ψn+2β2ℓnψn−η,¥ℑn−¥ψn≤ℑn−η2−β2−βℑn−ψn2−2βℓn¥ℑn−¥ψn,ψn−η+β2ℓn2¥ℑn−¥ψn2+2βℓn1−βℑn−η,¥ℑn−¥ψn+2βℓnψn−η,¥ℑn−¥ψn≤ℑn−η2−β2−βℑn−ψn2+β2ℓn2¥ℑn−¥ψn2+2βℓn1−βℑn−ψn,¥ℑn−¥ψn≤ℑn−η2−β2−βℑn−ψn2+β2ℓn2ρ2ℓn+12ℑn−ψn2+2βℓn1−βρℓn+1ℑn−ψn2=ℑn−η2−β2−β−2ρ1−βℓnℓn+1−βρ2ℓn2ℓn+12ℑn−ψn2=ℑn−η2−βΔnℑn−ψn2.
Applying (27) in (36), we have(37)ϕn−η2≤ϑn−η2+Λ2ϑn−1−ϑn2−2Λϑn−η,ϑn−1−ϑn−βΔnℑn−ψn2.
It is clear thatΩ⊂C1=ℸ. Assume that Ω⊂Cn for some n≥1. Then, η∈Cn and by (37), we have for all n≥1, η∈Cn+1. Thus, Ω⊂Cn+1 for all n≥1, i.e., PCn+1ϑ1 is well-defined and bounded.Part 2.
Illustrate thatϑn is bounded. Since Ω≠∅ and closed and convex subset of ℸ, there is a unique u∈Ω such that u=PΩϑ1. This leads to ϑn=PCnϑ1, Cn⊂Cn+1, and ϑn+1∈Cn for all n≥1, and we have(38)ϑn−ϑ1≤ϑn+1−ϑ1.
Furthermore, asΩ⊂Cn, for all n≥1, we obtain(39)ϑn−ϑ1≤u−ϑ1.
It follows by (38) and (39) that limn⟶∞ϑn−ϑ1 exists. Hence, ϑn is bounded.Part 3.
Fulfillment oflimn⟶∞ϑn=τ. By the definition of Cn, for m>n, we observe that ϑm=PCmϑ1∈Cm⊂Cn. From Lemma 2, we have(40)ϑm−ϑn2≤ϑm−ϑ12−ϑn−ϑ12.
By Part2, we conclude that limn,m⟶∞ϑm−ϑn2=0. Thus, ϑn is a Cauchy sequence. Hence, limn⟶∞ϑn=τ. Additionally, we get(41)limn⟶∞ϑn+1−ϑn=0.Part 4.
Prove thatτ∈Ω. It follows from (41) that(42)ℑn−ϑn=Λϑn−ϑn−1⟶0asn⟶∞.
Also, by (42) and condition (ii), we can write(43)ψn−ϑn≤ψn−ℑn+ℑn−ϑn⟶0asn⟶∞.
From triangle inequality on the norm and (42) and (43), we obtain(44)ℑn−ψn≤ℑn−ϑn+ψn−ϑn⟶0asn⟶∞.
Replacingη with ϑn in (36) and using (41) and (44), we have(45)ϕn−ϑn2≤Λ2ϑn−1−ϑn2−βΔnℑn−ψn2⟶0asn⟶∞.
Applying (41), (42), and (45), we can write(46)ϑn+1−ℑn≤ϑn+1−ϑn+ℑn−ϑn⟶0asn⟶∞,ϑn+1−ϕn≤ϑn+1−ϑn+ϕn−ϑn⟶0asn⟶∞,ϕn−ℑn≤ϕn−ϑn+ℑn−ϑn⟶0asn⟶∞.
It follows from (44) that(47)limn⟶∞℧ℓnℑn−ℑn=limn⟶∞ψn−ℑn=0.
Sinceliminfn⟶∞ℓn>0, there is ε>0 such that ℓn≥ε and ε∈0,2Λ for all n≥1. Then, by Lemma 3 (ii) and (47), we get(48)℧εℑn−ℑn≤2℧ℓnℑn−ℑn⟶0asn⟶∞.
From (45) and (46), since ϑn⟶τ as n⟶∞, we have also ℑn⟶τ as n⟶∞. Since ℧ε is a nonexpansive and continuous mapping, from (47), we conclude that τ∈Ω.Part 5.
Show thatτ=PΩϑ1. Since ϑn=PCnϑ1 and Ω⊂Cn, we can get(49)ϑ1−ϑn,ϑn−η≥0,∀η∈Ω.
Settingn⟶∞ in (49), we have(50)ϑ1−τ,τ−η≥0,∀η∈Ω.
This shows thatτ=PΩϑ1. This finishes the proof.


## 4. Solve a Minimization Problem

As an application of our theorem, we solve the following constrained convex minimization problem:(51)minϑ∈Cℶϑ,where ℶ:ℸ⟶ℝ is a convex function. We suppose that the function ℶ is differentiable such that ∇ℶ is an Λ−ism operator.It is easy to see that problem (51) is equivalent to the following problem:(52)minϑ∈ℸℶϑ+℘Cϑ,where ℘C is the indicator function of C. Thus, this problem becomes the problem of finding an element ϑ∗∈ℸ such that(53)∇ℶϑ∗+∂℘Cϑ∗∋0,where ∂℘C is the subdifferential of ℘C. We know that ∂℘C is a maximal monotone operator, and I+m∂℘C−1=PC for all m>0.For solving problem (51), we state the theorem in the following, which is similar to Theorem 1.Theorem 2.
Let the sequenceℓn be bounded below by minρ/L,ℓ0, where ρ∈0,1 and ℓ0>0. Given a parameter Λ≥0 such that 0<infnℓn≤supnℓn<2Λ. Let ϑn be the sequence in ℸ which is defined by ϑ0,ϑ1∈ℸ, C1=ℸ, 0<β<1, and(54)ℑn=ϑn+Λϑn−ϑn−1,ψn=PCℑn−ℓn∇ℶℑn,ϕn=1−βℑn+βψn+βℓn¥ℑn−¥ψn,where,ℓn+1=minℓn,ρℑn−ψn¥ℑn−¥ψn,if¥ℑn≠¥ψn,ℓn,else,Cn+1=η∈ℸ:ϕn−η2≤ϑn−η2+Λ2ϑn−1−ϑn2−2Λϑn−η,ϑn−1−ϑn−βΔℑn−ψn2,ϑn+1=PCn+1ϑ1,n≥1,where ¥:ℸ⟶ℸ is Λ−ism on a RHS ℸ, ϒ:ℸ⟶2ℸ is a maximally monotone operator, and Δ=2−β−2ρ1−βℓn/ℓn+1−βρ2ℓn2/ℓn+12. If Ω≠∅, then the sequence ϑn converges strongly to τ=PΩϑ1, provided that limn⟶∞ψn−ℑn=0.


## 5. Solve a Split Feasibility Problem

In this section, we investigated the application of our proposed methods to the split convex feasibility problem (SCFP). LetT:ℸ1⟶ℸ2 be a bounded linear operator and T∗ its adjoint defined on the two RHSs ℸ1 and ℸ2. Assume that C⊂ℸ1 and Q⊂ℸ2 are nonempty CCSs. The SCFP [54] take the shape as follows:(55)create a pointϑ∈Cso thatTϑ∈Q.In a HS, SFP was initiated by Censor and Elfving [54], and they used 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 SVFP [55, 56]. SFP is also used in a variety of disciplines such as dynamic emission tomographic image reconstruction, image restoration, and radiation therapy treatment planning [57–59]. Let us consider(56)¥ϑ≔∇12Tϑ−PQTϑ2=T∗I−PQTϑfor the metric projection PQ on to Q, the gradient ∇, and ϒ=∂iC. Due to the above construction, problem (55) has an inclusion format as described in (1). It can be seen that ¥ is Lipschitz continuous with constant L=T2, and ϒ is maximal monotone, see, e.g., [60].LetC be a nonempty CCS of a RHS ℸ, and a normal cone of C at ϑ∈C is defined by(57)NCϑ=z∈ℸ:z,y−ϑ≤0,∀y∈C.Supposeg:ℸ⟶−∞,+∞ is a proper, lower semicontinuous, and convex function. For each ϑ∈ℸ, the subdifferential ∂g of g is given by(58)∂gϑ=z∈ℸ:gy−gϑ≥z,y−ϑ,∀y∈C.For any nonempty CCSC of ℸ, the indicator function iC of C is defined by(59)iCϑ=0,ifϑ∈C∞,otherwise.It is obvious that the indicator functioniC is proper, convex, and lower semicontinuous on ℸ. A subdifferential ∂iC of iC is a maximal monotone operator, and(60)∂iCϑ=z∈ℸ:iCy−iCϑ≥z,y−ϑ,∀y∈C=z∈ℸ:z,y−ϑ≤0,∀y∈C=NCϑ.For eachϑ∈ℸ, now we define the resolvent of an indicator function ∂iC for each λ>0 in the following manner:(61)Jλ∂iC=Id+λ∂iC−1.Hence, we can observe that(62)y=Jλ∂iCϑ⟺ϑ∈y+λ∂iCy−1⟺ϑ−y∈λ∂iCy⟺y=PCϑ.Now, on the basis of the above, Algorithm1 may be reduced to the following scheme.Theorem 3.
Letϑn be a sequence generated by the following scheme: choose ϑ−1,ϑ0∈C, ρ∈0,1, Λ≥0, ℓ0>0, and 0<β<1.St. (i): computeℑn in the following way:(63)ℑn=ϑn+Λϑn−ϑn−1.St. (ii): calculate(64)ψn=PCℑn−ℓnT∗I−PQTℑn.Ifℑn=ψn, stop, and ℑn is a solution of problem (55); otherwise, continue to St. (iii).St. (iii): calculate(65)ϕn=1−βℑn+βψn+βℓnT∗I−PQTℑn−T∗I−PQTψn,whereℓn+1 is the stepsize sequence revised in the following way:(66)ℓn+1=minℓn,ρℑn−ψnT∗I−PQTℑn−T∗I−PQTψn,ifT∗I−PQTℑn≠T∗I−PQTψn,ℓn,otherwise.St. (iv): calculate(67)Cn+1=η∈ℸ:ϕn−η2≤ϑn−η2+Λ2ϑn−1−ϑn2−2Λϑn−η,ϑn−1−ϑn−βΔℑn−ψn2,whereΔ=2−β−2ρ1−βℓn/ℓn+1−βρ2ℓn2/ℓn+12.St. (v): compute(68)ϑn+1=PCn+1ϑ1,n≥1.Putn=n+1, and return to St. (i). If the solution set ΓSFP is nonempty, then the sequence ϑn converges weakly to an element of ΓSFP.


## 6. Numerical Discussion

This part is devoted to present a numerical solution to a SCFP in an infinite HS, which is a special inclusion problem as explained in Section5. The problem setting is taken from [61]. We provide the comparison of Algorithm 1 (Alg1) in [45] and our proposed Algorithm 1 (Alg2).Example 1.
Letℸ1=ℸ2=L20,2π be two HSs with an inner product(69)ϑ,y≔∫02πϑtytdt,∀ϑ,y∈L20,2π,and the induced norm defined by(70)ϑ≔∫02πϑt2dt,∀ϑ∈L20,2π.
Next, consider the feasible setC⊂ℸ1 as(71)C=ϑ∈ℸ1:∫02πϑtdt≤1,and Q⊂ℸ2 is(72)Q=ϑ∈ℸ2:∫02πϑt−sint2dt≤16.
Consider the mappingT:ℸ1⟶ℸ2 such that Tϑs=ϑs,ϑ∈ℸ1. Then, T∗ϑs=ϑs, and T=1. So, we shall solve the following problem:(73)createϑ∗∈Cso thatTϑ∗∈Q.
We can also observe that sinceTϑs=ϑs,ϑ∈ℸ1, the above problem is actually a CFP of the form(74)createϑ∗∈C∩Q.
Figures1–9 and Tables 1 and 2 show the numerical results by assuming Dn=ϑn−ϑn1≤10−6.Figure 1
Numerical conduct of Alg1 by choosing different values ofℓ0.Figure 2
Numerical conduct of Alg1 by choosing different values ofℓ0.Figure 3
Numerical conduct of Alg2 by choosing different values ofℓ0.Figure 4
Numerical conduct of Alg2 by choosing different values ofℓ0.Figure 5
Numerical comparison of Alg2 with Alg.1 by assuming values ofϑ−1=ϑ0=t.
(a)(b)Figure 6
Numerical comparison of Alg2 with Alg.1 by assuming values ofϑ−1=ϑ0=t2/5.
(a)(b)Figure 7
Numerical comparison of Alg2 with Alg1 by assuming values ofϑ−1=ϑ0=2ett5.
(a)(b)Figure 8
Numerical comparison of Alg2 with Alg1 by assuming values ofϑ−1=ϑ0=etsint.
(a)(b)Figure 9
Numerical comparison of Alg2 with Alg1 by assuming values ofϑ−1=ϑ0=t2−etcost.
(a)(b)Table 1
Numerical comparison of Alg2 with Alg1 by assuming different values ofℓ0.
Number of iterationsExecution time in secondsϑ−1=ϑ0ρΛℓ0Alg1Alg2Alg1Alg21/5expt/25/40.270.501.0056500.01360.01901/5expt/25/40.270.500.8062520.02190.01501/5expt/25/40.270.500.6072560.01860.02051/5expt/25/40.270.500.4083620.01600.01831/5expt/25/40.270.500.20104720.02520.0225Table 2
Numerical comparison of Alg2 with Alg1.
Number of iterationsExecution time in secondsϑ−1=ϑ0ρΛℓ0Alg1Alg2Alg1Alg2t0.330.350.5054470.04970.0184t2/50.330.350.5061480.13250.03902ett50.330.350.5071510.11660.0366etsint0.30.350.50103610.21930.0318t2−etcost0.330.350.50157890.33630.0467Remark 1.
It is well known that the success of any iterative method depends on two main things: first, the number of iterations: when the number of iterations is small, the method is successful in saving effort. Second, time factor: the method that needs less time in implementation is excellent than its counterpart, which needs a lot of time and is considered successful in saving time. So, from figures and tables, we observe that our algorithm needs fewer iterations and less time than Algorithm1 [45]. This illustrates that our method is successful in speeding up Algorithm 1 [45] and solving problem (55). Also, the performance of our algorithm is good because it saves time and effort in studding the convergence rate.



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