A SELF-ADAPTIVE ITERATIVE ALGORITHM FOR SOLVING THE SPLIT VARIATIONAL INEQUALITY PROBLEM IN HILBERT SPACES

http://jst.tnu.edu.vn 56 Email: jst@tnu.edu.vn

VARIATIONAL INEQUALITY PROBLEM IN HILBERT SPACES

Nguyen Minh Hieu¹, Tran Thi Huong^2*

1School of Applied Mathematics and Informatics - Hanoi University of Science and Technology

2TNU - University of Technology

ARTICLE INFO ABSTRACT

Received: 16/11/2021 The split variational inequality problem (SVIP) was first introduced by Censor et al. Up to now, there is a long list of works concerning algorithms to solve (SVIP). In this paper, we study the split variational inequality problem in Hilbert spaces. In order to solve this problem, we propose a self-adaptive algorithm. Our algorithm uses dynamic step-sizes, chosen based on information of the previous step and their strong convergence is proved. In comparison with the work by Censor et al. (Numer. Algor., 59:301–323, 2012), the new algorithm gives strong convergence results and does not require information about the spectral radius of the operator. And then, we give a numerical experiment to illustrate the performance of our algorithm.

Revised: 19/4/2022 Published: 27/4/2022

KEYWORDS

Split feasibility problem Variational inequality Hilbert spaces

Nonexpansive mapping Fixed point

THUẬT TOÁN LẶP TỰ THÍCH NGHI GIẢI BÀI TOÁN BẤT ĐẲNG THỨC BIẾN PHÂN TÁCH TRONG KHÔNG GIAN HILBERT

Nguyễn Minh Hiếu¹, Trần Thị Hương^2*

1Viện Toán ứng dụng và Tin học - Trường Đại học Bách khoa Hà Nội

2Trường Đại học Kỹ thuật Công nghiệp – ĐH Thái Nguyên

THÔNG TIN BÀI BÁO TÓM TẮT

Ngày nhận bài: 16/11/2021 Bài toán bất đẳng thức biến phân tách (SVIP) được nghiên cứu đầu tiên bởi Censor và các cộng sự. Đến nay, có rất nhiều công trình nghiên cứu các thuật toán để giải bài toán SVIP. Trong bài báo này, chúng tôi đề cập đến bài toán bất đẳng thức biến phân tách trong không gian Hilbert. Để giải bài toán, chúng tôi trình bày một thuật toán tự thích nghi, sử dụng cỡ bước được chọn dựa trên thông tin của các bước lặp trước đó, đồng thời chứng minh sự hội tụ mạnh của thuật toán. So với công trình nghiên cứu của tác giả Censor (Numer.

Algor., 59:301–323, 2012), thuật toán mới của chúng tôi cho kết quả hội tụ mạnh và không cần sử dụng bán kính phổ của toán tử. Cuối cùng, chúng tôi đưa ra ví dụ minh họa cho phương pháp đã đề xuất.

Ngày hoàn thiện: 19/4/2022 Ngày đăng: 27/4/2022

TỪ KHÓA

Bài toán chấp nhận tách Bất đẳng thức biến phân Không gian Hilbert Ánh xạ không giãn Điểm bất động

DOI: https://doi.org/10.34238/tnu-jst.5260

*Corresponding author. Email: tranthihuong@tnut.edu.vn

1. Introduction

LetH1andH2be two real Hilbert spaces with inner producth., .iand normk.k. Variational Inequality Problem(VIP) [1], [2] is the problem of finding a pointu^∗in a subsetCof a Hilbert space H such that

hAu^∗,u−u^∗i ≥0 ∀u∈C, (VIP(A,C)) whereA:C→H is a mapping, and we denote solution set of (VIP(A,C)) byS_(A,C).

TheSplit Feasibility Problem(SFP) proposed by Censor and Elfving [3] is finding a point

u^∗∈C and Fu^∗∈Q, (SFP)

whereCandQare nonempty closed convex subsets of real Hilbert spacesH1andH2, respectively andF:H1→H2is a bounded linear operator.

In this paper we discuss a self-adaptive algorithm for solving theSplit Variational Inequality Problemwhich was studied by Censor et al. in [4]

findu^∗∈S_(A,C) and Fu^∗∈S_(B,Q). (SVIP) To solve the (SVIP), Censor et al. [4] presented a weak convergence result whenA and Bare ηA,ηB-inverse strongly monotone operators onH1andH2, respectively.

In the present article, our aim is to introduce an iterative algorithm to solve the (SVIP) by using the viscosity approximation method [5], cyclic iterative methods [3], [6], [7] and a modification of theCQ–algorithm [8], [9]. We prove the strong convergence of the presented algorithm under some mild conditions. Particularly, in our method, the step size is selected in such a way that its implementation does not need any prior information on the norm of the transfer operators.

2. Preliminaries

In this section, we introduce some mathematical symbols, definitions, and lemmas which can be used in the proof of our main result.

Let H be a real Hilbert space with inner producth., .i and normk.k andCbe a nonempty, closed and convex subset ofH. In what follows, we writex^k*xto indicate that the sequence{x^k} converges weakly toxwhilex^k→xindicate that the sequence{x^k}converges strongly tox. It is known that in a Hilbert spaceH,

2hx,yi=kx+yk²− kxk²− kyk²=kxk²+kyk²− kx−yk², (1) and

kλx+ (1−λ)yk²=λkxk²+ (1−λ)kyk²−λ(1−λ)kx−yk², (2) for allx,y∈H andλ∈R(see, for example [10, Lemma 2.13], [11]). For eachx∈H there exists a mappingP_C:H →Csuch thatkx−P_Cxk ≤ kx−yk ∀x,y∈C.The mappingP_Cis called the metric projection ofH ontoC.

Lemma 2.1.(see [12]) (i) PCis a nonexpansive mapping.

(ii) P_Cx∈C ∀x∈H and P_Cx=x ∀x∈C.

(iii) x∈H,y=P_Cx if and only if y∈C andhx−y,z−yi ≤0 ∀z∈C.

Definition 2.1. An operator T :H →H is called a contraction operator with the contraction coefficientτ∈[0,1)ifkT x−Tyk ≤τkx−yk ∀x,y∈H.

It is easy to see that, ifT is a contraction operator, thenPCT is a contraction operator too. If τ≥0 we haveτ-Lipschitz continuous operator.

Definition 2.2. LetH1andH2be two Hilbert spaces and letF :H1→H2be a bounded linear operator. An operatorF^∗:H2→H1with the propertyhFx,yi=hx,F^∗yifor allx∈H1andy∈H2, is called an adjoint operator ofF.

The adjoint operator of a bounded linear operatorF on a Hilbert space always exists and is uniquely determined. Furthermore,F^∗is a bounded linear operator.

Definition 2.3.An operatorA:H →H is called anη-inverse strongly monotone operator with constantη>0 ifhAx−Ay,x−yi ≥ηkAx−Ayk² ∀x,y∈H.

It is easy to see that, if A is an η-inverse strongly monotone operator, then I^H −λA is a nonexpansive mapping forλ ∈(0,2η].

Lemma 2.2.(see [4])Let A:C→H beη-inverse strongly monotone on C andλ>0be a constant satisfying0<λ≤2η. Define the mapping T:C→C by taking

T x=P_C I^H −λA

x ∀x∈C. (3)

Then T is nonexpansive mapping on C, furthermore,Fix(T) =S_(A,C)is the set of fixed points of T , whereFix(T):={x∈C

T x=x}.

Lemma 2.3.(see [12])Assume that T be a nonexpansive mapping of a closed and convex subset C of a Hilbert spaceH intoH. Then the mapping I^H −T is demiclosed on C; that is, whenever{x^k} is a sequence in C which weakly converges to some point u^∗∈C and the sequence{(I^H −T)x^k} strongly converges to some y, it follows that(I^H −T)u^∗=y.

From Lemma , ifx^k*u^∗and(I^H −T)x^k→0, thenu^∗∈Fix(T).

Lemma 2.4.(See [2])Let{s_k}be a real sequence which does not decrease at infinity in the sense that there exists a subsequence{sk_n}such that sk_n ≤sk_n+1∀n≥0.Define an integer sequence by ν(k):=max

k₀≤n≤k|s_n<s_n+1 ,k≥k₀.Thenν(k)→∞as k→∞and for all k≥k₀, we have max{s_ν(k),sk} ≤s_ν(k)+1.

Lemma 2.5. (see [13])Let{s_k}be a sequence of nonnegative numbers satisfying the condition s_k+1≤(1−bk)sk+bkck,k≥0,where{bk}and{ck}are sequences of real numbers such that

(i) {b_k} ⊂(0,1)for all k≥0and∑^∞_k=1bk=∞, (ii) lim sup_k→∞c_k≤0.

Then,limk→∞sk=0.

3. Main Results

In this section, we use the viscosity approximation method and a modification of theCQ–

algorithm to establish the strong convergence of the proposed algorithm for finding the solution of the (SVIP). We consider the (SVIP) under the following conditions.

Assumption 3.1.

(A1) A:H1→H1isηA-inverse strongly monotone onH1. (A2) B:H2→H2isηB-inverse strongly monotone onH2. (A3) F:H1→H2be a bounded linear operator.

(A4) T:H1→H1is a contraction operator with the contraction coefficientτ∈[0,1).

(A5) The solution setΩ_SVIPof (SVIP) is not empty.

IfAandBsatisfy the properties(A1)and(A2), respectively, the solution setsS_(A,C)andS_(B,Q) are closed and convex. Here, for the sake of convenience, an empty set is considered to be closed and convex. Therefore, the solution setΩSVIPof the (SVIP) is also closed and convex.

Algorithm 1

Step 0. Select the initial pointx¹∈H1and the sequences{α_k},{β_k},{ρ_k},{κ_k}, andλ such that the conditions

{α_k} ⊂(0,1),αk→0 ask→∞, and

∞

∑

k=1

αk=∞, (C1)

0<λ≤2η,η=min{η_A,ηB}, (C2)

{β_k} ⊂[a,b]⊂(0,1), (C3)

{ρ_k} ⊂[c,d]⊂(0,1), {κ_k} ⊂(0,K), K>0 (C4) are satisfied. Setk:=1.

Step 1. Computey^k=βkx^k+ (1−βk)PC I^H1−λA x^k. Step 2. Computez^k=PQ I^H2−λB

Fy^k.

Step 3. Computev^k=y^k+γkF^∗(z^k−Fy^k), where the step sizeγk is defined by

γk=ρk

kz^k−Fy^kk² kF^∗(z^k−Fy^k)k²+κk

. (4)

Step 4. Computex^k+1=αkT x^k+ (1−αk)v^k. Step 5. Setk:=k+1 and go toStep 1.

Theorem 3.1.Suppose that all conditions in Assumption 3.1 are satisfied. Then the sequence{x^k} generated by Algorithm1converges strongly to the unique solution of theVIP(I^H1−T,Ω_SVIP).

Proof. SinceT is a contraction mapping,P_ΩSVIPT is a contraction too. By Banach contraction operator principle, there exists a unique pointu^∗∈Ω_SVIP such thatP_ΩSVIPTu^∗=u^∗. By Lemma 2.1(iii), we obtainu^∗is the unique solution to the VIP(I^H1−T,Ω_SVIP). Sinceu^∗∈Ω_SVIP,u^∗∈S_(A,C) andFu^∗∈S_(B,Q).

Letu∈Ω_SVIP,u∈S_(A,B). Since Lemma 2.2,u=P_C I^H1−λA

u. From Step 1 in Algorithm1, the nonexpansive property ofPC I^H1−λA

(see Lemma 2.2), and (2), we have that ky^k−uk²=

βk(x^k−u) + (1−βk)h

P_C I^H1−λA

x^k−P_C I^H1−λA ui

2

≤βkkx^k−uk²+ (1−βk)kx^k−uk²−βk(1−βk)

x^k−P_C I^H1−λA x^k

2

=kx^k−uk²−β_k(1−βk)

x^k−PC I^H1−λA x^k

2 (5)

≤ kx^k−uk². (6)

It follows from Step 3 in Algorithm1, the property of adjoint operatorF^∗, and (1) that kv^k−uk²=

y^k+γkF^∗ z^k−Fy^k

−u

2

=ky^k−uk²+γ_k²

F^∗ z^k−Fy^k

2+2γk

y^k−u,F^∗ z^k−Fy^k)

=ky^k−uk²+γ_k²

F^∗ z^k−Fy^k

2+2γk

Fy^k−Fu,z^k−Fy^k

=ky^k−uk²+γ_k²kF^∗ z^k−Fy^k k²+γk

z^k−Fu

2−

Fy^k−Fu

2−

z^k−Fy^k

2 . Sinceu∈Ω_SVIP,Fu∈S_(B,Q). It follows from Lemma 2.2 thatFu=PQ I^H2−λB

Fu. From Steps 2 and 3 in Algorithm1, the nonexpansive property ofPQ I^H2−λB

, (4), (C4), and the last inequality, we obtain

kv^k−uk²=ky^k−uk²+γ_k²

F^∗ z^k−Fy^k

2

+γk

PQ I^H2−λB

Fy^k−PQ I^H2−λB Fu

2−

Fy^k−Fu

2−

z^k−Fy^k

2

≤ ky^k−uk²+γ_k²

F^∗ z^k−Fy^k

2+γk

Fy^k−Fu

2−

Fy^k−Fu

2−

z^k−Fy^k

2

=ky^k−uk²+γ_k²

F^∗ z^k−Fy^k

2−γk

z^k−Fy^k

2

≤ ky^k−uk²+ρ_k²

z^k−Fy^k

4

F^∗(z^k−Fy^k)

2+κk

2

F^∗ z^k−Fy^k

2+κk

−ρk

z^k−Fy^k

4

F^∗(z^k−Fy^k)

2+κk

=ky^k−uk²−ρk(1−ρk)

z^k−Fy^k

4

F^∗(z^k−Fy^k)

2+κk

(7)

≤ ky^k−uk². (8)

It follows from the convexity of the norm functionk.konH1, the contraction property ofT with the contraction coefficientτ∈[0,1), (6), (8), and Step 4 in Algorithm1that

kx^k+1−uk=

αk T x^k−u

+ (1−αk)(v^k−u) ≤αk

T x^k−Tu +

Tu−u

+ (1−αk)kv^k−uk

≤τ αkkx^k−uk+αk

Tu−u

+ (1−αk)kx^k−uk

=

1−(1−τ)αk

kx^k−uk+ (1−τ)αk

Tu−u 1−τ

≤max n

kx^k−uk,

Tu−u 1−τ

o

≤ · · · ≤max n

kx⁰−uk,

Tu−u 1−τ

o .

This implies that the sequence{x^k}is bounded. SincePCandPQare nonexpansive mappings andF is the bounded linear operator, we also have the sequences{y^k},{z^k}, and{v^k}are bounded.

Now we claim that limn→∞kx^k−u^∗k=0, whereu^∗is the unique solution of the VIP(I^H1−T,ΩSVIP), that is,u^∗=P_ΩSVIPTu^∗. Indeed, from the convexity ofk.k², Step 4 in Algorithm1, (5), (7) withu replaced byu^∗, and the condition (C1), we get

kx^k+1−u^∗k²=

αk(T x^k−u^∗) + (1−αk)(v^k−u^∗)

2≤αk

T x^k−u^∗

2+ (1−αk)kv^k−u^∗k²

≤αk

T x^k−u^∗

2+ky^k−u^∗k²−ρk(1−ρk)

z^k−Fy^k

4

F^∗(z^k−Fy^k)

2+κk

≤αk

T x^k−u^∗

2+kx^k−u^∗k²−ρk(1−ρk)

z^k−Fy^k

4

F^∗(z^k−Fy^k)

2+κk

−βk(1−βk)

x^k−PC I^H1−λA x^k

2.

Hence,

ρk(1−ρk)

z^k−Fy^k

4

F^∗(z^k−Fy^k)

2+a_k

+βk(1−βk)

x^k−PC I^H1−λA x^k

2

≤

kx^k−u^∗k²− kx^k+1−u^∗k² +αk

T x^k−u^∗

2. (9)

Next, from Step 4 in Algorithm1and the contraction property ofT with the contraction coefficient τ∈[0,1), we have that

kx^k+1−u^∗k²=hαk(T x^k−u^∗) + (1−αk)(v^k−u^∗),x^k+1−u^∗i

= (1−αk)hv^k−u^∗,x^k+1−u^∗i+αkhT x^k−u^∗,x^k+1−u^∗i

≤1−αk

2

kv^k−u^∗k²+kx^k+1−u^∗k²

+αkhT x^k−Tu^∗,x^k+1−u^∗i+αkhTu^∗−u^∗,x^k+1−u^∗i

≤1−αk

2

kv^k−u^∗k²+kx^k+1−u^∗k² +αk

2

τkx^k−u^∗k²+kx^k+1−u^∗k²

+αkhTu^∗−u^∗,x^k+1−u^∗i.

This implies thatkx^k+1−u^∗k²≤(1−αk)kv^k−u^∗k²+αkτkx^k−u^∗k²+2αkhTu^∗−u^∗,x^k+1−u^∗i.

From (6), (8) withureplaced byu^∗, and the last inequality, we obtain kx^k+1−u^∗k²≤

1−(1−τ)αk

kx^k−u^∗k²+2αkhTu^∗−u^∗,x^k+1−u^∗i. (10)

We consider two cases.

Case 1. There exists an integerk₀≥0 such thatkx^k+1−u^∗k ≤ kx^k−u^∗kfor allk≥k₀.

Then, limk→∞kx^k−u^∗kexists. From the boundedness of the sequence{T x^k}, the conditions (C1), (C3), and (C4), it follows from (9) that

k→∞lim

x^k−PC I^H1−λA x^k

=0, (11) and

k→∞limkz^k−Fy^kk=0. (12)

From Step 1 in Algorithm1and (C3), we get

k→∞limkx^k−y^kk= (1−βk)lim

k→∞

x^k−PC I^H1−λA x^k

=0. (13) Hence,

k→∞lim

I^H1−PC I^H1−λA x^k

=0. (14) From Step 2 in Algorithm1and (12), we obtain

k→∞lim

I^H2−PQ I^H2−λB Fy^k

=0. (15) From Step 3 in Algorithm1, the property of adjoint operatorF^∗, and (12), we obtain

kv^k−y^kk=γkkF^∗(z^k−Fy^k)k →0 as k→∞. (16) It follows from (13) and (16) that

kx^k−v^kk →0 as k→∞. (17)

Using the boundedness of{v^k}and{T x^k}, Step 4 in Algorithm1, and the condition (C1), we also havekx^k+1−v^kk=αkkT x^k−v^kk →0 as k→∞.When combined with (17), this implies that

kx^k+1−x^kk →0 as k→∞. (18) Now we show that lim sup_k→∞hTu^∗−u^∗,x^k+1−u^∗i ≤0. Indeed, suppose that{x^kn}is a subsequence of{x^k}such that

lim sup

k→∞

hTu^∗−u^∗,x^k−u^∗i= lim

k_n→∞hTu^∗−u^∗,x^kn−u^∗i. (19) Since{x^kn}is bounded, there exists a subsequence{x^knl}of{x^kn}which converges weakly to some pointu^†. Without loss of generality, we may assume thatx^kn *u^†. We will prove thatu^†∈Ω_SVIP. Indeed, from (14), Lemmas 2.2 and 2.3, we obtainu^†∈S_(A,C).Moreover, sinceF is a bounded linear operator,Fx^kn *Fu^†. Using (17), Lemmas 2.2 and 2.3, we also obtainFu^†∈S_(B,Q). Hence, u^†∈Ω_SVIP. So, fromu^∗=P_ΩSVIPTu^∗, (19), and Lemma 2.1(iii) we deduce that

lim sup

k→∞

hTu^∗−u^∗,x^k−u^∗i=hTu^∗−u^∗,u^†−u^∗i ≤0, which combined with (18) gives

lim sup

k→∞

hTu^∗−u^∗,x^k+1−u^∗i ≤0. (20)

Now, the inequality (10) can be rewritten in the formkx^k+1−u^∗k²≤(1−bk)kx^k−u^∗k²+bkck, k≥0, where b_k = (1−τ)αk and c_k = _1−τ² hTu^∗−u^∗,x^k+1−u^∗i. Since the condition (C1) and τ∈[0,1), {b_k} ⊂(0,1) and∑^∞_k=1bk =∞. Consequently, fromτ ∈[0,1) and (20), we have that lim sup_k→∞ck≤0. Finally, by Lemma 2.5, limk→∞kx^k−u^∗k=0.

Case 2. There exists a subsequence{k_n}of{k}such thatkx^kn−u^∗k ≤ kx^kn+1−u^∗kfor alln≥0.

Hence, by Lemma 2.4, there exists an integer, nondecreasing sequence{ν(k)}fork≥k₀(for some k₀large enough) such thatν(k)→∞ask→∞,

kx^ν(k)−u^∗k ≤ kx^ν(k)+1−u^∗k and kx^k−u^∗k ≤ kx^ν(k)+1−u^∗k (21) for eachk≥0. From (10) withkreplaced byν(k), we have

0<kx^ν(k)+1−u^∗k²− kx^ν(k)−u^∗k²≤2α_ν(k)hTu^∗−u^∗,x^ν(k)+1−u^∗i.

Sinceα_ν(k)→0 and the boundedness of{x^ν(k)}, we conclude that

k→∞lim kx^ν(k)+1−u^∗k²− kx^ν(k)−u^∗k²

=0. (22)

By a similar argument to Case 1, we obtain

k→∞lim

I^H1−P_C I^H1−λA x^ν(k)

=0 and lim

k→∞

I^H2−P_Q I^H2−λB

Fy^ν(k) =0.

Also we get kx^ν(k)+1−u^∗k² ≤

1−(1−τ)α_ν(k)

kx^ν(k)−u^∗k²+2α_ν(k)hTu^∗−u^∗,x^ν(k)+1−u^∗i, where lim sup

n→∞

hTu^∗−u^∗,x^ν(k)+1−u^∗i ≤0.Since the first inequality in (21) andα_ν(k)> 0, we have that(1−τ)kx^ν(k)−u^∗k²≤2hTu^∗−u^∗,x^ν(k)+1−u^∗i.

Thus, from lim sup_n→∞hTu^∗−u^∗,x^ν(k)+1−u^∗i ≤0 andτ∈[0,1), we get lim

k→∞kx^ν(k)−u^∗k²=0.

This together with (22) implies that lim

k→∞kx^ν(k)+1−u^∗k² =0. Which together with the second inequality in (21) implies that lim

k→∞kx^k−u^∗k=0.This completes the proof.

4. Numerical Results

We give a numerical experiment to illustrate the performance of our algorithm. This result is performed in Python running on a laptop Dell Latitude 7480 Intel core i5, 2.40 GHz 8GB RAM.

Example 4.1. LetH1=R³andH2=R⁴. OperatorsA:R³→R³andB:R⁴→R⁴are defined by

Ax=





3 3 1

3 3 1

1 1 5







 x₁ x₂ x₃



,x=



 x₁ x₂ x₃



∈R³ and Bx=









1 1 1 1

1 1 1 1

1 1 3 1

1 1 1 1















 x₁ x₂ x₃ x₄







 , x=







 x₁ x₂ x₃ x₄









∈R⁴

that are inverse strongly monotone operator with constantηA=¹₇ andηB= ¹

3+√

3. Bounded linear operatorF:R³→R⁴,Fx=









1 0 2

1 0 2

0 0 −3

0 1 2











 x₁ x₂ x₃



. AndT x:R³→R³,T x=¹₂xis contractive operator

with constantτ=¹₂. LetCandQare defined by

C={x∈R³,ha₁,xi ≤b₁},witha₁=

−1 0 1>

,b₁=2;

Q={x∈R⁴,ha₂,xi ≤b₂}, witha₂=

1 0 1 0>

,b₂=3.

ΩSVIP = n

x=

t −t 0>

t ∈R:t≥ −2o

. The unique solution of VIP

I^R3−T,ΩSVIP

is x^∗=

0 0 0>

.Now, chooseαk=k^−0.5,λ =0.25,βk=0.5,ρk=0.25 andκk=0.1, tolerance ε=10⁻³and initial pointx¹=

1 3 1>

, we get x=

−6.78489854×10⁻⁴ 6.78489983×10⁻⁴ 2.71210451×10⁻¹⁰>

. This result archived within 11.9×10⁻³seconds.

Next, we used different choices of parameters. Table 1 shown below is the performance with differentαkparameter,λ =0.25, βk=0.5,ρk=0.25 andκk=0.1.

Table1: Result with differentαk

ε

αk=k^−0.5 αk=k^−0.8

kx−x^∗k time (s) k kx−x^∗k time (s) k

10⁻³ 0.96×10⁻³ 11.9×10⁻³ 53 0.99×10⁻³ 63.8×10⁻³ 632 10⁻⁶ 0.99×10⁻⁶ 33.9×10⁻³ 196 0.99×10⁻⁶ 857.7×10⁻³ 10688 10⁻⁹ 0.99×10⁻⁹ 54.8×10⁻³ 433 0.99×10⁻⁹ 7107.3×10⁻³ 64382

Then we changed the initial point, with the same choice of parameters, asαk=k^−0.5, λ =0.25, βk=0.5,ρk=0.25 andκk=0.1. The results are recorded in Table 2.

Table2: Result with different initial vector

ε

x¹=

1 1 1>

x¹=

9 9 9>

kx−x^∗k time (s) k kx−x^∗k time (s) k 10⁻³ 0.78×10⁻³ 2.9×10⁻³ 7 0.91×10⁻³ 3.9×10⁻³ 11 10⁻⁶ 0.93×10⁻⁶ 10.9×10⁻³ 51 0.99×10⁻⁶ 13.9×10⁻³ 98 10⁻⁹ 0.97×10⁻⁹ 34.9×10⁻³ 192 0.97×10⁻⁹ 41.8×10⁻³ 297

5. Conclusion

In this paper, we introduced a new algorithm (Algorithm1) and a new strong convergence theorem (Theorem 3.1) for solving the (SVIP) in a real Hilbert spaces without prior knowledge of operators norms. We consider a numerical example to illustrate the effectiveness of the proposed algorithm.

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