A Relaxed Self-Adaptive Projection Algorithm for Solving ...downloads.hindawi.com/journals/jfs/2020/6183214.pdf · B are the spectral radiuses of A∗A and B∗B, respectively. Since

Research ArticleA Relaxed Self-Adaptive Projection Algorithm for Solving theMultiple-Sets Split Equality Problem

Haitao Che 1 and Haibin Chen 2

1School of Mathematics and Information Science, Weifang University, Weifang, Shandong, China2School of Management Science, Qufu Normal University, Rizhao, Shandong, China

Correspondence should be addressed to Haibin Chen; [email protected]

Received 11 March 2020; Accepted 20 April 2020; Published 11 May 2020

Guest Editor: Chuanjun Chen

Copyright © 2020 Haitao Che and Haibin Chen. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

In this article, we introduce a relaxed self-adaptive projection algorithm for solving the multiple-sets split equality problem. Firstly,we transfer the original problem to the constrained multiple-sets split equality problem and a fixed point equation system isestablished. Then, we show the equivalence of the constrained multiple-sets split equality problem and the fixed point equationsystem. Secondly, we present a relaxed self-adaptive projection algorithm for the fixed point equation system. The advantage ofthe self-adaptive step size is that it could be obtained directly from the iterative procedure. Furthermore, we prove theconvergence of the proposed algorithm. Finally, several numerical results are shown to confirm the feasibility and efficiency ofthe proposed algorithm.

1. Introduction

Let H1,H2, andH3 be real Hilbert spaces. For i = 1, 2,⋯, t,j = 1, 2,⋯, r, Ci and Qj are nonempty closed convex subsetsof Hilbert spaces H1 and H2, respectively, and assume thatA : H1 ⟶H3, B : H2 ⟶H3 are two bounded linear opera-tors. The multiple-sets split equality problem (MSSEP) is tofind x and y satisfying the property

x ∈ C =\ti=1

Ci, y ∈Q =\rj=1

Qj such thatAx = By: ð1Þ

When B = I, MSSEP (1) reduces to the multiple-sets splitfeasibility problem

find a point x ∈ C =\ti=1

Ci, Ax ∈Q =\rj=1

Qj, ð2Þ

which is applied to intensity-modulated radiation ther-apy [1–11], signal processing [12–21], and image recon-struction [22–38]. Censor et al. [39] proposed the

proximity function pðxÞ to measure the distance of a pointto all sets

p xð Þ = 12〠

t

i=1li x − PCi

xð Þ�� 2 + 12〠

r

j=1λj Ax − PQj

Axð Þ�� 2, ð3Þ

where li > 0 for all i, and λ j > 0 for all j with ∑ti=1 li +

∑rj=1 λj = 1 . To solve (2), they considered the following

constrained MSSEP:

find a point x ∈Ω such that x solves 2ð Þ, ð4Þ

and then presented the projection method

xk+1 = PΩ xk − s∇p xð Þð Þ, ð5Þ

where s > 0 and Ω is an auxiliary simple nonempty closedconvex set with Ω ∩ S ≠∅, and S denotes the solution setof (2). The convergence of the projection method wasobtained under some mild conditions.

HindawiJournal of Function SpacesVolume 2020, Article ID 6183214, 12 pageshttps://doi.org/10.1155/2020/6183214

https://orcid.org/0000-0002-4632-0603

https://orcid.org/0000-0001-6416-0239

https://creativecommons.org/licenses/by/4.0/






https://doi.org/10.1155/2020/6183214

When t = r = 1, MSSEP (1) reduces to the split equalityproblem which was introduced by Moudafi [40] as follows:

find two points x ∈ C, y ∈Q such thatAx = By, ð6Þ

which is applied to the game theory [41] and optimal con-trol and approximation theory [42]. The following alter-nating CQ algorithm (ACQ) was introduced by Moudafi[40] as follows:

xk+1 = PC xk − γkA∗ Axk − Bykð Þð Þ,

yk+1 = PQ yk + βkB∗ Axk+1 − Bykð Þð Þ,

(ð7Þ

where γk, βk ∈ ðε, min fð1/λAÞ, ð1/λBÞg − εÞ for smallenough ε > 0, A∗ and B∗ denote the adjoint of A and B,respectively. λA and λB are the spectral radiuses of A∗Aand B∗B, respectively. Since the computation of PC andPQ onto a closed convex subset might be hard to beimplemented, Fukushima [43] suggested a way to computethe projection onto a level set of a convex function byconsidering a sequence of projections onto half-spacescontaining the original level set. Then, Moudafi [44] intro-duced the following relaxed alternating CQ algorithm(RACQ):

xk+1 = PCkxk − γkA

∗ Axk − Bykð Þð Þ,yk+1 = PQk

yk + βkB∗ Axk+1 − Bykð Þð Þ,

(ð8Þ

where Ck and Qk are two sequences of closed convex sets.Recently, Dang et al. [45] gave the following relaxed two-

point projection method to solve MSSEP (1):

xk+1 = PΩ1xk − γ 〠

t

i=1αi xk − PCi,k

xkð Þ� �

+ AT Axk − Bykð Þ ! !

,

yk+1 = PΩ2yk − γ 〠

r

j=1βj yk − PQj,k

ykð Þ� �

− BT Axk+1 − Bykð Þ ! !

,

8>>>>><>>>>>:

ð9Þ

where γ ∈ ð0, min ff1/21/2, 1/ð4kAk2Þð1/4kAk2Þ, 1/ð4kBk2ÞgÞ,Ci,k, i = 1, 2,⋯, r andQj,k, j = 1, 2,⋯, t are two sequences ofclosed convex sets corresponding to Ci and Qj, respec-tively. Ω1 ⊂H1 and Ω2 ⊂H2 are auxiliary simple sets. αi> 0 for all i, and βj > 0 for all j with ∑t

i=1αi +∑rj=1βj = 1.

Under some mild conditions, the weak convergence ofthe algorithm (9) was obtained.

Noting that the determination of the stepsize γ of algo-rithm (9) depends on the operator (matrix) norms ∥A∥ and∥B∥. This implies that if we implement the relaxed two-point projection method (9), one first need to calculate oper-ator norms of A and B, which is in general not an easy workin practice. To overcome this weakness, Lopez et al. [46] andZhao and Yang [47] introduced self-adaptive methods ofwhich the advantage of the methods is that the stepsizes donot need prior knowledge of the operator norms. Motivatedby them, we introduce a relaxed self-adaptive projection

algorithm for solving the multiple-sets split equality problem.First, we transfer the origin problem to the constrainedmultiple-sets split equality problem and establish the fixedpoint equation system. We show the equivalence of the con-strained multiple-sets split equality problem and the fixedpoint equation system. Second, based on the fixed pointequation system, we present a relaxed self-adaptive projec-tion algorithm for solving the constrained multiple-sets splitequality problem, and the convergence of the proposed algo-rithm is obtained. Finally, several numerical results areshown to confirm the feasibility and efficiency of the pro-posed algorithm.

The remainder of this paper is organized as follows. Sec-tion 2 shows some preliminaries and notations used for sub-sequent analysis. In Section 3, we transfer the origin problemto the constrained multiple-sets split equality problem andestablish the fixed point equation system and propose arelaxed self-adaptive projection algorithm for solving theconstrained multiple-sets split equality problem. The conver-gence of the proposed algorithm is obtained. In Section 4,several numerical results are shown to confirm the effective-ness of our algorithm.

2. Preliminaries

Throughout this paper, we use ⟶ and ⇀ to denote thestrong convergence and weak convergence, respectively. Wewrite ωwðxkÞ = fx : ∃xkj ⇀ xg to indicate the weak ω-limit

set of fxkg. For any x ∈H, there exists a unique nearest pointin C, denoted by PCx, such that

∥x − PCx∥ ≤ ∥x − y∥,∀y ∈ C: ð10Þ

It is well known that PC is nonexpansive and firmly non-expansive. Moreover, PC has the following well-known prop-erties (see for example [48]).

Lemma 1. Let C ⊂H be nonempty, closed and convex. Thenfor all x, y ∈H and z ∈ C,

(i) hx − PCx, z − PCxi ≤ 0

(ii) ∥PCx − PCy∥2 ≤ hPCx − PCy, x − yi ;

(iii) ∥PCx − z∥2 ≤ ∥x − z∥2−∥PCx − x∥2:

Definition 2. Letf : H ⟶ Rbe convex. The subdifferential of fat xis defined as

∂f xð Þ = ξ ∈H ∣ f yð Þ ≥ f xð Þ + ξ, y − xh i,∀y ∈Hf g: ð11Þ

An element of ∂f ðxÞ is said to be a subgradient.

Lemma 3. Suppose f : H ⟶ R is a convex function, then it issubdifferentiable everywhere and its subdifferentials are uni-formly bounded set of H:

2 Journal of Function Spaces

3. Algorithm and Its Convergence

In this section, we focus on a relaxed self-adaptive projectionalgorithm and obtain the convergence of the proposed algo-rithm. Following the idea of Censor et al. [39], we give twoadditional closed convex sets Ω1 ⊂H1 and Ω2 ⊂H2 and con-sider the constrained multiple-sets split equality problem

find x ∈Ω1, y ∈Ω2 such that x, yð Þ solves 1ð Þ, ð12Þ

where the sets Ci and Qj can be expressed by

Ci = x ∈H1 ∣ ci xð Þ ≤ 0f g,Qj = y ∈H2 ∣ qj ykð Þ ≤ 0

n o,

ð13Þ

ci : H1 ⟶ R and qj : H2 ⟶ R are convex functions forall i = 1, 2,⋯, t and j = 1, 2,⋯, r, and Γ denotes the solutionset of (32). Define

Ci,k = x ∈H1 ∣ ci xkð Þ + ξi,k, x − xk� �

≤ 0� �

, ð14Þ

where ξi,k ∈ ∂ciðxkÞ and

Qj,k = y ∈H2 ∣ qj ykð Þ + ηj,k, y − ykD E

≤ 0n o

, ð15Þ

where ηj,k ∈ ∂qjðykÞ. It is easily seen that Ci ⊂ Ci,k and Qj ⊂Qj,k for all k. Notice that Ci,k andQj,k are half-spaces and thusthe corresponding projections have closed-form expressions.Hence, we focus on the following multiple-sets split equalityproblem (CMSSEP):

find x ∈Ω1, y ∈Ω2 to solve x

∈ C =\ti=1

Ci,k, y ∈Q =\rj=1

Qj,ksuch thatAx = By:

ð16Þ

Now, we define the proximity function pkðx, yÞ:

pk x, yð Þ = 12〠

t

i=1αi x − PCi,k

xð Þ�� 2 + 1

2〠r

j=1βj y − PQj,k

yð Þ�� 2

+ 12 Ax − Byk k2,

ð17Þ

where αi > 0 for all i, and βj > 0 for all j with ∑ti=1 αi +∑r

j=1βj = 1.

Using the proximity function pkðx, yÞ, we can obtain thefollowing technical lemmas.

Lemma 4. Assume that (16) is consistent (i.e.,(16) has a solu-tion) and denotes its solution set by Γ. If ðx,yÞ ∈ Γ, then itsolves the fixed point equation system

x = PΩ1x − λ 〠

t

i=1αi x − PCi,k

xð Þ� �

+ AT Ax − Byð Þ ! !

,

y = PΩ2y − β 〠

r

j=1βj y − PQj,k

yð Þ� �

− BT Ax − Byð Þ ! !

:

8>>>>><>>>>>:

ð18Þ

Proof. To solve the problem (16), we consider the minimiza-tion problem

min pk x, yð Þ ∣ x ∈Ω1, y ∈Ω2f g: ð19Þ

(19) leads to the following unconstrained optimizationproblem:

minx∈Ω1,y∈Ω2

δΩ1xð Þ + δΩ2

yð Þ + pk x, yð Þ� �, ð20Þ

where δΩiis a indicator function of Ωi for i = 1, 2 defined by

δΩixð Þ =

0, x ∈Ωi,+∞, otherwise:

(ð21Þ

Note that ∂δΩ1ðxÞ =NΩ1

ðxÞ and ∂δΩ2ðyÞ =NΩ2

ðyÞ,where NΩ1

and NΩ2are the normal cone of the convex sets

Ω1 and Ω1, respectively. From the optimality conditions of(20), it yields

0 ∈ 〠t

i=1αi x − PCi,k

xð Þ� �

+ AT Ax − Byð Þ + ∂δΩ1xð Þ,

0 ∈ 〠r

j=1βj y − PQj,k

yð Þ� �

− BT Ax − Byð Þ + ∂δΩ2yð Þ,

8>>>>><>>>>>:

ð22Þ

which means that, for λ > 0, β > 0,

x − λ 〠t

i=1αi x − PCi,k

xð Þ� �

+ AT Ax − Byð Þ !

= x + λ∂δΩ1xð Þ,

y − β 〠r

j=1βj y − PQj,k

yð Þ� �

− BT Ax − Byð Þ !

= y + β∂δΩ2yð Þ,

ð23Þ

that is,

x = I + λNΩ1

−1 x − λ 〠t

i=1αi x − PCi,k

xð Þ� �


,

y = I + βNΩ2

−1 y − β 〠r

j=1βj y − PQj,k

yð Þ� �


:

ð24Þ

3Journal of Function Spaces

Since ðI + λNΩ1Þ−1 = PΩ1

and ðI + βNΩ2Þ−1 = PΩ2

, weobtain

x = PΩ1x − λ 〠

t

i=1αi x − PCi,k

xð Þ� �


,

y = PΩ2y − β 〠

r

j=1βj y − PQj,k

yð Þ� �


:

8>>>>><>>>>>:

ð25Þ

Thus, the desired result can be obtained.The following lemma reveals that ESEP (16) is equivalent

to the fixed point equation system (18).

Lemma 5. Assume that the problem (16) is consistent. ðx∗, y∗Þ∈ Γ solves ESEP (2) if and only if ðx∗, y∗Þ solves the fixed pointequation system (18).

Proof. From Lemma 4, we reveal that ðx∗, y∗Þ can solve (16);it also can solve (18). Next, we will prove that ðx∗, y∗Þ cansolve (18), it also can solve (16). Obviously, one has x∗

∈Ω1, and y∗ ∈Ω2. It follows from the proposition of pro-jection that

x∗ − λ 〠t

i=1αi x∗ − PCi,k

x∗ð Þ� �

+ AT Ax∗ − By∗ð Þ !

− x∗, u − x∗* +

≤ 0, u ∈ Γ,

y∗ − β 〠r

j=1βj y∗ − PQj,k

y∗ð Þ� �

− BT Ax∗ − By∗ð Þ !

− y∗, v − y∗* +

≤ 0, v ∈ Γ:

8>>>>><>>>>>:

ð26Þ

which means

−λ 〠t


x∗ð Þ� �

+ AT Ax∗ − By∗ð Þ !

, u − x∗* +

≤ 0, u ∈ Γ,

−β 〠r


y∗ð Þ� �

− BT Ax∗ − By∗ð Þ !

, v − y∗* +

≤ 0, v ∈ Γ:

8>>>>><>>>>>:

ð27Þ

Hence, from Lemma 3, we add two inequalities toobtain

〠t


x∗ð Þ�� 2 + 〠

r


y∗ð Þ�� 2

+ Ax∗ − By∗, Bv − Au + Ax∗ − By∗h i ≤ 0:ð28Þ

Furthermore, from Au = Bv, we deduce

∥x∗ − PCi,kx∗ð Þ∥ = 0, for i = 1, 2,⋯, t,

y∗ − PQj,ky∗ð Þ

�� 2 = 0, for j = 1, 2,⋯, r,

Ax∗ − By∗k k2 = 0:

ð29Þ

Thus, ðx∗, y∗Þ solves ESEP (16). This completes theproof.

Based on (18), we can introduce a relaxed self-adaptiveprojection algorithm to solve (16), with σk ∈ ð0, 1Þ.

Alggorithm 6. Let x0 ∈H1, y0 ∈H2 be arbitrary. We calculatethe ðk + 1Þth iterate via the following formula

uk = PΩ1xk − λk 〠

t

i=1αi xk − PCi,k

xkð Þ� �


,

xk+1 = γkxk + 1 − γkð Þuk,

vk = PΩ2yk − λk 〠

r

j=1βj yk − PQj,k

ykð Þ� �

− BT Axk − Bykð Þ ! !

,

yk+1 = γkyk + 1 − γkð Þvk,

8>>>>>>>>>>><>>>>>>>>>>>:

ð30Þ

where the stepsize λk is chosen by

with σk ∈ (1, 0).

Next, we will focus on the convergence analysis ofAlgorithm 6.

Theorem 7. Assume limk→∞

γk = 0, ∑∞k=1 γk =∞ and σk ∈ ½M1,

M2� ⊂ ð0, 1Þ, then the sequence ðxk, ykÞ generated by Algo-rithm 6 converges to a solution of (1).

Proof. Taking ðx∗, y∗Þ ∈ Γ, one has

Ax∗ = By∗: ð32Þ

From (30) and the fact that the projection is nonexpan-sive, we have

λk = 2σk∑t

i=1 αi∥xk − PCi,kxkð Þ∥2 +∑r

j=1 βj∥yk − PQj,kykð Þ∥2+∥Axk − Byk∥

2

∥∑ti=1 αi xk − PCi,k

xkð Þ� �

+ AT Axk − Bykð Þ∥2+∥∑rj=1 βj yk − PQj,k

ykð Þ� �

− BT Axk − Bykð Þ∥2= 4σk

pk xk,ykð Þ∥∇pk xk, ykð Þ∥2 , ð31Þ


uk − x∗k k2 = PΩ1xk − λk 〠

t

i=1αi xk − PCi,k

xkð Þ� �


− x∗��

��2

≤ xk − λk 〠t

i=1αi xk − PCi,k

xkð Þ� �

+ AT Axk − Bykð Þ !

− x∗��

��2

= xk − x∗∥2 + λkð Þ2∥〠t

i=1αi xk − PCi,k

xkð Þ� �

+ AT Axk − Bykð Þ��

��2

− 2λk 〠t

i=1αi xk − PCi,k

xkð Þ� �

+ AT Axk − Bykð Þ, xk − x∗* +

:

ð33Þ

Since

−2λk 〠t

i=1αi xk − PCi,k

xkð Þ� �

+ AT Axk − Bykð Þ, xk − x∗* +

= −2λk 〠t

i=1αi xk − PCi,k

xkð Þ� �

, xk − x∗* +

− 2λk AT Axk − Bykð Þ, xk − x∗� �

= −2λk 〠t

i=1αi xk − PCi,k

xkð Þ, xk − x∗D E

− 2λk Axk − Byk, Axk − Ax∗h i

≤ −2λk 〠t

i=1αi∥xk − PCi,k

xkð Þ∥2 − λk∥Axk − Byk∥2

− λk∥Axk − Ax∗∥2 + λk∥Byk − Ax∗∥2,ð34Þ

together with (33), we deduce

∥uk − x∗∥2 ≤ ∥xk − λk 〠t

i=1αi xk − PCi,k

xkð Þ� �


− x∗∥2

= ∥xk − x∗∥2 + λkð Þ2∥〠t

i=1αi xk − PCi,k

xkð Þ� �

+ AT Axk − Bykð Þ∥2

− 2λk 〠t


xkð Þ∥2 − λk∥Axk − Byk∥2

− λk∥Axk − Ax∗∥2 + λk∥Byk − Ax∗∥2:

ð35Þ

Similarly, we have

∥vk − y∗∥2 = ∥PΩ2yk − λk 〠

r

j=1βj yk − PQj,k

ykð Þ� �

− BT Axk − Bykð Þ ! !

− y∗∥2

≤ ∥yk − λk 〠r

j=1βj yk − PQj,k

ykð Þ� �

− BT Axk − Bykð Þ !

− y∗∥2

≤ ∥yk − y∗∥2 + λkð Þ2∥〠r

j=1βj yk − PQj,k

ykð Þ� �

− BT Axk − Bykð ÞÞ∥2

− 2λk 〠t

i=1βj∥yk − PQj,k

ykð Þ∥2 − λk∥Byk − By∗∥2

− λk∥Axk − Byk∥2 + λk∥Axk − By∗∥2:

ð36Þ

From (35) and (36), it follows

∥uk − x∗∥2 + ∥vk − y∗∥2 ≤ ∥xk − x∗∥2 + ∥yk − y∗∥2

− λk 2 〠t


xkð Þ∥2 + 〠t

i=1βj∥yk

− PQj,kykð Þ∥2+∥Axk − Byk∥

2!

− λk ∥〠t

i=1αi xk − PCi,k

xkð Þ� �


+ ∥〠r

j=1βj yk − PQj,k

ykð Þ� �

− BT Axk − Bykð Þ∥2!!

,

ð37Þ

which together with (31) means

∥uk − x∗∥2 + ∥vk − y∗∥2 ≤ ∥xk − x∗∥2 + ∥yk − y∗∥2: ð38Þ

Furthermore, it follows from (31) and (38) that

∥xk+1 − x∗∥2 + ∥yk+1 − y∗∥2 = ∥γkxk + 1 − γkð Þuk− x∗∥2 + ∥γkyk + 1 − γkð Þvk − y∗∥2

≤ γk∥xk − x∗∥2 + 1 − γkð Þ∥uk − x∗∥2

+ γk∥yk − y∗∥2 + 1 − λkð Þ∥vk − y∗∥2

≤ γk ∥xk − x∗∥2 + ∥yk − y∗∥2

+ 1 − γkð Þ ∥uk − x∗∥2 + ∥vk − y∗∥2

≤ ∥xk − x∗∥2 + ∥yk − y∗∥2,

ð39Þ

By induction, one has

∥xk+1 − x∗∥2 + ∥yk+1 − y∗∥2 ≤ ∥x0 − x∗∥2 + ∥y0 − y∗∥2: ð40Þ

Hence, fxng and fyng are bounded. Following (31), (36),and (39), we have

∥xk+1 − x∗∥2 + ∥yk+1 − y∗∥2 ≤ γk ∥xk − x∗∥2 + ∥yk − y∗∥2

+ 1 − γkð Þ ∥uk − x∗∥2 + ∥vk − y∗∥2

≤ ∥xk − x∗∥2

+ ∥yk − y∗∥2 − 1 − γkð Þλk 2 〠t


xkð Þ∥2

+ 〠t

i=1βj∥yk − PQj,k

ykð Þ∥2 + ∥Axk − Byk∥2!

− λk ∥〠t

i=1αi xk − PCi,k

xkð Þ� �


+ ∥〠r

j=1βj yk − PQj,k

ykð Þ� �

− BT Axk − Bykð Þ∥2!!

:

ð41Þ


Without loss of generality, we can assume that there isσ > 0 such that 4ð1 − γkÞσkð1 − σkÞ > σ for all k. Settingsk = ∥xk − x∗∥2 + ∥yk − y∗∥2, together with (41), we havethe following inequality

σpk xk,ykð Þð Þ2

∥∇pk xk, ykð Þ∥2 + sk+1 − sk ≤ 0: ð42Þ

Since sk is eventually decreasing, we obtain sk as con-vergent. From (42), we have lim

k→∞pkðxk, ykÞ = 0: Further-

more,

limk→∞

∥xk − PCi,kxkð Þ∥2 = 0, f ori = 1, 2,⋯, t, ð43Þ

limk→∞

∥yk − PQj,kykð Þ∥2 = 0, f orj = 1, 2,⋯, r, ð44Þ

limk→∞

∥Axk − Byk∥2 = 0: ð45Þ

Furthermore,

∥xk+1 − xk∥ = ∥γkxk + 1 − γkð Þuk − xk∥ = 1 − γkð Þ∥uk − xk∥

≤ 1 − γkð Þλk 〠t


xkð Þ∥+∥AT Axk − Bykð Þ∥ !

,

ð46Þ

which with (41), (45), and the assumption on γk means

limk→∞

∥xk+1 − xk∥2 = 0: ð47Þ

Note that

∥xk+1 − uk∥ = ∥γkxk + 1 − γkð Þuk − uk∥ = γk∥xk − uk∥, ð48Þ

we have

limk→∞

∥xk+1 − uk∥2 = 0: ð49Þ

(47) and (49) imply

limk→∞

∥xk − uk∥2 = 0: ð50Þ

Similarly, we have

limk→∞

∥yk+1 − yk∥2 = 0,

limk→∞

∥yk+1 − vk∥2 = 0,

limk→∞

∥yk − vk∥2 = 0:

ð51Þ

Thus, fxkg and fykg are asymptotically regular.Notice that

〠t

i=1αi xk − PCi,k

xkð Þ� �

+ AT Axk − Bykð ÞÞ∥2��

+ ∥〠r

j=1βj yk − PQj,k

ykð Þ� �

− BT Axk − Bykð ÞÞ��2

≤2 〠t

i=1αi∥ xk, − , PCi,k

; xkð Þ�

∥2 + ∥A∥2∥Axk − Byk∥2

+ 〠r

j=1βj∥yk − PQj,k

ykð Þ∥2 + ∥B∥2∥Axk − Byk∥2!

≤2max 1,∥A∥2 + ∥B∥2� �

〠t

i=1αi∥ xk, − , PCi,k

; xkð Þ�

∥2

+ ∥Axk − Byk∥2 + 〠

r

j=1βj∥yk − PQj,k

ykð Þ∥2!,

ð52Þ

which implies that

λk ≥ σk1

max 1,∥A∥2 + ∥B∥2� � : ð53Þ

Moreover, it follows from (22) that

∥xk+1 − xk

λk∥ = 1 − γkð Þ 1

λk∥uk − xk∥≤ 1 − γkð Þ

� 〠t


xkð Þ∥ + ∥AT Axk − Bykð Þ∥ !

,

ð54Þ

which with (43), (45), and the assumption on γk yields

limk→∞

∥xk+1 − xk

λk∥ = 0: ð55Þ

Similarly, one has

limk→∞

∥yk+1 − yk

λk∥ = 0: ð56Þ

Let �x and �y be, respectively, weak cluster points ofthe sequences fxkg and fykg, then there exist two subse-quences of fxkg and fykg (again labeled fxkg and fykgwhich converge weakly to �x and �y). Next, we will showthat ð�x, �yÞ ∈ Γ. It follows from (30) that


xk+1 − xkλk 1 − γkð Þ − λk 〠

t

i=1αi xk − PCi,k

xkð Þ� �


∈NΩ1

xk+1 − γkxk1 − γk

� �,

yk+1 − ykλk 1 − γkð Þ − λk 〠

r

j=1βj yk − PQj,k

ykð Þ� �

− BT Axk − Bykð Þ !

∈NΩ2

yk+1 − γkyk1 − γk

� �:

ð57Þ

From the graphs of the maximal monotone operators,NC andNQ are weakly-strongly closed, and by passing tothe limit in the last inclusions, we obtain that �x ∈Ω1 and�y ∈Ω2.

On the other hand, from Lemma 1 and the definition ofCi,k, one has

ci xkð Þ ≤ ξi,k, xk − PCi,kxkð Þ

D E≤ ∥ξi,k∥∥xk − PCi,k

xkð Þ∥≤M1∥xk − PCi,k

xkð Þ∥,ð58Þ

whereM satisfies ∥ξi,k∥≤M1 for all k. The lower semicontinu-ity of function ciðxÞ and (41) assert that

ci �xð Þ ≤ lim infk→∞

ci xkð Þ ≤ 0: ð59Þ

Thus, �x ∈ Ci for i = 1, 2,⋯, t. Likewise, we can obtain

qj xkð Þ ≤ ηj,k, yk − PQj,kykð Þ

D E≤ ∥ηj,k∥∥yk − PQj,k

ykð Þ∥≤M2∥yk − PQj,k

ykð Þ∥,ð60Þ

where M2 satisfies ∥ηj,k∥≤M2 for all k. The lower semiconti-nuity of function qjðxÞ and (42) lead to

qi �yð Þ ≤ lim infk→∞

qi ykð Þ ≤ 0: ð61Þ

Thus, �y ∈Qj for j = 1, 2,⋯, r. Moreover, the weak con-vergence of Axk − Byk to A�x − B�y and the lower semicontinu-ity of the squared norm imply

∥A�x − B�y∥≤lim infk→∞

∥Axk − Byk∥ = 0, ð62Þ

hence, ð�x, �yÞ ∈ Γ. This completes the proof.

4. Numerical Examples

We are in a position to show numerical examples to demon-strate the performance and convergence of Algorithm 6. Thewhole programs are written in MATLAB 7.0. All the numer-ical results are carried out on a personal Lenovo computerwith Intel®Core™ i7-7500U CPU 2.70GHz and RAM

4.00GB. We denote the vector with all elements 1 by e inwhat follows.

Example 8. Let

A =3 −1 22 1 03 0 3

0BB@

1CCA, B =

4 −4 2 13 −1 4 35 1 0 4

0BB@

1CCA ð63Þ

C1 = fx ∈ R3 ∣ x1 + 5x22 + 4x3 ≤ 0g, C2 = fx ∈ R3 ∣ 3x1 +10x3 ≤ 0g,Q1 = fy ∈ R4 ∣ 2y1 − 3y2 − 2y3 +4y4 ≤ 0g, andQ2 =fy ∈ R4 ∣ 2y21 − y2 + 4y3 − 3y4 ≤ 0g. Find x ∈ C = C1 ∩ C2, y ∈Q =Q1 ∩Q2 such that Ax = By.

Example 9. Let

A =

0:2620 0:0268 0:2589

0:5697 0:5004 0:0458

0:3595 0:8270 0:2464

0BBB@

1CCCA,

B =

0:6607 0:0130 0:0335 0:9213

0:3294 0:7180 0:4060 0:9840

0:6594 0:3911 0:7163 0:9834

0BBB@

1CCCA

ð64Þ

C1 = fx ∈ R3 ∣ x41 + x22 − 2x23 − 1 ≤ 0g, C2 = fx ∈ R3 ∣ 2x21+ x32 − 3x23 − 1 ≤ 0g,Q1 = fy ∈ R4 ∣ 2y31 − y22 + 2y33 + 6y4 − 2 ≤0g, andQ2 = fy ∈ R4 ∣ 2y21 + 3y23 + 2y24 − 2 ≤ 0g. Find x ∈ C =C1 ∩ C2, y ∈Q =Q1 ∩Q2 such that Ax = By:

Example 10. Let A = ðaijÞJ×N and B = ðbijÞJ×M . C1 = fx1 ∈ RN

∣∥x1∥≤2g, C2 = fx2 ∈ RN ∣−e ≤ x2 ≤ 3eg:Q1 = fy1 ∈ RM∣−2e ≤y1 ≤ 6eg, and Q2 = fy2 ∈ RM ∣∥y2∥≤4g, where faijg, fbijg ∈ð0, 1Þ are all generated randomly; J , N and M are pos-itive integers. Find x ∈ C = C1 ∩ C2, y ∈Q =Q1 ∩Q2 suchthat Ax = By:

In this example, we consider J = 10,N = 10, andM = 20 ;J = 20,N = 30, andM = 40 ; and J = 40,N = 50, andM = 60and three initial values:

(i) Case 1 x = onesðN , 1Þ, y = onesðM, 1Þ ;(ii) Case 2 x = 10 ∗ onesðN , 1Þ, y = 10 ∗ onesðM, 1Þ ;(iii) Case 3 x = −10 ∗ onesðN , 1Þ, y = −10 ∗ onesðM, 1Þ:We take Ω1 = C1,n, Ω2 =Q1,n when the algorithm iterates

to step n, γk = 1/20k, σk = ð1/4Þ + ð1/2kÞ, α1 = α2 = β1 = β2 =1/4 in Algorithm 6. In the following tables and figures, wedenote Algorithm 6 and the algorithm in reference [45] byQSPA and RTPPM, respectively. And we set }n}, }s} and }

x∗, } and }y∗} to express the number of iteration, CPU timein seconds, and the final solution, respectively. Init. denotethe initial points, and pkðx, yÞ ≤ ε = 10−4 is used as the stop


criterion. The numerical results can be seen from Tables 1–3and Figures 1–4. For Figures 3 and 4, take J = 20,N = 30,andM = 40 in Example 10.

From Tables 1–3, we can see that the iterative numberand CPU time of Algorithm 6 is less algorithm RTPPM.Figures 1–4 indicate that Algorithm 6 is more stable thanRTPPM.

Furthermore, for testing the stationary property of itera-tive number, we carry 500 experiments for the initial pointwhich are presented randomly, such as

x1 = rand 3, 1ð Þ, y1 = rand 4, 1ð Þ, ð65Þ

in Example 9, the results can be found in Figure 1.

Table 3: The numerical results of Example 10.

QSPR with λn QSPR with 0.5 λn RTPPM with λnJ N M n s n s n s

Case 1

10 10 20 30 0:003122 48 0.004302 808 0:05093320 30 40 37 0.005304 94 0.011647 1994 0.258627

40 50 60 91 0.013088 188 0.028659 4014 1.286561

Case 2

10 10 20 56 0.011269 125 0.031055 3236 0.201379

20 30 40 107 0.038957 171 0.039266 1762 0.233102

40 50 60 295 0.041663 351 0.100628 5084 1.673274

Case 3

10 10 20 67 0.008644 98 0.016412 812 0.051558

20 30 40 118 0.015576 178 0.028107 1953 0.270206

40 50 60 233 0.058625 302 0.108850 4176 1.360695


Init. QSPR RTPPM

x1 = 0, 0, 0ð ÞT n = 14, s = 0:000671 n = 7684, s = 0:198442

y1 = 1, 0,−1, 1ð ÞTx∗ = 0:0196,−0:0222,−0:0055ð ÞT x∗ = 0:0342,−0:0734,−0:0153ð ÞT

y∗ = 0:0345,0:0119,0:0071,−0:0357ð ÞT y∗ = 0:0509,0:0012,−0:0023,−0:0495ð ÞT

x1 = 0, 1, 1ð ÞT n = 15, s = 0:000636 n = 49, s = 0:001585

y1 = 0, 0, 0, 0ð ÞTx∗ = −0:0750,0:0232,0:0181ð ÞT x∗ = −0:2420,−0:0354,0:0589ð ÞT

y∗ = −0:0279,0:0205,0:0014,−0:0109ð ÞT y∗ = −0:0721,0:0426,−0:0254,−0:0566ð ÞT

x1 = 1, 1, 1ð ÞT n = 258, s = 0:008365 n = 34587, s = 0:816461

y1 = 1, 1,−1,−1ð ÞTx∗ = −0:1554,−0:0125,−0:0255ð ÞT x∗ = 0:0836,−0:1013,−0:0566ð ÞT

y∗ = −0:0951,0:0345,0:0180,−0:0257ð ÞT y∗ = 0:0496,−0:0180,0:0025,−0:0371ð ÞT


Init. QSPR RTPPM

x1 = 1, 1, 1ð ÞT n = 16, s = 0:001268 n = 478, s = 0:029797

y1 = 1, 1, 1, 1ð ÞTx∗ = 0:1710,0:1525,0:1824ð ÞT x∗ = 1:2294,0:7761,0:9712ð ÞT

y∗ = 0:1110,0:1218,0:1283,0:0100ð ÞT y∗ = 0:6171,0:7022,0:6533,0:1692ð ÞT

x1 = 10 1, 1, 1ð ÞT n = 37, s = 0:001877 n = 1687, s = 0:079442

y1 = 10 1, 1, 1, 1ð ÞTx∗ = 0:2973,0:3380,0:7617ð ÞT x∗ = 1:5257,0:4196,1:5156ð ÞT

y∗ = 0:6560,0:3316,0:2638,−0:1878ð ÞT y∗ = 0:9622,0:8416,0:2119,0:1513ð ÞT

x1 = 100 1, 1, 1ð ÞT n = 63, s = 0:003101 n = 2651, s = 0:110352

y1 = 100 1, 1, 1, 1ð ÞTx∗ = 0:5363,−0:1663,2:0146ð ÞT x∗ = 1:5276,0:3916,1:5165ð ÞT

y∗ = 0:9380,0:0704,−0:1862,0:0354ð ÞT y∗ = 0:9713,0:8464,0:1785,0:1454ð ÞT


On the other initial point, such as

x1 = rand 3, 1ð Þ ∗ 10, y1 = rand 4, 1ð Þ ∗ 10, ð66Þ


Similarly, we carry 500 experiments for the initial pointwhich are presented randomly, such as

x1 = rand N , 1ð Þ, y1 = rand M, 1ð Þ, ð67Þ


Number of experiment

6

8

10

12

14

16

18

Num

ber o

f ite

ratio

n fo

r QSP

A

0

50

100

150

200

250

300

350

400

450

Num

ber o

f ite

ratio

n fo

r RTP

PM

0 50 100 150 200 250 300 350 400 450 500Number of experiment

0 50 100 150 200 250 300 350 400 450 500

Figure 1: The iteration number of QSPA and RTPPM.

15

20

25

30

35

40

Num

ber o

f ite

ratio

n fo

r QSP

A

0

500

1000

1500

2000

2500

3000

3500

4000

Num

ber o

f ite

ratio

n fo

r RTP

PM

Number of experiment0 50 100 150 200 250 300 350 400 450 500



50

60

70

80

90

100

110

120

130

140

Num

ber o

f ite

ratio

n fo

r QSP

A

1200

1400

1600

1800

2000

2200

2400

2600

Num

ber o

f ite

ratio

n fo

r RTP

PM





On the other initial point, such as

x1 = rand N , 1ð Þ ∗ 10, y1 = rand M, 1ð Þ ∗ 10, ð68Þ


Data Availability

The data used to support the findings of this study are avail-able from the corresponding author upon request.

Conflicts of Interest

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Authors’ Contributions

Each author equally contributed to this paper and read andapproved the final manuscript.

Acknowledgments

This project is supported by the Natural Science Foundationof China (11401438, 11671228, 11601261, and 11571120),Shandong Provincial Natural Science Foundation(ZR2019MA022), and Project of Shandong Province HigherEducational Science and Technology Program (Grant No.J14LI52).

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SPA

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