A new algorithm for the split feasibility problem with multiple output

sets in Hilbert spaces

YAXUAN ZHANG,YUMING GUAN

College of Science

Civil Aviation University of China

Tianjin 300300

CHINA

Abstract: - In this paper, we study the split feasibility problem with multiple output sets in Hilbert spaces. We

propose a new self-adaptive algorithm combing with ball-relaxation and inertial acceleration, and prove its strong

convergence. Numerical simulations are provided to illustrate the effectiveness of the proposed algorithm.

Key-Words: - split feasibility problem with multiple output sets, self-adaptive step size, ball-relaxation, inertial

acceleration, strong convergence.

Received: October 7, 2022. Revised: December 10, 2022. Accepted: January 9, 2023. Published: February 2, 2023.

1 Introduction

The multiple-sets split feasibility problem (MSSFP)

is to find 𝑥∗∈𝐻1such that

𝑥∗∈

𝑡



𝑖=1

𝐶𝑖, 𝐴𝑥∗∈

𝑟



𝑗=1

𝑄𝑗,(1)

where 𝐶𝑖, 𝑖 =1,2,· · · , 𝑡 ⊂𝐻1,𝑄𝑗, 𝑗 =1,2,· · · , 𝑟 ⊂

𝐻2are nonempty, closed and convex subsets of

Hilbert spaces 𝐻1and 𝐻2, respectively, and 𝐴:

𝐻1→𝐻2is a bounded linear operator.

It is obviously that if 𝑟=𝑡=1, the MSSFP is

reduced to the split feasibility problem (SFP).

The SFP and the MSSFP were first proposed by

Censor and Elfving in [1] and [2] for modeling cer-

tain inverse problems, which have been widely used

in many application fields, such as, medical image re-

construction, [1, 3, 4], intensity-modulated radiation

therapy (IMRT), [5, 6], and gene regulatory network

inference, [7], etc. Many authors have also made a

continuation of the study on the MSSFP and its vari-

ant form, for instance, see, [8–16].

Recently, Reich et al. proposed the split feasibil-

ity problem with multiple output sets in [14]. Let

𝐻,𝐻𝑗, 𝑗 =1,2,· · · , 𝑟, be real Hilbert spaces and let

𝐴𝑗:𝐻→𝐻𝑗, 𝑗 =1,2,· · · , 𝑟, be bounded linear op-

erators. Let 𝐶and 𝑄𝑗be nonempty, closed and con-

vex subsets of 𝐻and 𝐻𝑗, 𝑗 =1,2,· · · , 𝑟, respectively.

Find an element 𝑥∗, such that

𝑥∗∈𝑆=𝐶∩ (

𝑟



𝑗=1

𝐴−1

𝑗(𝑄𝑗)).(2)

They also provided algorithms for solving this prob-

lem.

In this paper, we study a slightly generalized

multiple-sets split feasibility problem with multi-

ple output sets: Let 𝐻,𝐻𝑗, 𝑗 =1,2,· · · , 𝑟, be

real Hilbert spaces and let 𝐴𝑗:𝐻→𝐻𝑗, 𝑗 =

1,2,· · · , 𝑟, be bounded linear operators. Let 𝐶𝑖and

𝑄𝑗be nonempty, closed and convex subsets of 𝐻and

𝐻𝑗, 𝑗 =1,2,· · · , 𝑟, respectively. Find an element 𝑥∗,

such that

𝑥∗∈𝑆=

𝑡



𝑖=1

𝐶𝑖∩ (

𝑟



𝑗=1

𝐴−1

𝑗(𝑄𝑗)).(3)

In other words, the aim is to find an 𝑥∗∈𝐶𝑖such that

𝐴𝑗𝑥∗∈𝑄𝑗for all 𝑖=1,2,· · · , 𝑡, 𝑗 =1,2· · · , 𝑟.

If 𝑡=1, the problem (3) reduces to the problem

(2). If 𝐴𝑗≡𝐴, 𝐻 𝑗≡𝐻1, the problem (3) reduces to

the MSSFP (1).

Many iterative methods have been proposed for

solving the SFP. One of the well-known algorithms

is the CQ method proposed by Byrne, [3], which is

formulated as follows

𝑥𝑛+1=𝑃𝐶(𝑥𝑛−𝛼𝑛𝐴∗(𝐼−𝑃𝑄)𝐴𝑥𝑛),(4)

where the step size 𝛼𝑛∈ (0,2

∥𝐴∥2), and 𝑃𝐶and 𝑃𝑄

stand for the metric projection onto 𝐶and 𝑄, respec-

tively.

Since the projections onto a general nonempty

closed convex subset is hard to be implemented, Yang

[15] proposed the half-space relaxation projection CQ

algorithm. Yu et al. [16] introduced the ball-relaxed

projection CQ algorithms.

Since the norm estimation of || 𝐴𝑗|| for step size is

hard to get, Several choice of the self-adaptive step

size have been presented, see for instance, Yang [17],

López et al. [18], Gibali et a.l. [19], etc.

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2023.22.12

Yaxuan Zhang, Yuming Guan

E-ISSN: 2224-2880

100

Volume 22, 2023

To achieve a faster convergence of the algorithms,

many references have investigated the inertial tech-

nique, see for example, Suantai et al. [20], etc.

In this paper, we adopt the ball-relaxation, a new

self-adaptive step size and inertial acceleration tech-

nique to the algorithm solving the problem (3). Since

the orthogonal projections onto balls and the self-

adaptive step size can be directly calculated, the pro-

posed algorithm is easy to implement.

The rest is outlined as follows. Some useful con-

cepts and lemmas for our analysis are reviewed in the

next section. In section 3, we present our algorithm

and prove its strong convergence. Finally, in section

4, we exhibit a numerical example in order to illus-

trate our results and observe the performance of our

algorithm.

2 Preliminaries

In this section, we introduce some definitions and ba-

sic lemmas that will be used in the sequel. Let 𝐻be a

real Hilbert space, and its inner product and norm be

expressed by ⟨·,·⟩ and ∥ · ∥, respectively. Besides, we

use the symbol 𝑥𝑛→𝑥(𝑥𝑛⇀ 𝑥)to express that the

sequence {𝑥𝑛}converges strongly (weakly) to 𝑥.

Definition 2.1 Let 𝐶be a nonempty closed convex

subset of 𝐻. Then the mapping 𝑇:𝐶→𝐻is said to

be:

(1) nonexpansive if

∥𝑇𝑥 −𝑇 𝑦∥ ≤ ∥𝑥−𝑦∥,∀𝑥, 𝑦 ∈𝐶. (5)

(2) firmly nonexpansive if

∥𝑇𝑥 −𝑇 𝑦∥2≤ ∥𝑥−𝑦∥2− ∥(𝐼−𝑇)𝑥− (𝐼−𝑇)𝑦∥2,

∀𝑥, 𝑦 ∈𝐶, (6)

or equivalently if

∥𝑇𝑥 −𝑇 𝑦∥2≤ ⟨𝑇𝑥 −𝑇 𝑦, 𝑥 −𝑦⟩,∀𝑥, 𝑦 ∈𝐶, (7)

where 𝐼is the identity operator.

Definition 2.2 Let 𝐶be a nonempty, closed and con-

vex subset of 𝐻. The metric projection 𝑃𝐶:𝐻→𝐶

defined by

𝑃𝐶(𝑥)=arg min

𝑦∈𝐶

∥𝑥−𝑦∥2, 𝑥 ∈𝐶. (8)

Definition 2.3 Let 𝑓:𝐻→ (−∞,+∞] be a proper

function. Then 𝑓is said to be weakly lower semicon-

tinuous at 𝑥if 𝑥𝑛⇀ 𝑥 implies

𝑓(𝑥) ≤ lim inf

𝑛→∞ 𝑓(𝑥𝑛).(9)

𝑓is lower semicontinuous on 𝐻if it is lower semicon-

tinuous at every point 𝑥∈𝐻and 𝑓is weakly lower

semicontinuous on 𝐻if it is weakly lower semicon-

tinuous at every point 𝑥∈𝐻.

Lemma 2.1 [21] Let 𝐶be a nonempty closed and

convex subset of 𝐻. Then for all 𝑥, 𝑦 ∈𝐻and 𝑧∈𝐶,

we have the following statements:

(1) ⟨𝑥−𝑃𝐶(𝑥), 𝑦 −𝑃𝐶(𝑥)⟩ ≤ 0;

(2)𝑃𝐶and 𝐼−𝑃𝐶are both firmly nonexpansive;

(3) ⟨𝑥, 𝑦⟩=1

2∥𝑥∥2+1

2∥𝑦∥2−1

2∥𝑥−𝑦∥2;

(4) ∥𝑥+𝑦∥2≤ ∥𝑥∥2+2⟨𝑦, 𝑥 +𝑦⟩.

Lemma 2.2 [21] Let 𝑓:𝐻→ (−∞,+∞] be a

strongly convex function with constant 𝜆. Then for

all 𝑥, 𝑦 ∈𝐻,

𝑓(𝑦) ≥ 𝑓(𝑥) + ⟨𝜉, 𝑦 −𝑥⟩ + 𝜆

2∥𝑦−𝑥∥2, 𝜉 ∈𝜕 𝑓 (𝑥)

(10)

Lemma 2.3 [4] Let 𝐻1and 𝐻2be real Hilbert spaces

and 𝑓:𝐻1→Ris given by 𝑓(𝑥)=1

2∥(𝐼−𝑃𝑄)𝐴𝑥 ∥2

where 𝑄is closed convex subset of 𝐻2and

𝐴:𝐻1→𝐻2be a bounded linear operator.

Then

(1)the function 𝑓is convex and weakly lower

semicontinuous on 𝐻1;

(2) ∇ 𝑓(𝑥)=𝐴∗(𝐼−𝑃𝑄)𝐴𝑥, for 𝑥∈𝐻1;

(3) ∇ 𝑓is ∥𝐴∥2-Lipschitzian continuous, i.e.,

∥∇ 𝑓(𝑥) − ∇ 𝑓(𝑦)∥ ≤ ∥ 𝐴∥2∥𝑥−𝑦∥,∀𝑥, 𝑦 ∈𝐻1.

Lemma 2.4 [22] Assume that {𝑠𝑛}is a sequence of

nonnegative real numbers such that

𝑠𝑛+1≤ (1−𝛼𝑛)𝑠𝑛+𝛼𝑛𝛿𝑛, 𝑛 ≥1,(11)

𝑠𝑛+1≤𝑠𝑛−𝜂𝑛+𝛾𝑛, 𝑛 ≥1,(12)

where {𝛼𝑛}is a sequence in (0,1),{𝜂𝑛}is a sequence

of nonnegative real numbers, {𝛿𝑛}and {𝛾𝑛}are two

sequences in Rsuch that

(1)∞

𝑛=1𝛼𝑛=∞;

(2)lim

𝑛→∞ 𝛾𝑛=0;

(3)lim

𝑘→∞ 𝜂𝑛𝑘=0implies lim sup

𝑘→∞

𝛿𝑛𝑘≤0for any sub-

sequence {𝑛𝑘}of {𝑛}.

Then lim

𝑛→∞ 𝑠𝑛=0.

3 Algorithm and its convergence

In this section, we introduce ball-relaxed algorithm

with a new self-adaptive step size and inertial acceler-

ation for solving the problem (3) and prove its strong

convergence.

Set

𝐶𝑖={𝑥∈𝐻:𝑐𝑖(𝑥) ≤ 0},(13)

𝑄𝑗={𝑦∈𝐻𝑗:𝑞𝑗(𝑦) ≤ 0},(14)

where 𝑐𝑖(𝑥), 𝑖 =1,2,· · · , 𝑡 and 𝑞𝑗(𝑦), 𝑗 =1,2,· · · , 𝑟

are convex, weakly lower semi-continuous functions,

respectively.

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2023.22.12

Yaxuan Zhang, Yuming Guan

E-ISSN: 2224-2880

101

Volume 22, 2023

If 𝑐𝑖(𝑥), 𝑖 =1,2,· · · , 𝑡 and 𝑞𝑗(𝑦), 𝑗 =1,2,· · · , 𝑟

are 𝜆𝑖- and 𝛾𝑗- strongly convex, define a series of sets

𝐶𝑏

𝑖,𝑛 and 𝑄𝑏

𝑗,𝑛,𝑛≥1, by

𝐶𝑏

𝑖,𝑛 ={𝑥∈𝐻:𝑐𝑖(𝑥𝑛) + ⟨𝜉𝑛

𝑖, 𝑥 −𝑥𝑛⟩

+𝜆𝑖

2∥𝑥−𝑥𝑛∥2≤0},(15)

𝑄𝑏

𝑗,𝑛 ={𝑦∈𝐻𝑗:𝑞𝑗(𝐴𝑗𝑥𝑛) + ⟨𝜁𝑛

𝑗, 𝑦 −𝐴𝑗𝑥𝑛⟩

+𝛾𝑗

2∥𝑦−𝐴𝑗𝑥𝑛∥2≤0},(16)

where 𝜉𝑛

𝑖∈𝜕𝑐𝑖(𝑥𝑛),𝑖=1,2,· · · , 𝑡 and 𝜁𝑛

𝑗∈

𝜕𝑞 𝑗(𝐴𝑗𝑥𝑛), 𝑗 =1,2,· · · 𝑟.

It is easy to verify that 𝐶𝑏

𝑖,𝑛, 𝑖 =1,2,· · · , 𝑡 and

𝑄𝑏

𝑗,𝑛, 𝑗 =1,2,· · · , 𝑟 are closed balls containning 𝐶

and 𝑄, respectively, see, [23].

Define that

𝑑𝑛=max

𝑖=1,··· ,𝑡 ∥𝑥−𝑃𝐶𝑖(𝑥)∥,

𝑣𝑛=max

𝑗=1,··· ,𝑟 ∥𝐴𝑗𝑥−𝑃𝑄𝑗(𝐴𝑗𝑥) ∥,(17)

Then the problem (3) is equivalent to the following

minimization problem:

min 𝑓(𝑥)=1

𝑡



𝑖=1

𝑙𝑖∥𝑥−𝑃𝐶𝑖(𝑥)∥2

𝑟



𝑗=1

𝜆𝑗∥𝐴𝑗𝑥−𝑃𝑄𝑗(𝐴𝑗𝑥)∥2.(18)

where 𝑙𝑖, 𝑖 =1,· · · , 𝑡 and 𝜆𝑗, 𝑗 =1,· · · , 𝑟 are all pos-

itive constants such that 𝑡

𝑖=1𝑙𝑖+𝑟

𝑗=1𝜆𝑗=1.

Using (17), the problem (18) is equivalent to the

following minimization problem:

min 𝑓(𝑥)=1

2Φ2

𝑛.(19)

where Φ𝑛=max{𝑑𝑛, 𝑣𝑛}.

In the sequel, we assume that the following three

assumptions hold.

(A1) The solution set 𝑆of (3) is nonempty.

(A2) The functions 𝑐𝑖:𝐻→Rand 𝑞𝑗:𝐻𝑗→

Rdefined in (13) and (14) are 𝜆𝑖- and 𝛾𝑗- strongly

convex lower semicontinuous functions.

(A3) For any 𝑥∈𝐻and 𝑦𝑗∈𝐻𝑗, at least one sub-

gradient 𝜉𝑖∈𝜕𝑐𝑖(𝑥)and 𝜁𝑗∈𝜕𝑞 𝑗(𝑦𝑗)can be calcu-

lated. The subdifferentials 𝜕𝑐𝑖and 𝜕𝑞 𝑗are bounded

on the bounded sets.

Algorithm 3.1 For any initial point 𝑥0, 𝑥1∈𝐻, the

sequence {𝑥𝑛}be defined as follows:

1. Compute 𝑦𝑛

𝑦𝑛=𝑥𝑛+𝛽𝑛(𝑥𝑛−𝑥𝑛−1).(20)

2. Compute 𝑑𝑛and define 𝐿𝑛

𝑑𝑛=max

𝑖=1,··· ,𝑡 ∥𝑦𝑛−𝑃𝐶𝑏

𝑖,𝑛 (𝑦𝑛)∥,(21)

𝐿𝑛={𝑖∈ {1,2,· · · , 𝑡}:∥𝑦𝑛−𝑃𝐶𝑏

𝑖,𝑛 (𝑦𝑛)∥ =𝑑𝑛}.

(22)

3. Compute 𝑣𝑛and define 𝐻𝑛

𝑣𝑛=max

𝑗=1,··· ,𝑟 ∥𝐴𝑗𝑦𝑛−𝑃𝑄𝑏

𝑗,𝑛 (𝐴𝑗𝑦𝑛)∥,(23)

𝐻𝑛={𝑗∈ {1,2,· · · , 𝑟}:∥𝐴𝑗𝑦𝑛−𝑃𝑄𝑏

𝑗,𝑛 (𝐴𝑗𝑦𝑛)∥ =𝑣𝑛}.

(24)

4. If 𝑑𝑛≥𝑣𝑛, then choose 𝑖𝑛∈𝐿𝑛and let Δ = 𝐼,

𝜇𝑛=𝑃𝑏

𝐶𝑖𝑛,𝑛 𝑦𝑛,𝑓𝑛(𝑦𝑛)=1

2∥(𝐼−𝑃𝐶𝑏

𝑖𝑛,𝑛 )𝑦𝑛∥2;

else choose 𝑗𝑛∈𝐻𝑛and let Δ = 𝐴𝑗𝑛,𝜇𝑛=

𝑃𝑏

𝑄𝑗𝑛,𝑛 𝐴𝑗𝑛𝑦𝑛,𝑓𝑛(𝑦𝑛)=1

2∥(𝐼−𝑃𝑄𝑏

𝑗𝑛,𝑛 )𝐴𝑗𝑛𝑦𝑛∥2.

5. Compute 𝑥𝑛+1

𝑥𝑛+1=𝛼𝑛𝑔(𝑦𝑛) + (1−𝛼𝑛)(𝑦𝑛−𝜏𝑛∇𝑓𝑛(𝑦𝑛)),(25)

where

𝜏𝑛=𝜌𝑛

∥Δ𝑦𝑛−𝜇𝑛∥2

∥Δ∗(Δ𝑦𝑛−𝜇𝑛)∥2+𝜃𝑛

,(26)

𝛼𝑛∈ (0,1),𝜌𝑛∈ (0,2)and {𝜃𝑛} is a bounded

sequence of positive real numbers. 𝑔:𝐻→𝐻

is a strict contraction mapping 𝐻into itself with

the contraction coefficient 𝑐∈ [0,1).

Now we establish the strong convergence for Al-

gorithm 3.1.

Theorem 3.1 Let 𝐻and 𝐻𝑗be real Hilbert spaces,

𝐶𝑖, 𝑖 =1,2,· · · , 𝑡 and 𝑄𝑗, 𝑗 =1,2,· · · , 𝑟 be

nonempty, closed and convex subsets of 𝐻and 𝐻𝑗,

respectively. Let 𝐴𝑗:𝐻→𝐻𝑗, 𝑗 =1,2,· · · , 𝑟 be

bounded linear operators with their adjoint denoted

by 𝐴∗

𝑗. Assume that {𝛼𝑛},{𝛽𝑛}and 𝜌𝑛satisfy the

following conditions:

(C1) lim

𝑛→∞ 𝛼𝑛=0and ∞

𝑛=1𝛼𝑛=∞;

(C2) inf

𝑛∈N𝜌𝑛(2−𝜌𝑛)>0;

(C3) {𝛽𝑛} ⊂ [0, 𝛽], where 𝛽∈ [0,1)and

lim

𝑛→∞

𝛽𝑛

𝛼𝑛∥𝑥𝑛−𝑥𝑛−1∥=0;

(C4) 0<inf{𝜃𝑛} ≤ sup{𝜃𝑛}<+∞.

Then the sequence {𝑥𝑛}generated by Algorithm 3.1

converges strongly to 𝑧∈𝑆, and 𝑧is the unique solu-

tion to the variational inequality:

⟨(𝐼−𝑔)(𝑧), 𝑦 −𝑧⟩ ≥ 0,∀𝑦∈𝑆, (27)

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2023.22.12

Yaxuan Zhang, Yuming Guan

E-ISSN: 2224-2880

102

Volume 22, 2023

Proof Note that 𝑔:𝐻→𝐻is contractive, so 𝑃𝑆𝑔

is also contractive, thus 𝑃𝑆𝑔has a unique fixed point

𝑧, which by Lemma 2.1(1) is the unique solution of

(27).

Notice that

∥𝑥𝑛+1−𝑧∥

=∥𝛼𝑛𝑔(𝑦𝑛)+(1−𝛼𝑛)(𝑦𝑛−𝜏𝑛∇𝑓𝑛(𝑦𝑛)) − 𝑧∥

≤𝛼𝑛∥𝑔(𝑦𝑛) − 𝑧∥

+ (1−𝛼𝑛)∥𝑦𝑛−𝜏𝑛∇𝑓𝑛(𝑦𝑛) − 𝑧∥.

(28)

Next, we consider the following two cases.

Case A: 𝑑𝑛≥𝑣𝑛.

In this case, 𝜏𝑛=

𝜌𝑛| | (𝐼−𝑃𝐶𝑏

𝑖𝑛,𝑛

)𝑦𝑛| |2

| | (𝐼−𝑃𝐶𝑏

𝑖𝑛,𝑛

)𝑦𝑛| |2+𝜃𝑛,𝑓𝑛(𝑦𝑛)=

2||(𝐼−𝑃𝐶𝑏

𝑖𝑛,𝑛 )𝑦𝑛||2and ∇𝑓𝑛(𝑦𝑛)=(𝐼−𝑃𝐶𝑏

𝑖𝑛,𝑛 )𝑦𝑛.

Applying Lemma 2.1 (2-3) and the nonexpansivity of

𝑃𝐶𝑏

𝑖𝑛,𝑛 that

∥𝑦𝑛−𝜏𝑛∇𝑓𝑛(𝑦𝑛) − 𝑧∥2

=∥𝑦𝑛−𝑧∥2+𝜏2

𝑛∥∇ 𝑓𝑛(𝑦𝑛)∥2

−2𝜏𝑛⟨∇ 𝑓𝑛(𝑦𝑛), 𝑦𝑛−𝑧⟩

=∥𝑦𝑛−𝑧∥2+𝜏2

𝑛∥(𝐼−𝑃𝐶𝑏

𝑖𝑛,𝑛 )𝑦𝑛∥2

−2𝜏𝑛⟨(𝐼−𝑃𝐶𝑏

𝑖𝑛,𝑛 )𝑦𝑛, 𝑦𝑛−𝑧⟩

=∥𝑦𝑛−𝑧∥2+𝜏2

𝑛∥(𝐼−𝑃𝐶𝑏

𝑖𝑛,𝑛 )𝑦𝑛∥2

−2𝜏𝑛⟨(𝐼−𝑃𝐶𝑏

𝑖𝑛,𝑛 )𝑧− (𝐼−𝑃𝐶𝑏

𝑖𝑛,𝑛 )𝑦𝑛, 𝑦𝑛−𝑧⟩

≤ ∥𝑦𝑛−𝑧∥2+𝜏2

𝑛∥(𝐼−𝑃𝐶𝑏

𝑖𝑛,𝑛 )𝑦𝑛∥2

−2𝜏𝑛∥𝐼−𝑃𝐶𝑏

𝑖𝑛,𝑛 )𝑧− (𝐼−𝑃𝐶𝑏

𝑖𝑛,𝑛 )𝑦𝑛∥2

=∥𝑦𝑛−𝑧∥2+𝜏2

𝑛∥(𝐼−𝑃𝐶𝑏

𝑖𝑛,𝑛 )𝑦𝑛∥2

−2𝜏𝑛∥(𝐼−𝑃𝐶𝑏

𝑖𝑛,𝑛 )𝑦𝑛∥2

≤ ∥𝑦𝑛−𝑧∥2+𝜌2

𝑛

∥(𝐼−𝑃𝐶𝑏

𝑖𝑛,𝑛 )𝑦𝑛∥4

(∥(𝐼−𝑃𝐶𝑏

𝑖𝑛,𝑛 )𝑦𝑛∥2+𝜃𝑛)2

(∥(𝐼−𝑃𝐶𝑏

𝑖𝑛,𝑛 )𝑦𝑛∥2+𝜃𝑛)

−2𝜌𝑛

∥(𝐼−𝑃𝐶𝑏

𝑖𝑛,𝑛 )𝑦𝑛∥4

∥(𝐼−𝑃𝐶𝑏

𝑖𝑛,𝑛 )𝑦𝑛∥2+𝜃𝑛

=∥𝑦𝑛−𝑧∥2−𝜌𝑛(2−𝜌𝑛)

∥(𝐼−𝑃𝐶𝑏

𝑖𝑛,𝑛 )𝑦𝑛∥4

∥(𝐼−𝑃𝐶𝑏

𝑖𝑛,𝑛 )𝑦𝑛∥2+𝜃𝑛

(29)

Case B: 𝑑𝑛< 𝑣𝑛.

In this case, 𝜏𝑛=

𝜌𝑛∥ (𝐼−𝑃𝑄𝑏

𝑗𝑛,𝑛

)𝐴𝑗𝑛𝑦𝑛∥2

∥𝐴∗

𝑗𝑛(𝐼−𝑃𝑄𝑏

𝑗𝑛,𝑛

)𝐴𝑗𝑛𝑦𝑛∥2+𝜃𝑛,

𝑓𝑛(𝑦𝑛)=1

2||(𝐼−𝑃𝑄𝑏

𝑗𝑛,𝑛 )𝐴𝑗𝑛𝑦𝑛||2and ∇𝑓𝑛(𝑦𝑛)=

𝐴∗

𝑗𝑛(𝐼−𝑃𝑄𝑏

𝑗𝑛,𝑛 )𝐴𝑗𝑛𝑦𝑛. Similar with the deduction of

Case A, we obtain that

∥𝑦𝑛−𝜏𝑛∇𝑓𝑛(𝑦𝑛) − 𝑧∥2

≤ ∥𝑦𝑛−𝑧∥2−𝜌𝑛(2−𝜌𝑛)

∥ (𝐼−𝑃𝑄𝑏

𝑗𝑛,𝑛

)𝐴𝑗𝑛𝑦𝑛∥4

∥𝐴∗

𝑗𝑛(𝐼−𝑃𝑄𝑏

𝑗𝑛,𝑛

)𝐴𝑗𝑛𝑦𝑛∥2+𝜃𝑛.

(30)

From (C2), (29) and (30) we have that

∥𝑦𝑛−𝜏𝑛∇𝑓𝑛(𝑦𝑛) − 𝑧∥ ≤ ∥𝑦𝑛−𝑧∥.(31)

By (20), we also have

∥𝑦𝑛−𝑧∥=∥𝑥𝑛+𝛽𝑛(𝑥𝑛−𝑥𝑛−1) − 𝑧∥

≤ ∥𝑥𝑛−𝑧∥ + 𝛽𝑛∥𝑥𝑛−𝑥𝑛−1∥.(32)

Combining (28), (31) and (32)

∥𝑥𝑛+1−𝑧∥ ≤ 𝛼𝑛||𝑔(𝑦𝑛) − 𝑔(𝑧)|| + 𝛼𝑛||𝑔(𝑧) − 𝑧||

+ (1−𝛼𝑛)||𝑦𝑛−𝜏𝑛∇𝑓𝑛(𝑦𝑛) − 𝑧||

≤𝛼𝑛𝑐||𝑦𝑛−𝑧|| + 𝛼𝑛||𝑔(𝑧) − 𝑧||

+ (1−𝛼𝑛)||𝑦𝑛−𝑧||

≤ [1−𝛼𝑛(1−𝑐)]∥𝑥𝑛−𝑧∥

+ [1−𝛼𝑛(1−𝑐)] 𝛽𝑛∥𝑥𝑛−𝑥𝑛−1∥

+𝛼𝑛∥𝑔(𝑧) − 𝑧∥

=[1−𝛼𝑛(1−𝑐)] ∥𝑥𝑛−𝑧∥ + 𝛼𝑛(1−𝑐)

∥𝑔(𝑧) − 𝑧∥ + 1−𝛼𝑛(1−𝑐)

𝛼𝑛𝛽𝑛∥𝑥𝑛−𝑥𝑛−1∥

1−𝑐.

(33)

According to (C3), we see that 𝑡𝑛=

[1−𝛼𝑛(1−𝑐) ]𝛽𝑛∥𝑥𝑛−𝑥𝑛−1∥

𝛼𝑛→0. Hence the sequence 𝑡𝑛

is bounded. There exists some 𝑀 > 0such that

∥𝑥𝑛+1−𝑧∥ ≤ [1−𝛼𝑛(1−𝑐)]∥𝑥𝑛−𝑧∥ + 𝛼𝑛(1−𝑐)𝑀

≤max{∥𝑥𝑛−𝑧∥, 𝑀}

≤max{∥𝑥0−𝑧∥, 𝑀}.

(34)

We conclude that {𝑥𝑛}is bounded and hence {𝑦𝑛}is

bounded.

According to the definition of 𝑦𝑛, it holds that

∥𝑦𝑛−𝑧∥2=∥𝑥𝑛+𝛽𝑛(𝑥𝑛−𝑥𝑛−1) − 𝑧∥2

=∥𝑥𝑛−𝑧∥2+2𝛽𝑛⟨𝑥𝑛−𝑥𝑛−1, 𝑥𝑛−𝑧⟩

+𝛽2

𝑛∥𝑥𝑛−𝑥𝑛−1∥2.

(35)

By Lemma 2.1(3), the following equation holds.

⟨𝑥𝑛−𝑥𝑛−1, 𝑥𝑛−𝑧⟩=−1

2∥𝑥𝑛−1−𝑧∥2+1

2∥𝑥𝑛−𝑧∥2

2∥𝑥𝑛−𝑥𝑛−1∥2.(36)

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2023.22.12

Yaxuan Zhang, Yuming Guan

E-ISSN: 2224-2880

103

Volume 22, 2023

Substituting the equality (36) into (35), we obtain

∥𝑦𝑛−𝑧∥2=∥𝑥𝑛−𝑧∥2+𝛽𝑛(−∥𝑥𝑛−1−𝑧∥2

+ ∥𝑥𝑛−𝑧∥2+ ∥𝑥𝑛−𝑥𝑛−1∥2)

+𝛽2

𝑛∥𝑥𝑛−𝑥𝑛−1∥2

≤ ∥𝑥𝑛−𝑧∥2+𝛽𝑛(∥𝑥𝑛−𝑧∥2− ∥𝑥𝑛−1−𝑧∥2)

+2𝛽𝑛∥𝑥𝑛−𝑥𝑛−1∥2

≤ ∥𝑥𝑛−𝑧∥2+𝛽𝑛∥𝑥𝑛−𝑥𝑛−1∥ (∥𝑥𝑛−𝑧∥

− ∥𝑥𝑛−1−𝑧∥) + 2𝛽𝑛∥𝑥𝑛−𝑥𝑛−1∥2

=∥𝑥𝑛−𝑧∥2+𝐸𝑛,

(37)

where 𝐸𝑛=𝛽𝑛∥𝑥𝑛−𝑥𝑛−1∥(∥𝑥𝑛−𝑧∥ − ∥𝑥𝑛−1−𝑧∥) +

2𝛽𝑛∥𝑥𝑛−𝑥𝑛−1∥2.

According to the boundedness of {𝜃𝑛}and {𝑦𝑛},

put

𝐿=max{sup

𝑛

{∥(𝐼−𝑃𝐶𝑏

𝑖𝑛,𝑛 )𝑦𝑛∥2+𝜃𝑛},

sup

𝑛

{∥ 𝐴∗

𝑗𝑛(𝐼−𝑃𝑄𝑏

𝑗𝑛,𝑛 )𝐴𝑗𝑛𝑦𝑛∥2+𝜃𝑛}}.(38)

It follows from (29) and (30) that

Φ4

𝑛≤𝐿

𝜌𝑛(1−𝜌𝑛)(∥𝑦𝑛−𝑧∥2−∥𝑦𝑛−𝜏𝑛∇𝑓𝑛(𝑦𝑛)−𝑧∥2),

(39)

where Φ𝑛=max{𝑑𝑛, 𝑣𝑛}.

Using (31) and (39), we have

∥𝑥𝑛+1−𝑧∥2

=⟨𝛼𝑛𝑔(𝑦𝑛)+(1−𝛼𝑛)(𝑦𝑛−𝜏𝑛∇𝑓𝑛(𝑦𝑛)) − 𝑧, 𝑥𝑛+1−𝑧⟩

=(1−𝛼𝑛)⟨𝑦𝑛−𝜏𝑛∇𝑓𝑛(𝑦𝑛) − 𝑧, 𝑥𝑛+1−𝑧⟩

+𝛼𝑛⟨𝑔(𝑦𝑛) − 𝑧, 𝑥𝑛+1−𝑧⟩

≤1−𝛼𝑛

2(∥𝑥𝑛+1−𝑧∥2+ ∥𝑦𝑛−𝜏𝑛∇𝑓𝑛(𝑦𝑛) − 𝑧∥2)

+𝛼𝑛⟨𝑔(𝑦𝑛) − 𝑔(𝑧), 𝑥𝑛+1−𝑧⟩

+𝛼𝑛⟨𝑔(𝑧) − 𝑧, 𝑥𝑛+1−𝑧⟩

≤1−𝛼𝑛

2(∥𝑥𝑛+1−𝑧∥2+ ∥𝑦𝑛−𝜏𝑛∇𝑓𝑛(𝑦𝑛) − 𝑧∥2)

+𝛼𝑛

2(𝑐2∥𝑦𝑛−𝑧∥2+ ∥𝑥𝑛+1−𝑧∥2)

+𝛼𝑛⟨𝑔(𝑧) − 𝑧, 𝑥𝑛+1−𝑧⟩

≤1−𝛼𝑛

2(∥𝑥𝑛+1−𝑧∥2+ ∥𝑦𝑛−𝑧∥2−𝜌𝑛(1−𝜌𝑛)Φ4

𝑛

𝐿)

+𝛼𝑛

2(𝑐∥𝑦𝑛−𝑧∥2+ ∥𝑥𝑛+1−𝑧∥2)

+𝛼𝑛⟨𝑔(𝑧) − 𝑧, 𝑥𝑛+1−𝑧⟩,

(40)

which is rearranged to obtain

∥𝑥𝑛+1−𝑧∥2

≤ (1−𝛼𝑛)∥𝑦𝑛−𝑧∥2

− (1−𝛼𝑛)𝜌𝑛(1−𝜌𝑛)Φ4

𝑛

𝐿+𝛼𝑛𝑐∥𝑦𝑛−𝑧∥2

+2𝛼𝑛⟨𝑔(𝑧) − 𝑧, 𝑥𝑛+1−𝑧⟩

=[1−𝛼𝑛(1−𝑐)] ∥𝑦𝑛−𝑧∥2

− (1−𝛼𝑛)𝜌𝑛(1−𝜌𝑛)Φ4

𝑛

𝐿

+2𝛼𝑛⟨𝑔(𝑧) − 𝑧, 𝑥𝑛+1−𝑧⟩.

(41)

Substituting (37) into (41) yields that

∥𝑥𝑛+1−𝑧∥2≤ [1−𝛼𝑛(1−𝑐)](∥𝑥𝑛−𝑧∥2+𝐸𝑛)

− (1−𝛼𝑛)𝜌𝑛(1−𝜌𝑛)Φ4

𝑛

𝐿

+2𝛼𝑛⟨𝑔(𝑧) − 𝑧, 𝑥𝑛+1−𝑧⟩

=[1−𝛼𝑛(1−𝑐)] ∥𝑥𝑛−𝑧∥2

+ [1−𝛼𝑛(1−𝑐)]𝐸𝑛

− (1−𝛼𝑛)𝜌𝑛(1−𝜌𝑛)Φ4

𝑛

𝐿

+2𝛼𝑛⟨𝑔(𝑧) − 𝑧, 𝑥𝑛+1−𝑧⟩.

(42)

Set

𝑠𝑛=∥𝑥𝑛−𝑧∥2;

𝛾𝑛=[1−𝛼𝑛(1−𝑐)]𝐸𝑛+2𝛼𝑛⟨𝑔(𝑧) − 𝑧, 𝑥𝑛+1−𝑧⟩;

𝛿𝑛=[1−𝛼𝑛(1−𝑐)] 𝐸𝑛

𝛼𝑛(1−𝑐)

1−𝑐⟨𝑔(𝑧) − 𝑧, 𝑥𝑛+1−𝑧⟩;

𝜂𝑛=(1−𝛼𝑛)𝜌𝑛(1−𝜌𝑛)Φ4

𝑛

𝐿.

(43)

From (42) and (43), we derive that

𝑠𝑛+1≤ [1−𝛼𝑛(1−𝑐)]𝑠𝑛+𝛼𝑛(1−𝑐)𝛿𝑛, 𝑛 ≥1,

(44)

𝑠𝑛+1≤𝑠𝑛−𝜂𝑛+𝛾𝑛, 𝑛 ≥1.(45)

Let {𝑛𝑘}be a subsequence of {𝑛}such that

lim sup

𝑘→∞

𝜂𝑛𝑘≤0.(46)

That is

lim sup

𝑘→∞

(1−𝛼𝑛𝑘)𝜌𝑛𝑘(1−𝜌𝑛𝑘)

Φ4

𝑛𝑘

𝐿≤0,(47)

which by conditions (C1) and (C2) implies

lim

𝑘→∞

Φ𝑛𝑘=0.(48)

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2023.22.12

Yaxuan Zhang, Yuming Guan

E-ISSN: 2224-2880

104

Volume 22, 2023

By the definition of Φ𝑛, it indicates that

lim

𝑘→∞ ∥(𝐼−𝑃𝐶𝑏

𝑖𝑛𝑘,𝑛𝑘

)𝑦𝑛𝑘∥=0,(49)

lim

𝑘→∞ ∥(𝐼−𝑃𝑄𝑏

𝑗𝑛𝑘,𝑛𝑘

)𝐴𝑗𝑛𝑦𝑛𝑘∥=0.(50)

The definitions of 𝑖𝑛𝑘and 𝑗𝑛𝑘ensure that

lim

𝑘→∞ ∥(𝐼−𝑃𝐶𝑏

𝑖,𝑛𝑘

)𝑦𝑛𝑘∥=0, 𝑖 =1,2,· · · , 𝑡, (51)

lim

𝑘→∞ ∥(𝐼−𝑃𝑄𝑏

𝑗,𝑛𝑘

)𝐴𝑗𝑦𝑛𝑘∥=0, 𝑗 =1,2,· · · , 𝑟.

(52)

Since 𝜕𝑞 𝑗, 𝑗 =1,2,· · · , 𝑟 are bounded on

bounded sets , there exists a constant 𝜇 > 0such

that ∥𝜁𝑛𝑘

𝑗∥ ≤ 𝜇, 𝑗 =1,2,· · · , 𝑟, 𝑘 ∈N, where

𝜁𝑛𝑘

𝑗∈𝜕𝑞 𝑗(𝑦𝑛𝑘). Note that 𝑃𝑄𝑏

𝑗,𝑛𝑘

𝐴𝑗𝑦𝑛𝑘∈𝑄𝑏

𝑗,𝑛𝑘and

from (52) we obtain

𝑞𝑗(𝐴𝑗𝑦𝑛𝑘) ≤ ⟨𝜁𝑛𝑘

𝑗, 𝐴 𝑗𝑦𝑛𝑘−𝑃𝑄𝑏

𝑗,𝑛𝑘

𝐴𝑗𝑦𝑛𝑘⟩

−𝛾𝑗

2∥𝐴𝑗𝑦𝑛𝑘−𝑃𝑄𝑏

𝑗,𝑛𝑘

𝐴𝑗𝑦𝑛𝑘∥2

≤ ∥𝜁𝑛𝑘

𝑗∥∥ 𝐴𝑗𝑦𝑛𝑘−𝑃𝑄𝑏

𝑗,𝑛𝑘

𝐴𝑗𝑦𝑛𝑘∥

≤𝜇∥(𝐼−𝑃𝑄𝑏

𝑗,𝑛𝑘

)𝐴𝑗𝑦𝑛𝑘∥ → 0, 𝑘 → ∞.

(53)

Since {𝑦𝑛𝑘}is bounded, there exists a subsequence

{𝑦𝑛𝑘𝑚} ⊂ {𝑦𝑛𝑘}such that 𝑦𝑛𝑘𝑚⇀ 𝑥∗and

lim sup

𝑘→∞

⟨𝑔(𝑧) − 𝑧, 𝑦𝑛𝑘−𝑧⟩=lim

𝑚→∞⟨𝑔(𝑧) − 𝑧, 𝑦𝑛𝑘𝑚−𝑧⟩.

(54)

Since 𝑞𝑗(·) is convex and weakly lower semicon-

tinuous and that 𝐴𝑗𝑦𝑛𝑘𝑚⇀ 𝐴 𝑗𝑥∗, by (53) we have

𝑞𝑗(𝐴𝑗𝑥∗) ≤ lim inf

𝑚→∞ 𝑞𝑗(𝐴𝑗𝑦𝑛𝑘𝑚) ≤ 0.(55)

Hence 𝐴𝑗𝑥∗∈𝑄𝑗.

By the definition of 𝐶𝑏

𝑖,𝑛𝑘, the assumption (A3) and

(51), there exists a constant 𝜀 > 0such that

𝑐𝑖(𝑦𝑛𝑘) ≤ ⟨𝜉𝑛𝑘

𝑖, 𝑦𝑛𝑘−𝑃𝐶𝑏

𝑖,𝑛𝑘

(𝑦𝑛𝑘)⟩

−𝜆𝑖

2∥𝑦𝑛𝑘−𝑃𝐶𝑏

𝑖,𝑛𝑘

(𝑦𝑛𝑘)∥2

≤𝜀∥𝑦𝑛𝑘−𝑃𝐶𝑏

𝑖,𝑛𝑘

(𝑦𝑛𝑘)∥ → 0, 𝑘 → ∞.

(56)

By the weakly lower semi-continuity of 𝑐𝑖and

𝑦𝑛𝑘𝑚⇀ 𝑥∗, we have

𝑐𝑖(𝑥∗) ≤ lim inf

𝑚→∞ 𝑐𝑖(𝑦𝑛𝑘𝑚) ≤ 0.(57)

Therefore 𝑥∗∈𝐶𝑖(𝑖=1, ..., 𝑡). So 𝑥∗∈𝑆.

From Lemma 2.1(1) and (54) we obtain:

lim sup

𝑘→∞

⟨𝑔(𝑧) − 𝑧, 𝑦𝑛𝑘−𝑧⟩

=lim

𝑚→∞⟨𝑔(𝑧) − 𝑧, 𝑦𝑛𝑘𝑚−𝑧⟩

=⟨𝑔(𝑧) − 𝑧, 𝑥∗−𝑧⟩ ≤ 0.

(58)

On the other hand, by (C3), we have

∥𝑦𝑛−𝑥𝑛∥=𝛽𝑛∥𝑥𝑛−𝑥𝑛−1∥ → 0, 𝑛 → ∞.(59)

So we have

∥𝑥𝑛+1−𝑥𝑛∥

=∥𝛼𝑛(𝑔(𝑦𝑛) − 𝑥𝑛)+(1−𝛼𝑛)(𝑦𝑛−𝜏𝑛∇𝑓𝑛(𝑦𝑛) − 𝑥𝑛)∥

≤𝛼𝑛∥𝑔(𝑦𝑛) − 𝑥𝑛∥+(1−𝛼𝑛)∥𝑦𝑛−𝑥𝑛∥

+ (1−𝛼𝑛)𝜏𝑛∥∇ 𝑓𝑛(𝑦𝑛)∥.

(60)

Note that

𝜏𝑛=𝜌𝑛

Φ2

𝑛

||∇ 𝑓𝑛(𝑦𝑛)||2+𝜃𝑛

≤2

inf{𝜃𝑛}Φ2

𝑛→0, 𝑛 → ∞.

(61)

Thus

∥𝑥𝑛+1−𝑥𝑛∥ → 0, 𝑛 → ∞.(62)

From (58), (59) and (62), we derive that

lim sup

𝑘→∞

⟨𝑔(𝑧) − 𝑧, 𝑥𝑛𝑘+1−𝑧⟩ ≤ 0.(63)

Then (C3) and (63) implies that

lim sup

𝑘→∞

𝛿𝑛𝑘≤0.(64)

Using Lemma 2.4 , we conclude that 𝑥𝑛→𝑧. The

proof is complete. □

4 Numerical experiments

In this section, we provide some numerical experi-

ments to show the efficiency of Algorithm 3.1 and

compare the convergence rates of our algorithm and

other algorithms. The codes are written in Matlab

R2018b and run on Inter(R) Core(TM) i9-12900H

CPU @ 2.50 GHz , RAM 16.00 GB.

The inertial coefficient 𝛽𝑛is given by

𝛽𝑛=𝜀𝑛

||𝑥𝑛−𝑥𝑛−1|| ,||𝑥𝑛−𝑥𝑛−1|| >1

𝜀𝑛,||𝑥𝑛−𝑥𝑛−1|| ≤ 1.

(65)

It is easy to check that 𝛽𝑛≤𝜀𝑛and that 𝛽𝑛||𝑥𝑛−

𝑥𝑛−1|| ≤ 𝜀𝑛. It also can be proved, see [24], that if

𝜀𝑛≥0and ∞

𝑛=0𝜀𝑛<+∞, the corresponding algo-

rithms are strongly convergent.

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2023.22.12

Yaxuan Zhang, Yuming Guan

E-ISSN: 2224-2880

105

Volume 22, 2023

Example 4.1 Consider the following problem: find

an element 𝑥∗∈R𝑁such that

𝑥∗∈𝑆=

𝑡



𝑖=1

𝐶𝑖∩ (

𝑟



𝑗=1

𝐴−1

𝑗(𝑄𝑗)),(66)

where the closed convex subsets 𝐶𝑖and 𝑄𝑗are given

𝐶𝑖={𝑥∈R𝑁:

𝑁



𝑘=𝑖

10 𝑘−1

𝑁−1𝑥2

𝑘−1≤0},(67)

𝑄𝑗={𝑦∈R𝑁(𝑗+1):

𝑁(𝑗+1)



𝑘=𝑗

10 𝑘−1

𝑁(𝑗+1) −1𝑦2

𝑘−1≤0}.

(68)

It is obvious that 𝐶𝑖and 𝑄𝑗are ellipsoids (see

[25]), and 𝑐𝑖(𝑥)=𝑁

𝑘=𝑖10 𝑘−1

𝑁−1𝑥2

𝑘−1and 𝑞𝑗(𝑦)=

𝑁(𝑗+1)

𝑘=𝑗10 𝑘−1

𝑁(𝑗+1) −1𝑦2

𝑘−1are both 2-strongly convex

functions, see [16]).

Set 𝑁=5,𝑡=10, 𝑟 =20, 𝜌𝑛=0.1, 𝛼𝑛=

𝑛, 𝜀𝑛=1

𝑛1.2, 𝜃𝑛=0.1, 𝑔(𝑥)=0.8𝑥, and 𝑒denotes

the vector of corresponding dimension of which the

coordinates are all 1. Let 𝐴𝑗:R𝑁→R𝑁(𝑗+1)be

bounded linear operators, of which the elements are

randomly generated in the closed interval [0,10]. Set

TOL =1

𝑡+𝑟(

𝑡



𝑖=1

∥𝑥𝑛−𝑃𝐶𝑖𝑥𝑛∥2

𝑟



𝑗=1

∥𝐴𝑗𝑥𝑛−𝑃𝑄𝑗𝐴𝑗𝑥𝑛∥2)(69)

for all 𝑛≥1. Note that if at the 𝑛th step, TOL =0,

then 𝑥𝑛∈𝑆, that is, 𝑥𝑛is a solution to this problem.

We use TOL <10−3as a stopping criteria.

First, we consider the impact of inertial term on

the convergence rate under different initial values 𝑥0

and 𝑥1. We use Alg.3.1 to refer to Algorithm 3.1 and

Alg.3.2 to refer to Algorithm 3.1 without inertial term.

The results of numerical experiments are reported in

Table 1 and Fig.1.

From Table 1 and Fig.1, we can see that Alg.3.1

has advantage over Alg.3.2 in both the iteration num-

ber and the CPU time. This shows that the inertial

perturbation can improve the convergence of the al-

gorithms.

Next, we consider the impact of 𝜌𝑛in the self-

adaptive step size on the convergence rate. For 𝜌𝑛=

0.1, 𝜌𝑛=0.3,𝜌𝑛=0.6, 𝜌𝑛=0.9, 𝜌 =1.2, 𝜌 =1.5,

and 𝜌=1.9, with other parameters retaining the same

values as above, we examine the convergence of the

sequence {𝑥𝑛}which is generated by Algorithm 3.1.

The results as shown in Fig.2(e).

123456

Number of iterations

10-2

10-1

TOL

Alg.3.1

Alg.3.2

(a) 𝑥0=𝑥1=1

50 ∗𝑟 𝑎𝑛𝑑 (𝑁 , 1)

0 5 10 15 20 25 30 35

Number of iterations

10-2

10-1

100

101

102

TOL

Alg.3.1

Alg.3.2

(b) 𝑥0=𝑥1=1

20 ∗𝑒

0 5 10 15 20 25 30 35 40

Number of iterations

10-2

10-1

100

101

102

103

TOL

Alg.3.1

Alg.3.2

10 ∗𝑒

0 10 20 30 40 50 60

Number of iterations

10-2

10-1

100

101

102

103

104

105

TOL

Alg.3.1

Alg.3.2

(d) 𝑥0=𝑥1=𝑒

Fig.1: Comparison of Alg.3.1 and Alg.3.2 under

different choices of initial values.

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2023.22.12

Yaxuan Zhang, Yuming Guan

E-ISSN: 2224-2880

106

Volume 22, 2023

Table 1: The numerical results for Alg.3.1 and

Alg.3.2

Alg.3.1 Alg.3.2

Initial Point 𝑛Time(s) 𝑛Time(s)

𝑥0=𝑥1=1

50 ∗𝑟𝑎𝑛𝑑 (𝑁, 1)4 0.0223 7 0.0330

𝑥0=𝑥1=1

20 ∗𝑒28 0.0823 33 0.0941

𝑥0=𝑥1=1

10 ∗𝑒35 0.0986 40 0.1218

𝑥0=𝑥1=𝑒54 0.1500 59 0.1495

0 10 20 30 40 50 60

Number of iterations

10-2

10-1

100

101

102

103

104

105

TOL

n=0.1

n=0.3

n=0.6

n=0.9

n=1.1

n=1.5

n=1.9

(e) Different choices of 𝜌𝑛

0 50 100 150 200 250 300 350

Number of iterations

10-2

10-1

100

101

102

103

104

105

TOL

Alg.3.1

Alg.3.3

(f) Different choices of step sizes

Fig.2: Comparison of different choices of 𝜌𝑛and

step sizes.

It seems that Algorithm 3.1 converges faster if 𝜌𝑛

takes values around the midpoint of the interval (0,2).

Finally, we consider the convergence rate under

different choices of self-adaptive step sizes. We de-

note by Alg.3.3 the algorithm the same as ours except

that the the step size is the ordinary self-adaptive one:

𝜏𝑛=𝜌𝑛

𝑎1

𝑎2

,(70)

where

𝑎1=

𝑡



𝑖=1

||(𝐼−𝑃𝐶𝑏

𝑖,𝑛 )𝑦𝑛||2+

𝑟



𝑗=1

||(𝐼−𝑃𝑄𝑏

𝑗,𝑛 )𝐴𝑗𝑦𝑛||2,

(71)

𝑎2=||

𝑡



𝑖=1

(𝐼−𝑃𝐶𝑏

𝑖,𝑛 )𝑦𝑛+

𝑟



𝑗=1

𝐴∗

𝑗(𝐼−𝑃𝑄𝑏

𝑗,𝑛 )𝐴𝑗𝑦𝑛||2,

(72)

with other parameters retaining the same values as

above, we provide the results in Fig.2(f).

From Fig.2(f), we see that our algorithm with step

size defined as in (26) is more effective in that it

used fewer iterates in the experiment. The reason

may be that we applied the largest distance among

∥𝐴𝑗𝑦𝑛−𝑃𝑄𝑏

𝑗,𝑛 (𝐴𝑗𝑦𝑛)∥, 𝑖 =1,2,· · · , 𝑡 and ∥𝐴𝑗𝑦𝑛−

𝑃𝑄𝑏

𝑗,𝑛 (𝐴𝑗𝑦𝑛)∥, 𝑗 =1,2,· · · , 𝑟 while Alg.3.3 use the

sum of them.

Acknowledgments

The authors would like to thank the editors

and reviewers for their valuable comments on the

manuscript.

References:

[1] Y. Censor, T. Elfving, A multi projection algo-

rithm using Bregman projections in a product

space, Numer. Algorithms, Vol.8, No.3, 1994,

pp. 221-239.

[2] Y. Censor, T. Elfving, N. Kopf, T. Bortfeld, The

multiple-sets split feasibility problem and its ap-

plication for inverse problem, Inverse Probl.,

Vol.21, 2005, pp. 2071-2084.

[3] C. Byrne, Iterative oblique projection onto con-

vex sets and the split feasibility problem, Inverse

Probl., Vol.18, No.2, 2002, pp. 441-453.

[4] C. Byrne, A unified treatment of some iter-

ative algorithms in signal processing and im-

age reconstruction, Inverse Probl., Vol.20, No.1,

2004, pp. 103-120.

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2023.22.12

Yaxuan Zhang, Yuming Guan

E-ISSN: 2224-2880

107

Volume 22, 2023

[5] Y. Censor, A. Segal, Iterative projection meth-

ods in biomedical inverse problems. Mathe-

matical methods in biomedical imaging and

intensity-modulated radiation therapy (IMRT),

Vol.10, 2008, pp. 656.

[6] Y. Censor, T. Bortfeld, B. Martin, A. Trofi-

mov, A unified approach for inversion problems

in intensity-modulated radiation therapy, Phys.

Med. Biol., Vol.51, No.10, 2006, 2353-65.

[7] J. Wang, Y. Hu, C. Li, J.C. Yao, Linear conver-

gence of CQ algorithms and applications in gene

regulatory network inference, Inverse Probl.,

Vol.33, No.5, 2017, pp. 055017.

[8] Y. Censor, A. Segal, The split common fixed

point problem for directed operators, J. Convex

Anal., Vol.26, No.5, 2010, pp. 55007.

[9] A. Moudafi, The split common fixed point

problem for demicontractive mappings, Inverse

Probl., Vol.26, No.5, 2010, pp. 055007 .

[10] S. Tuyen, T.M. Reich, N.M. Trang, Parallel it-

erative methods for solving the split common

fixed point problem in Hilbert spaces, Numer.

Funct. Anal. Optim., Vol.41, No.7, 2020, pp.

778-805.

[11] Y. Censor, A. Gibali, S. Reich, Algorithms for

the split variational inequality problems, Numer.

Algorithms, Vol.59, 2012, pp. 301-323.

[12] S. Reich, T.M. Tuyen, Iterative methods for

solving the generalized split common null point

problem in Hilbert spaces, Optimization, Vol.69,

No.5, 2020, pp. 1013-1038.

[13] S. Reich, T.M. Tuyen, A new algorithm for

solving the split common null point problem

in Hilbert spaces, Numer. Algorithms, Vol.83,

2020, pp. 789-805.

[14] S. Reich, T.M. Tuyen, The split feasibility prob-

lem with multiple output sets in Hilbert spaces,

Optim. Lett., Vol.14, 2020, pp. 2335-2353.

[15] Q.Z. Yang, The relaxed CQ algorithm solv-

ing the split feasibility problem, Inverse Probl.,

Vol.20, 2004, pp. 1261-1266.

[16] H. Yu, W.R. Zhan, F.H. Wang, The ball-relaxed

CQ algorithms for the split feasibility problem,

Optimization, Vol.67, No.10, 2018, pp. 1687-

1699.

[17] Q.Z. Yang, On variable-step relaxed projection

algorithm for variational inequalities, J. Math.

Anal. Appl., Vol.302, 2005, pp. 166-179.

[18] G. López, V. Martín-Márquez, F.H. Wang, H.K.

Xu, Solving the split feasibility problem with-

out prior knowledge of matrix norms, Inverse

Probl., Vol.28, 2012, pp. 374-389.

[19] A. Gibali, L.W. Liu, Y.C. Tang, Note on the

modified relaxation CQ algorithm for the split

feasibility problem, Optim. Lett., Vol.12, 2018,

pp. 817-830.

[20] S. Suantai, N. Pholasa, P. Cholamjiak, Re-

laxed CQ algorithms involving the inertial tech-

nique for multiple-sets split feasibility problems,

RACSAM, Vol.113, 2019, pp. 1081-1099.

[21] H.H. Bauschke, P.L. Combettes, Convex Anal-

ysis and Monotone Operator Theory in Hilbert

Space, Springer: London, UK, 2011.

[22] S. He, C. Yang, Solving the variational in-

equality problem defined on intersection of

finite level sets, Abstr. Appl. Anal., 2013,

https://doi.org/10.1155/2013/942315.

[23] G.H. Taddele, P. Kumam, A.G. Gebrie, J.

Abubakar, Ball-relaxed projection algorithms

for multiple-sets split feasibility problem, Opti-

mization, Vol.2, 2021, pp. 1-31.

[24] Y.Y. Li, Y.X. Zhang, Bounded Perturbation

Resilience of Two Modified Relaxed CQ Al-

gorithms for the Multiple-Sets Split Feasibil-

ity Problem. Axioms, Vol.10, 2021, pp. 197,

https://doi.org/10.3390/axioms10030197.

[25] Y.H. Dai, Fast algorithms for projection on an

ellipsoid, SIAM J Optim., Vol.16, 2006, pp. 986-

1006.

5Contribution of individual authors

Yaxuan Zhang proposed the algorithm and checked

the correctness of the manuscript.

Yuming Guan proved the convergence theorem of the

algorithm and carried out the numerical simulation.

6Sources of funding

This work is supported by the national science foun-

dation of China (grant no. NSFC 11705279).

Creative Commons Attribution License 4.0

(Attribution 4.0 International, CC BY 4.0)

This article is published under the terms of the Creative

Commons Attribution License 4.0

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

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2023.22.12

Yaxuan Zhang, Yuming Guan

E-ISSN: 2224-2880

108

Volume 22, 2023

Conflict of Interest

The authors have no conflicts of interest to declare

that are relevant to the content of this article.