Xiaoyin Li, Hongwei Liu, Jiangli Cheng and Dongyao Zhang

(School of Mathematics and Statistics, Xidian University, Xi’an 710126, China)

Abstract: Many approaches inquiring into variational inequality problems have been put forward, among which subgradient extragradient method is of great significance. A novel algorithm is presented in this article for resolving quasi-nonexpansive fixed point problem and pseudomonotone variational inequality problem in a real Hilbert interspace. In order to decrease the execution time and quicken the velocity of convergence, the proposed algorithm adopts an inertial technology. Moreover, the algorithm is by virtue of a non-monotonic step size rule to acquire strong convergence theorem without estimating the value of Lipschitz constant. Finally, numerical results on some problems authenticate that the algorithm has preferable efficiency than other algorithms.

Keywords: inertial method; fixed point; variational inequality; strong convergence; subgradient extragradient method

0 Introduction

His presumed to be a real Hilbert interspace, which possesses a given non-null, closed convex subsetC.In this article, variational inequality problem is mainly considered, which is hereinafter abbreviated as VIP. It can be represented as seeking a dotx*∈Cthat satisfies the below expression:

〈x-x*,F(x*)〉≥0, ?x∈C

(1)

where 〈·,·〉 is known as transvection and ‖·‖ is referred to as norm. The answer set of (1) is written as sol(F,C).As a crucial part and extension of many problems in nonlinear analysis, VIP has appeared in diverse physical models, economic science, transportation, engineering, optimal control and other problems[1-4]. Because of its significance and applications, considerable interest has been attached to research methods for approximating solutions of VIP. In the last few years, the projection method has been widely noticed by some authors[5-16], who have proposed a series of variant forms about this method and obtained the convergence of these forms.

Fixed point problem (FPP), a well-explored subject in functional analysis, is another major and related problem.It is of great importance to study the common solutions of FPP and VIP due to the fact that constraint conditions of some mathematical models can be converted into fixed point and variational inequality problems. These examples can be discovered in various practical matters such as image recovery and signal processing[17-18]. This article focuses on solving FPP and VIP, in other words, seeking a dotx*to satisfy Eq.(2).

x*∈Fix(U)∩sol(F,C)

(2)

whereU:H→Hand Fix(U)={x∈H:Ux=x}.In recent years, lots of methods to solve problem (2) have been proposed and discussed[19-26].

On the basis of extragradient method[9], Nadezhkina and Takahashi[20]put forward an iterative approach to solve problem (2) and confirmed the theorem of weak convergence. The most distinguishing characteristic of this approach is the projection in each iteration, which is based on practicable setCand demands to be carried out twice. However, whenCis equipped with a comparatively complicated structure, the calculation cost will be expensive. After that, Censor et al.[10]presented a modified approach. The target is to substitute the second projection ontoCfor a specific semi-space which can be computed explicitly. In addition, Kraikaew and Saejung[22]proposed another improved method for solving problem (2) by combining Halpern iterative method with subgradient extergradient method, and the strong convergence result was testified.

Looking from another aspect, inertial technique was applied by Alvarez and Attouch[27]to acquire the inertial proximal point algorithm. Meanwhile, the termθk(xk-xk-1) was introduced, which is known as the inertia. Since inertial type algorithm can be viewed as a procedure, which is conducive to promote the convergence rate of sequences, fast iterative algorithms based on inertial technique have been extensively studied by some authors[15,21,23-24].

Inspired by the above research results[15-16,24-25], an innovative algorithm is presented in this paper for resolving quasi-nonexpansive fixed point and pseudomonotone variational inequality problem. Main contributions of this algorithm are as follows. A rule of non-monotonic step size is employed which demands no information of Lipschitz constant. In addition, the algorithm is constructed by using the subgradient extragradient method which is highly applied, and the inertial termθk(xk-xk-1) is utilized which is beneficial to speeding up the convergence and the viscosity method which ensures the strong convergence. Comparing the proposed algorithm in this paper with those in Refs. [16] and [25], pseudomonotone variational inequalities are considered, and under the weaker condition, it is proved that the proposed algorithm is provided with the property of strong convergence. Through the analysis of algorithms[11-12], it can be found that the step size composition of the new algorithm is more concise and does not make use of linesearch, which means that no calculation of additional projections is needed. Furthermore, numerical experiments show that the new algorithm has preferable runtime and number of iterations than the presented ones.

The synopsis of this article is arranged as shown below. Several correlative and fundamental results are given in Section 1. Algorithm A and the proof of convergence constitute Section 2. Ultimately, some classical numerical problems and comparisons are provided in Section 3.

1 Preliminaries

A small number of elementary concepts and lemmas are expounded in this portion, which can be directly applied later.Following this, strong convergence and weak convergence of {ξn} for dotξare expressed asξn→ξandξn?ξ, respectively.

Definition1[7]Givenξ,η∈H, the mappingFis called as follows:

(i) Monotone, if

〈F(ξ)-F(η),ξ-η)〉≥0

(ii)Pseudomonotone, if

〈F(η),ξ-η)〉≥0?〈F(ξ),ξ-η)〉≥0

(iii) L-Lipschitz continuous, if

?L>0, s.t.‖F(xiàn)(ξ)-F(η)‖≤L‖ξ-η‖

(iv) Sequentially weakly continuous, if

?{ξn}?H,ξn?ξ?F(ξn)?F(ξ)

Remark1For each monotone mapping, according to Definition 1, it is not difficult to observe that it is pseudomonotone. However, the opposite implication is not true.

Definition2[28]LetU:H→Hand Fix(U)≠?.

(i)Uis referred to as a contraction, if

?l∈(0,1), s.t.‖Uξ-Uη‖≤l‖ξ-η‖,

?ξ,η∈H

Ifl=1,Uis nonexpansive.

(ii)Uis quasi-nonexpansive, if

‖Uξ-η‖≤‖ξ-η‖, ?ξ∈H,η∈Fix(U)

(iii)I-Uis said to be demiclosed at zero, if

?{ξn}?H,ξn?ξ,

(I-U)ξn→0?ξ∈Fix(U)

Remark2With respect to each nonexpansive mapping, it is common knowledge that it is quasi-nonexpansive only when the disaggregation of fixed points is non-null.

Lemma1[28]Cis supposed to be a known non-null closed convex subset. For any dotξ∈H, the below expressions are true:

(i)ζ=PCξ?〈ζ-ξ,ζ-η〉≤0, ?η,ζ∈C

(ii)‖ξ+η‖2≤‖ξ‖2+2〈η,ξ+η〉, ?η∈H

(iii) ‖PC(ξ)-η‖2+‖ξ-PC(ξ)‖2≤‖ξ-η‖2, ?η∈C

(iv) 〈ξ-PC(ξ),η-PC(ξ)〉≤0, ?η∈C

Lemma2[29]F:C→His presumed to be a pseudomonotone succession, whenx*∈Cis an answer of inequality (1), it is equivalent to seeking dotx*that satisfies the below formula:

〈x-x*,F(x)〉≥0, ?x∈C

2 Algorithm and Convergence Analysis

Novel Algorithm A is stated and studied in this part. First, it is assumed that the presented algorithm satisfies the below conditions.

(C1)Cis a known non-null, closed convex aggregation and Fix(U)∩sol(F,C)≠?;

(C2)Fis sequentially weakly continuous overC, pseudomonotone and L-Lipschitz continuous inH;

(C3)I-Uis demiclosed at zero andUis quasi-nonexpansive;

(C4)fis a contraction onHsuch that contraction coefficientl∈(0,1);

(C5){qn}, {tn} and {sn} are successions obtained from (0,1) and these successions satisfyqn+tn+sn=1;

At present, a clear and accurate description of the algorithm is given. The form is listed below:

AlgorithmA

Step1: Defineun=τn(xn-xn-1)+xn, and figure up

yn=PC(un-λnF(un))

Step2:WithTn={x∈H:〈yn-x,un-yn-λnF(un)〉≥0}, reckon up

zn=PTn(un-λnF(yn))

Step3:Setxn+1=qnUzn+tnzn+snf(xn), and update

Putn=n+1 and turn to Step 1.

Remark3According to conditions listed above, expression (3) is acquired:

(3)

Indeed, choose parameterτnto be

Consequently, with respect to any positive integernand the selection of parameterτnmentioned above, it is known thatτn‖xn-1-xn‖≤θn.By the condition (C6), the following formula can be procurable:

Next, some lemmas are proposed, which lay a preliminary foundation for the convergence property of Algorithm A.

ProofFis known to be L-Lipschitz continuous, whenF(un)-F(yn)≠0, it can be derived that

Through the notion ofλn+1and induction, it can be attained that the lower bound and upper bound of sequence {λn} are min{λ1,μ/L} andλ1+P, respectively. Bringinghn+1=λn+1-λninto the equation, it also can be seen from the concept ofλn+1that

(4)

(5)

For eachy∈C,p∈sol(F,C) can be testified. Indeed, according to Lemma 1(iv) and the notion ofynk, it can be derived that

〈unk-λnkF(unk)-ynk,y-ynk〉≤0

This signifies that

〈F(unk)-F(ynk),y-ynk〉+〈F(ynk),y-ynk〉≥

(6)

(7)

rk+〈F(ynj),y-ynj〉≥0, ?j≥Nk

(8)

Since{rk} decreases, it is easy to get {Nk} is increasing.

For anyk, putting

there is 〈dNk,F(yNk)〉=1.By Eq.(8), it is deduced that

〈F(yNk),rkdNk+y-yNk〉≥0

Through utilizing the pseudomonotonicity ofF, the formula below can be attained:

〈F(rkdNk+y),rkdNk+y-yNk〉≥0

(9)

Furthermore, according to the concept of sequentially weakly continuity andynk?p, it is derived thatF(ynk)?F(p).AssumeF(p)≠0 (if not,pis a solution). Apparently, applying lower semicontinuity of norm mapping, it is easy to obtain:

Lemma6Suppose that successions {zn}, {yn} and {un} are produced by Algorithm A, the expression given below holds:

ProofTakev∈sol(F,C).It follows fromzn=PTn(un-λnF(yn)) and Lemma 1(iii) that

‖zn-v‖2=‖PTn(un-λnF(yn))-v‖2≤

‖un-λnF(yn)-v‖2-‖un-λnF(yn)-zn‖2=

‖v-un+λnF(yn)‖2-‖zn-un+λnF(yn)‖2=

‖v-un‖2+2λn〈v-un,F(yn)〉-‖zn-un‖2-

2λn〈zn-un,F(yn)〉=‖v-un‖2-‖zn-un‖2+

2λn〈v-zn,F(yn)〉=‖v-un‖2-‖zn-un‖2+

2λn〈yn-zn,F(yn)〉+2λn〈v-yn,F(yn)〉

Byyn∈Candv∈sol(F,C), it can be deduced that 〈yn-v,F(v)〉≥0.From the pseudomonotonicity ofF, the formula given below can be attained:

〈yn-v,F(yn)〉≥0

(10)

According tozn∈Tn, there is

〈zn-yn,un-yn-λnF(un)〉≤0

Using the above inequality,Eq.(10) and Cauchy-Schwarz inequality, the expression below can be attained:

‖zn-v‖2≤‖v-un‖2-‖zn-un‖2+

2λn〈yn-zn,F(yn)〉=‖v-un‖2-‖zn-yn‖2-

‖yn-un‖2+2〈zn-yn,un-yn-λnF(yn)〉=

‖v-un‖2-‖zn-yn‖2-‖yn-un‖2+

2λn〈zn-yn,F(un)-F(yn)〉+2〈zn-yn,un-

yn-λnF(un)〉≤‖v-un‖2-‖zn-yn‖2-

‖yn-un‖2+2λn‖zn-yn‖‖F(xiàn)(un)-

F(yn)‖

(11)

Combining the definition ofλn+1and Eq.(11), there is

‖zn-v‖2≤‖v-un‖2-‖zn-yn‖2-

F(yn)‖≤‖v-un‖2-‖zn-yn‖2-

Lemma7Succession {xn} is provided by Algorithm A. If the presumptions (C1)-(C6) are given, it can be inferred that {xn} is bounded.

1-μ(λn/λn+1)>0

Using Lemma 6, we get

(12)

Using the notion ofun, it can be deduced that

‖v-un‖=‖v-(τn(xn-xn-1)+xn)‖≤τn‖xn-xn-1‖+‖v-xn‖=

(13)

Utilizing Eq.(3), the existence of positive numberM1can be inferred, together with

(14)

From Eqs.(12), (13) and (14), the expression below is obtained:

‖zn-v‖≤‖v-un‖≤‖v-xn‖+snM1, ?n≥N

(15)

Combining Eq.(15) and the formula ofxn+1, there is

‖xn+1-v‖=‖qnUzn+tnzn+snf(xn)-v‖≤

qn‖Uzn-v‖+tn‖zn-v‖+sn(‖f(xn)-f(v)‖+

‖f(v)-v‖)≤(tn+qn)‖zn-v‖+sn(l‖xn-v‖+

‖f(v)-v‖)≤(1-sn)(‖v-xn‖+snM1)+

sn(l‖v-xn‖+‖f(v)-v‖)≤sn(‖f(v)-v‖+

M1)+(1-(1-l)sn)‖v-xn‖=(1-

Therefore, it is concluded that succession{xn} is bounded. Meanwhile, this signifies that {yn}, {un}, {f(xn)}, {zn} and {Uzn} have the same property.

Theorem1Succession {xn} can be provided by Algorithm A. If conditions (C1)- (C6) are given, {xn} to dotvmust be strong convergence, wherev=PFix(U)∩sol(F,C)f(v) andv∈Fix(U)∩sol(F,C).

ProofThree parts listed below are the process of this proof.

Part1The expression below can be illustrated:

tnqn‖Uzn-zn‖2+(1-sn)(1-μ(λn/λn+1))·

(‖yn-un‖2+‖zn-yn‖2)≤

‖xn-v‖2-‖xn+1-v‖2+snM4

Indeed, using inequality (15), there is

‖v-un‖2≤(‖v-xn‖+snM1)2=

‖v-xn‖2+snM2

(16)

for someM2>0.

On the other hand,from the expression ofxn+1, it can be deduced that

‖xn+1-v‖2=‖qnUzn+tnzn+snf(xn)-v‖2=

‖qn(Uzn-v)+tn(zn-v)+sn(f(xn)-v)‖2≤

qn‖Uzn-v‖2+tn‖zn-v‖2+sn‖f(xn)-v‖2-

tnqn‖Uzn-zn‖2≤(tn+qn)‖zn-v‖2+

sn(‖f(xn)-f(v)‖+‖f(v)-v‖)2-tnqn‖Uzn-zn‖2≤

(1-sn)‖zn-v‖2+sn(l‖v-xn‖+

‖f(v)-v‖)2-tnqn‖Uzn-zn‖2≤

(1-sn)‖zn-v‖2+sn‖v-xn‖2-

tnqn‖Uzn-zn‖2+sn(‖f(v)-v‖2+

2‖f(v)-v‖‖v-xn‖)≤sn‖v-xn‖2+

(1-sn)‖zn-v‖2-tnqn‖Uzn-zn‖2+

snM3

(17)

for someM3>0.

Combining formulas (16), (17) and Lemma 6, the expression below can be derived:

‖xn+1-v‖2≤sn‖xn-v‖2-tnqn‖Uzn-zn‖2+

(‖yn-un‖2+‖zn-yn‖2))≤‖xn-v‖2-

tnqn‖Uzn-zn‖2+snM2+snM3-(1-sn)(1-

(18)

That is

tnqn‖Uzn-zn‖2+(1-sn)(1-μ(λn/λn+1))·

(‖yn-un‖2+‖zn-yn‖2)≤‖xn-v‖2-

‖xn+1-v‖2+snM4

(19)

whereM4∶=M2+M3.

Part2The expression below can be attained:

‖xn+1-v‖2≤(1-(1-l)sn)‖xn-v‖2+

Indeed, using the formula ofun, there is

‖un-v‖2=‖τn(xn-xn-1)+xn-v‖2≤

(τn‖xn-xn-1‖+‖xn-v‖)2=

‖xn-v‖2+3Mτn‖xn-xn-1‖

(20)

According to Lemma 1(ii), inequality(12) and (20), it can be concluded

‖xn+1-v‖2=‖qn(Uzn-v)+tn(zn-v)+

sn(f(xn)-f(v))+sn(f(v)-v)‖2≤‖qn(Uzn-v)+

tn(zn-v)+sn(f(xn)-f(v))‖2+2sn〈xn+1-v,f(v)-v〉≤

(1-sn)‖zn-v‖2+sn‖f(xn)-f(v)‖2+

2sn〈xn+1-v,f(v)-v〉≤(1-sn)‖un-v‖2+

snl‖xn-v‖2+2sn〈xn+1-v,f(v)-v〉≤(1-

sn)‖xn-v‖2+snl‖xn-v‖2+2sn〈xn+1-v,

f(v)-v〉+3Mτn‖xn-xn-1‖=2sn〈xn+1-v,

f(v)-v〉+(1-(1-l)sn)‖xn-v‖2+3Mτn‖xn-

xn-1‖=(1-(1-l)sn)‖xn-v‖2+(1-l)sn·

(21)

Part3It can be proved that the succession {‖xn-v‖} strongly converges to zero.

(22)

Combining Part 1 and inequality (22), it can be infered that

‖xnk-v‖2)≤0

(23)

This implies that

‖unk-ynk‖→0 (k→∞)

(24)

‖ynk-znk‖→0(k→∞)

(25)

‖Uznk-znk‖→0(k→∞)

(26)

By the definition ofun, there is

(27)

Combining Eqs.(24)-(27), the following expressions are derived:

‖xnk-ynk‖≤‖xnk-unk‖+‖unk-ynk‖→0，

(k→∞)

(28)

‖xnk+1-znk‖≤snk‖f(xnk)-znk‖+tnk‖znk-

znk‖+qnk‖Uznk-znk‖→0，(k→∞)

(29)

‖znk-unk‖≤‖znk-ynk‖+‖ynk-unk‖→0，

(k→∞)

(30)

‖xnk-xnk+1‖≤‖xnk-unk‖+‖unk-znk‖+

‖znk-xnk+1‖→0，(k→∞)

(31)

For{xnk} is bounded, it is not hard to infer that succession {xnkj}?{xnk} and pointx*∈Hexist, together withxnkj?x*.Then there is

(32)

According to Lemma 5, it is derived thatx*∈Fix(U)∩sol(F,C).Sincev=PFix(U)∩sol(F,C)f(v), Eq.(33) can be obtained:

(33)

Combining Eqs.(31) and (33), the result below is deduced:

xnk,f(v)-v〉+〈xnk-v,f(v)-v〉)≤

(34)

3 Numerical Experiments

On this portion, the validation of Algorithm A is verified by several numerical experiments and comparisons. In the numerical implementations, the first two inquire into the test problem of finite dimensional spaces, and the last one is about infinite dimensional spaces. In experimental results described in the tables below, the runtime (time) and the number of iterations (ite.) are written down in seconds. In addition, ‖xn-yn‖≤εis used as stopping condition for all algorithms, whereεis the iteration accuracy. With respect to Algorithm A, the parameters listed below are taken:

μ=0.2,λ1=0.09,f(x)=0.15x

θn=6/(n+1)4,pn=(n+1)-2

tn=(n+1)-1,sn=(n3+1)-1

Problem1The first problem is a classical problem which was considered in Refs. [24] and [25]. LetH=R.Presume thatU:H→His expressed asUx=(x/2)sinxandF:H→His expressed asFx=x+sinx.In this problem, presetC=[-5,5] and start pointx0=x1=1.F, by simple calculation, can be verified to be monotone and consequently pseudomonotone onC.Meanwhile,Uis not nonexpansive but quasi-nonexpansive onC.For Algorithm 3.3[11], setσ=0.9,γ=0.9,θn=0.2 andαn=(n3+1)-1.For Algorithm 3.1[25], takeμ=0.2,τ1=0.09,f(x)=0.15x,αn=(n3+1)-1andβn=(n+1)-1.Different choices ofεare given, and the results of the experiment are illustrated in Table 1.

Table 1 Experimental results for Problem 1

Problem2The second problem was applied in Refs. [12] and [13]. Suppose thatH=R2.LetF:R2→R2be represented as

Table 2 Experimental results for Problem 2

Table 3 Experimental results for Problem 3

4 Conclusions

Based on several classical approaches, a novel algorithm is presented. In particular, the suggested algorithm adopts a non-monotonic step size rule so that its convergence has no requirement to reckon Lipschitz constant. When parameters satisfy certain conditions, strong convergence is confirmed. Eventually, the testification of algorithm validation is undertaken by some numerical problems.