Two modifications of the inertial Tseng extragradient method with self-adaptive step size for solving monotone variational inequality problems

Timilehin Opeyemi Alakoya; Lateef Olakunle Jolaoso; Oluwatosin Temitope Mewomo

doi:10.1515/dema-2020-0013

Open Access Published by De Gruyter Open Access September 15, 2020

Two modifications of the inertial Tseng extragradient method with self-adaptive step size for solving monotone variational inequality problems

Timilehin Opeyemi Alakoya , Lateef Olakunle Jolaoso and Oluwatosin Temitope Mewomo

From the journal Demonstratio Mathematica

https://doi.org/10.1515/dema-2020-0013

Abstract

In this work, we introduce two new inertial-type algorithms for solving variational inequality problems (VIPs) with monotone and Lipschitz continuous mappings in real Hilbert spaces. The first algorithm requires the computation of only one projection onto the feasible set per iteration while the second algorithm needs the computation of only one projection onto a half-space, and prior knowledge of the Lipschitz constant of the monotone mapping is not required in proving the strong convergence theorems for the two algorithms. Under some mild assumptions, we prove strong convergence results for the proposed algorithms to a solution of a VIP. Finally, we provide some numerical experiments to illustrate the efficiency and advantages of the proposed algorithms.

Keywords: extragradient method; inertial; monotone; variational inequality; Lipschitz-continuous

MSC 2010: 65K15; 47J25; 65J15; 90C33

1 Introduction

Let H be a real Hilbert space with the inner product 〈⋅,⋅〉 and the induced norm ∥⋅∥. Let C be a nonempty, closed, and convex subset in H. In this article, we consider the classical variational inequality problem (VIP), which is to find a point x†∈C such that

(1)〈Ax†,y−x†〉≥0,∀ y∈C,

where A:H→H is a given operator. The solution set of VIP (1) is denoted by VI(C,A).

Variational inequality theory is an important tool in economics, engineering, mathematical programming, transportation, and in other fields (see, for example, [1,2,3,4,5,6,7,8]). Many numerical methods have been constructed for solving variational inequalities and related optimization problems, see [9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27] and references therein.

One of the most popular methods for solving the problem (VIP) is the extragradient method (EGM). This method was introduced by Korpelevich [28] in 1976 as follows:

(2)x0∈C,yn=PC(xn−λAxn),xn+1=PC(xn−λAyn),n≥0,

where λ∈(0,1L) and PC denotes the metric projection from H onto C. The EGM was first introduced for solving saddle point problems, after which the method was further extended to VIPs in both the Euclidean spaces and Hilbert spaces. The convergence of the EGM only requires that the operator A is monotone and L-Lipschitz continuous. If the solution set VI(C,A) is nonempty, then the sequence {xn} generated by algorithm (2) converges weakly to an element in VI(C,A).

In recent years, the EGM (2) has received great attention by many authors, who improved it in various ways (see, for instance, [9,17,29,30,31,32] and references therein). In order to obtain the strong convergence of the EGM in real Hilbert spaces, Maingé [33] proposed a modified version of the algorithm as follows:

(3)x0∈H,yn=PC(xn−λnAxn),tn=PC(xn−λnAyn),xn+1=tn−αnFtn,

where A:H→H is monotone on C and L-Lipschitz continuous on H and F:H→H is Lipschitz continuous and strongly monotone on C such that VI(C,A)≠∅. Maingé proved that if the parameters satisfy the conditions: λn∈[a,b]⊂(0,1L),αn∈[0,1),limn→∞αn=0, and ∑n=0∞αn=∞, then the sequence {xn} converges strongly to x†∈VI(C,A), where x† is the solution of the following VIP

〈Ax†,y−x†〉≥0,∀ y∈VI(C,F).

One of the drawbacks of the EGM and its modified version above is that in each iteration in the algorithm, two projections are made onto the closed convex set C. However, projections onto a general closed and convex set are not easily executed, a fact that might affect the efficiency and applicability of the method.

In order to overcome the aforementioned drawback, Censor et al. [9] presented the subgradient extragradient method, in which the second projection onto C is replaced by a projection onto a specific constructible half-space which can be easily calculated. Their algorithm is of the form:

(4)yn=PC(xn−λAxn),Tn={w∈H|〈xn−λAxn−yn,w−yn〉≤0},xn+1=PTn(xn−λAyn),∀ n≥0,

where λ∈(0,1L).

Tseng in [32] proposed another method for solving the VIP (1), which uses only one projection in each iteration. This method is known as the Tseng extragradient method (TEGM) and is presented as follows:

(5)x0∈H,yn=PC(xn−λAxn),xn+1=yn−λ(Ayn−Axn),∀ n≥0,

where λ∈(0,1L).

Another shortcoming of algorithms (2), (3), (4), and (5) is the choice of stepsize. The stepsize plays an essential role in the convergence properties of iterative methods. In the aforementioned algorithms, the stepsizes are defined to be dependent on the Lipschitz constant L of the monotone operator. In this case, a prior knowledge or estimate of the Lipschitz constant is required. However, in many cases, this parameter is unknown or difficult to estimate. Moreover, the stepsize defined by the constant is often very small and slows down the convergence rate of iterative methods. In practice, a larger stepsize can often be used and yields better numerical results.

Yang and Liu [34] inspired by the TEGM and the viscosity method with a simple step size proposed the following algorithm for solving VIP (1):

Algorithm 1.1

Step 0. Take λ0>0,x0∈H,μ∈(0,1).

Step 1. Given the current iterate xn, compute

yn=PC(xn−λnF(xn)).

If xn=yn, then stop: xn is a solution. Otherwise, go to Step 2.

Step 2. Compute

xn+1=αnf(xn)+(1−αn)zn

and

λn+1=minμ∥xn−yn∥∥F(xn)−F(yn)∥, λn,ifF(xn)−F(yn)≠0,λn,otherwise,

where zn=yn+λn(F(xn)−F(yn)). Set n≔n+1 and return to Step 1,

where F:H→H is monotone and Lipschitz continuous with constant L>0, f:H→H is a strict contraction mapping with constant ρ∈[0,1), and {αn}⊂(0,1). They proved the strong convergence of the algorithm without any prior knowledge of the Lipschitz constant of the mapping.

Very recently, Thong et al. [35] introduced a new algorithm which was a combination of the modified TEGM and the viscosity method with inertial technique. The proposed algorithm is presented as follows.

Algorithm 1.2

Initialization: Let x0,x1∈H be arbitrary.

Iterative steps: Calculate xn+1 as follows:

Step 1. Set wn=xn+αn(xn−xn−1) and compute

yn=PC(wn−λAwn).

If yn=wn, then stop and yn is a solution of the VIP. Otherwise, go to Step 2.

Step 2. Compute

xn+1=βnf(xn)+(1−βn)zn,

where zn=yn−λ(Ayn−Awn). Set n≔n+1 and go to Step 1,

where A:H→H is monotone and Lipschitz continuous with constant L>0, f:H→H is a contraction mapping with contraction parameter, λ∈(0,1L),{αn}⊂[0,α) for some α>0 and {βn}⊂(0,1) satisfying limn→∞βn=0,∑n=1∞βn=∞. Under certain mild assumptions, they proved that the proposed algorithm converges strongly to a solution of the VIP (1).

In this work, we propose iterative schemes to remedy the drawbacks highlighted above. Motivated by the works of Yang et al. [34] and Thong et al. [35] and the current research interest in this direction, we propose two new inertial-type algorithms for solving the VIP (1) based on the TEGM and Moudafi’s viscosity scheme which does not require a prior knowledge of the Lipschitz constant of the monotone operator. The inertial term αn(xn−xn−1) introduced can be regarded as a procedure for speeding up the convergence properties (see, for example, [22,23,33,36,37,38,39]). The first algorithm requires the computation of only one projection onto the feasible set per iteration while the second algorithm needs the computation of only one projection onto a half-space, which is easy to compute. Under some mild conditions, we prove strong convergence theorems for the algorithms without any prior knowledge of the Lipschitz constant of the monotone operator. Finally, we provide some numerical experiments to show the efficiency and advantages of the proposed algorithms. The numerical illustrations show that our proposed algorithms with inertial effects converge faster than the original algorithms without inertial effects.

2 Preliminaries

Let H be a real Hilbert, for a nonempty, closed, and convex subset C of H, the metric projection PC:H→C is defined, for each x∈H, as the unique element PCx∈C such that

∥x−PCx∥=inf{∥x−z∥:z∈C}.

It is known that PC is nonexpansive. We denote the weak and strong convergence of a sequence {xn} to a point x∈H by xn⇀x and xn→x, respectively.

Definition 2.1

A function f:H→ℝ is said to be weakly lower semicontinuous (w-lsc) at x∈H, if

f(x)≤liminfn→∞f(xn)

holds for an arbitrary sequence {xn}n=0∞ in H satisfying xn⇀x.

Lemma 2.2

[40,41] Letδ∈(0,1), forx,y∈H, we have the following statements:

|〈x,y〉|≤∥x∥∥y∥;
∥x+y∥2≤∥x∥2+2〈y,x+y〉;
∥x+y∥2=∥x∥2+2〈x,y〉+∥y∥2;
∥δx+(1−δ)y∥2=δ∥x∥2+(1−δ)∥y∥2−δ(1−δ)∥x−y∥2.

Lemma 2.3

[42] Let C be a nonempty closed convex subset of a real Hilbert spaceH. For anyx∈Handz∈C, we have

z=PCx⇔〈x−z,z−y〉≥0forally∈C.

Lemma 2.4

[42] Let C be a closed convex subset in a real Hilbert spaceH, andx∈H. Then,

∥PCx−PCy∥2≤〈PCx−PCy,x−y〉forally∈C;
∥PCx−y∥2≤∥x−y∥2−∥x−PCx∥2forally∈C.

Lemma 2.5

[43] Let{an}be a sequence of nonnegative real numbers,{αn}be a sequence in(0,1)with∑n=1∞αn=∞, and{bn}be a sequence of real numbers. Assume that

an+1≤(1−αn)an+αnbn,foralln≥1,

iflimsupk→∞bnk≤0for every subsequence{ank}of{an}satisfyingliminfk→∞(ank+1−ank)≥0, thenlimn→∞an=0.

Lemma 2.6

[44] IfA:C→His a continuous and monotone mapping, thenx∗is a solution of (1) if and only ifx∗is a solution of the following problem:

findx∗∈Csuchthat〈Ay, y−x∗〉≥0,∀ y∈C.

3 Main results

In this work, we consider the VIP (1) under the following assumptions:

The solution set of (1) denoted by VI(C,A) is nonempty.
The mapping A is monotone, i.e.,
(6)〈Ax−Ay, x−y〉≥0,∀ x,y∈H.
The mapping A is Lipschitz-continuous with constant L>0, i.e., there exists L>0 such that

(7)∥Ax−Ay∥≤L∥x−y∥ ∀ x,y∈H.

We take f:H→H to be a strict contraction mapping with contraction parameter k∈[0,1). Let {αn}⊂[0,α) for some α>0 and {βn}⊂(0,1) satisfy the following conditions:

(8)limn→∞βn=0,∑n=1∞βn=∞andlimn→∞αnβn∥xn−xn−1∥=0.

Now, the first algorithm is presented as follows.

Algorithm 3.1

Step 0. Take x0,x1∈H arbitrarily, λ0>0,μ∈(0,1).

Step 1. Set wn=xn+αn(xn−xn−1) and compute

yn=PC(wn−λnAwn).

If yn=wn, then stop and yn is the solution of the VIP (1). Otherwise, go to Step 2.

Step 2. Compute

xn+1=βnf(xn)+(1−βn)zn

and

λn+1=minμ∥wn−yn∥∥Awn−Ayn∥,λn,ifAwn−Ayn≠0,λn,otherwise,

where zn=yn+λn(Awn−Ayn). Set n≔n+1 and return to Step 1.

Lemma 3.2

The sequence{λn}generated by Algorithm 3.1 is monotonically decreasing with lower boundminμLλ0.

Proof

Repeating the proof as in [34] and replacing {xn} by {wn}, we obtain the desired result.□

Remark 3.3

It is clear that the limit of {λn} exists, and we denote λ=limn→∞λn. It then follows that λ>0.

Now, we prove the boundedness of the sequence {xn} generated by Algorithm 3.1.

Lemma 3.4

Let{xn}be a sequence generated by Algorithm 3.1. Then,{xn}is bounded.

Proof

Suppose p∈VI(C,A). Then, by the definition of {zn} and using Lemma 2.2, we obtain

(9)∥zn−p∥2=∥yn−λn(Ayn−Awn)−p∥2=∥yn−p∥2+λn2∥Ayn−Awn∥2−2λn〈Ayn−Awn,yn−p〉=∥yn−wn∥2+∥wn−p∥2+2〈yn−wn,wn−p〉+λn2∥Ayn−Awn∥2−2λn〈Ayn−Awn,yn−p〉=∥wn−p∥2+∥yn−wn∥2−2〈yn−wn,yn−wn〉+2〈yn−wn,yn−p〉+λn2∥Ayn−Awn∥2−2λn〈Ayn−Awn,yn−p〉=∥wn−p∥2−∥yn−wn∥2+2〈yn−wn,yn−p〉+λn2∥Ayn−Awn∥2−2λn〈Ayn−Awn,yn−p〉.

Recalling that yn=PC(wn−λnAwn), by Lemma 2.3, we obtain

〈yn−wn+λnAwn,yn−p〉≤0,

which is equivalent to

(10)〈yn−wn,yn−p〉≤−λn〈Awn,yn−p〉.

Combining (9) and (10), we have

(11)∥zn−p∥2≤∥wn−p∥2−∥yn−wn∥2−2λn〈Awn,yn−p〉+λn2∥Ayn−Awn∥2−2λn〈Ayn−Awn,yn−p〉≤∥wn−p∥2−∥yn−wn∥2+λn2∥Ayn−Awn∥2−2λn〈yn−p,Ayn〉=∥wn−p∥2−∥yn−wn∥2+λn2μ2λn+12∥yn−wn∥2−2λn〈yn−p,Ayn〉=∥wn−p∥2−∥yn−wn∥2+λn2μ2λn+12∥yn−wn∥2−2λn〈yn−p,Ayn−Ap〉−2λn〈yn−p,Ap〉≤∥wn−p∥2−1−λn2μ2λn+12||yn−wn||2.

Now, consider the limit

(12)limn→∞1−λn2μ2λn+12=1−μ2>0.

Hence, there exists N≥0 such that ∀ n≥N, we have that 1−λn2μ2λn+12>0. Thus, it follows that ∀ n≥N, we have

(13)∥zn−p∥≤∥wn−p∥.

From the definition of {wn}, we obtain

(14)∥wn−p∥=∥xn+αn(xn−xn−1)−p∥≤∥xn−p∥+αn∥xn−xn−1∥=∥xn−p∥+βnαnβn∥xn−xn−1∥.

From the condition αnβn∥xn−xn−1∥→0, it follows that there exists a constant M1>0 such that

(15)αnβn∥xn−xn−1∥≤M1,∀ n≥1.

Hence, combining (13), (14) and (15), we have

(16)∥zn−p∥≤∥wn−p∥≤∥xn−p∥+βnM1.

From the definition of {xn}, we have

(17)∥xn+1−p∥=∥βnf(xn)+(1−βn)zn−p∥=∥βn(f(xn)−p)+(1−βn)(zn−p)∥≤βn∥f(xn)−p∥+(1−βn)∥zn−p∥≤βn∥f(xn)−f(p)∥+βn∥f(p)−p∥+(1−βn)∥zn−p∥≤βnk∥xn−p∥+βn∥f(p)−p∥+(1−βn)∥zn−p∥.

Substituting (16) into (17), we obtain

∥xn+1−p∥≤(1−(1−k)βn)∥xn−p∥+βnM1+βn∥f(p)−p∥=(1−(1−k)βn)∥xn−p∥+(1−k)βnM1+∥f(p)−p∥1−k≤max∥xn−p∥,M1+∥f(p)−p∥1−k⋮≤max∥xN−p∥,M1+∥f(p)−p∥1−k.

This implies that the sequence {xn} is bounded. It also follows that {zn},{f(xn)},{wn}, and {yn} are bounded.□

Lemma 3.5

Assume that{wn}and{yn}are sequences generated by Algorithm (3.1) such thatlimn→∞∥wn−yn∥=0. If{wnk}converges weakly to somez0∈H, thenz0∈VI(C,A).

Proof

By the hypotheses of the lemma, we have that ynk⇀z0 and z0∈C. Since A is monotone, then by the definition of ynk and by applying Lemma 2.3, we obtain

〈ynk−wnk+λnkAwnk, z−ynk〉≥0,∀ z∈C.

This implies that

0≤〈ynk−wnk, z−ynk〉+λnk〈Awnk, z−ynk〉=〈ynk−wnk, z−ynk〉+λnk〈Awnk, z−wnk〉+λnk〈Awnk, wnk−ynk〉≤〈ynk−wnk, z−ynk〉+λnk〈Az, z−wnk〉+λnk〈Awnk, wnk−ynk〉.

Letting k→∞, applying the facts that limk→∞∥ynk−wnk∥=0,{ynk} is bounded and limk→∞λnk=λ>0, we have

〈Az, z−z0〉≥0,∀ z∈C.

Applying Lemma 2.6, we have that z0∈VI(C,A).□

Lemma 3.6

Let{xn}, {wn}, {yn}, {λn}, {βn}, andμbe as defined in Algorithm 3.1 andM4>0be some constant. Then, the following inequality holds:

(18)(1−βn)1−λn2μ2λn+12∥yn−wn∥2≤∥xn−x∗∥2−∥xn+1−x∗∥2+βnM4,

wherex∗∈VI(C,A).

Proof

Let x∗∈VI(C,A), then by the definition of {xn} and using Lemma 2.2, we have

(19)∥xn+1−x∗∥2=∥βnf(xn)+(1−βn)zn−x∗∥2=βn∥f(xn)−x∗∥2+(1−βn)∥zn−x∗∥2−βn(1−βn)∥f(xn)−zn∥2≤βn∥f(xn)−x∗∥2+(1−βn)∥zn−x∗∥2≤βn(∥f(xn)−f(x∗)∥+∥f(x∗)−x∗∥)2+(1−βn)∥zn−x∗∥2≤βn(k∥xn−x∗∥+∥f(x∗)−x∗∥)2+(1−βn)∥zn−x∗∥2≤βn(∥xn−x∗∥+∥f(x∗)−x∗∥)2+(1−βn)∥zn−x∗∥2=βn∥xn−x∗∥2+βn(2∥xn−x∗∥⋅∥f(x∗)−x∗∥+∥f(x∗)−x∗∥2)+(1−βn)∥zn−x∗∥2≤βn∥xn−x∗∥2+(1−βn)∥zn−x∗∥2+βnM2,

for some M2>0. Substituting (11) into (19), we get

(20)∥xn+1−x∗∥2≤βn∥xn−x∗∥2+(1−βn)∥wn−x∗∥2−(1−βn)1−λn2μ2λn+12∥yn−wn∥2+βnM2.

From (16), we obtain

(21)∥wn−x∗∥2≤(∥xn−x∗∥+βnM1)2=∥xn−x∗∥2+βn(2M1∥xn−x∗∥+βnM12)≤∥xn−x∗∥2+βnM3,

for some M3>0. Combining (20) and (21), we obtain

∥xn+1−x∗∥2≤βn∥xn−x∗∥2+(1−βn)∥xn−x∗∥2+βnM3−(1−βn)1−λn2μ2λn+12∥yn−wn∥2+βnM2=∥xn−x∗∥2+βnM3−(1−βn)1−λn2μ2λn+12∥yn−wn∥2+βnM2.

Hence, we have that

(1−βn)1−λn2μ2λn+12∥yn−wn∥2≤∥xn−x∗∥2−∥xn+1−x∗∥2+βnM4,

where M4≔M2+M3.□

Now, we prove the convergence of Algorithm 3.1.

Theorem 3.7

Assume that (A1), (A2), and (A3) hold and the sequence{αn}is chosen such that it satisfies (8). Then, the sequence{xn}generated by Algorithm 3.1 converges strongly to an elementx∗∈VI(C,A), wherex∗=PVI(C,A)∘f(x∗).

Proof

Let x∗∈VI(C,A), then using (13) and Lemma 2.2, we obtain

(22)∥xn+1−x∗∥2=∥βnf(xn)+(1−βn)zn−x∗∥2=∥βn(f(xn)−f(x∗))+(1−βn)(zn−x∗)+βn(f(x∗)−x∗)∥2≤∥βn(f(xn)−f(x∗))+(1−βn)(zn−x∗)∥2+2βn〈f(x∗)−x∗,xn+1−x∗〉=βn∥(f(xn)−f(x∗)∥2+(1−βn)∥(zn−x∗)∥2−βn(1−βn)∥(f(xn)−f(x∗))−(zn−x∗)2∥+2βn〈f(x∗)−x∗,xn+1−x∗〉≤βn∥(f(xn)−f(x∗)∥2+(1−βn)∥(zn−x∗)∥2+2βn〈f(x∗)−x∗,xn+1−x∗〉≤βnk2∥xn−x∗∥2+(1−βn)∥(zn−x∗)∥2+2βn〈f(x∗)−x∗,xn+1−x∗〉≤βnk∥xn−x∗∥2+(1−βn)∥(wn−x∗)∥2+2βn〈f(x∗)−x∗,xn+1−x∗〉.

By the definition of {wn} and using Lemma 2.2, we have

(23)∥wn−x∗∥2=∥xn+αn(xn−xn−1)−x∗∥2=∥xn−x∗∥2+2αn〈xn−x∗,xn−xn−1〉+αn2∥xn−xn−1∥2≤∥xn−x∗∥2+2αn∥xn−x∗∥⋅∥xn−xn−1∥+αn2∥xn−xn−1∥2.

Combining (22) and (23), we obtain

(24)∥xn+1−x∗∥2≤(1−(1−k)βn)∥xn−x∗∥2+2αn∥xn−x∗∥⋅∥xn−xn−1∥+αn2∥xn−xn−1∥2+2βn〈f(x∗)−x∗,xn+1−x∗〉=(1−(1−k)βn)∥xn−x∗∥2+(1−k)βn⋅21−k〈f(x∗)−x∗,xn+1−x∗〉+αn∥xn−xn−1∥(2∥xn−x∗∥+αn∥xn−xn−1∥)≤(1−(1−k)βn)∥xn−x∗∥2+(1−k)βn⋅21−k〈f(x∗)−x∗,xn+1−x∗〉+3Mαn∥xn−xn−1∥=(1−(1−k)βn)∥xn−x∗∥2+(1−k)βn21−k〈f(x∗)−x∗,xn+1−x∗〉+3M1−k⋅αnβn∥xn−xn−1∥,

where M≔supn∈N{∥xn−x∗∥,αn∥xn−xn−1∥}>0.

Next, we claim that the sequence {∥xn−x∗∥} converges to zero. In order to establish this, by Lemma 2.5, it suffices to show that limsupk→∞〈f(x∗)−x∗,xnk+1−x∗〉≤0 for every subsequence {∥xnk−x∗∥} of {∥xn−x∗∥} satisfying

liminfk→∞(∥xnk+1−x∗∥−∥xnk−x∗∥)≥0.

Now, suppose that {∥xnk−x∗∥} is a subsequence of {∥xn−x∗∥} such that

liminfk→∞(∥xnk+1−x∗∥−∥xnk−x∗∥)≥0.

Then, it follows that

liminfk→∞(∥xnk+1−x∗∥2−∥xnk−x∗∥2)=liminfk→∞{(∥xnk+1−x∗∥−∥xnk−x∗∥)(∥xnk+1−x∗∥+∥xnk−x∗∥)}≥0.

Then from Lemma 3.6, and using the facts that limk→∞1−λnk2μ2λnk+12=1−μ2>0 and limk→∞βnk=0, we obtain

(25)limk→∞∥ynk−wnk∥=0.

From (25), we get

(26)∥znk−wnk∥=∥ynk+λnk(Awnk−Aynk)−wnk∥≤∥ynk−wnk∥+λnk∥Awnk−Aynk∥≤∥ynk−wnk∥+λnk×μλnk+1∥wnk−ynk∥=1+λnk×μλnk+1∥ynk−wnk∥→0.

Also, we have that

(27)∥xnk+1−znk∥=∥βnkf(xnk)+(1−βnk)znk−znk∥=βnk∥f(xnk)−znk∥→0

and

(28)∥xnk−wnk∥=∥xnk−[xnk+αnk(xnk−xnk−1)]∥=αnk∥xnk−xnk−1∥=βnk⋅αnkβnk∥xnk−xnk−1∥→0.

Applying (26), (27), and (28), we obtain

(29)∥xnk+1−xnk∥≤∥xnk+1−znk∥+∥znk−wnk∥+∥wnk−xnk∥→0.

Since {xnk} is bounded, there exists a subsequence {xnkj} that converges weakly to some z0∈H, such that

(30)limsupk→∞〈f(x∗)−x∗,xnk−x∗〉=limj→∞〈f(x∗)−x∗,xnkj−x∗〉=〈f(x∗)−x∗,z0−x∗〉.

By Lemma 3.5, and (25) and (28), we have z0∈VI(C,A). Since the solution set VI(C,A) is a closed, convex subset and f is a strict contraction, the mapping PVI(C,A)∘f is a contraction mapping. Hence, by the Banach contraction mapping principle, there exists a unique element x∗∈VI(C,A) such that x∗=PVI(C,A)∘f(x∗). By Lemma 2.3, we have

(31)〈f(x∗)−x∗, z−x∗〉≤0,∀ z∈VI(C,A).

Hence, it follows from (31) that

(32)limsupk→∞〈f(x∗)−x∗, xnk−x∗〉=〈f(x∗)−x∗, z0−x∗〉≤0.

Combining (29) and (32), we have

(33)limsupk→∞〈f(x∗)−x∗, xnk+1−x∗〉≤limsupk→∞〈f(x∗)−x∗, xnk+1−xnk〉+limsupk→∞〈f(x∗)−x∗, xnk−x∗〉=〈f(x∗)−x∗, z0−x∗〉≤0.

Thus, by (33), limn→∞αnβn∥xn−xn−1∥=0, (24) and Lemma 2.5, we have limn→∞∥xn−x∗∥=0 as required.□

We next propose our second algorithm. Suppose C is a nonempty convex set which satisfies the following conditions:

The set C is given by
C={x∈H:h(x)≤0},
where h:H→ℝ is a convex and subdifferentiable function on C.
h is weakly lower semicontinuous.
For any x∈H, at least one subgradient ξ∈∂h(x) can be calculated, where ∂h(x) is defined as follows:
∂h(x)={z∈H:h(u)≥h(x)+〈u−x,z〉, ∀u∈H}.
In addition, ∂h(x) is bounded on bounded sets.
Define the set Cn by the following half-space:

Cn={x∈H:h(wn)+〈ξn,x−wn〉≤0},

where ξn∈∂h(wn). By the definition of the subgradient, it is clear that C⊆Cn.

We now present the following algorithm using the half-space defined above.

Let f:H→H be a strict contraction mapping with contraction parameter k∈[0,1). Let {αn}⊂[0,α) for some α>0 and {βn}⊂(0,1) satisfying the following conditions:

limn→∞βn=0,∑n=1∞βn=∞andlimn→∞αnβn∥xn−xn−1∥=0.

Let {xn} be a sequence generated by the following iterative process.

Algorithm 3.8

Step 0. Take x0,x1∈H arbitrarily, λ0>0,μ∈(0,1).

Step 1. Set wn=xn+αn(xn−xn−1) and compute

yn=PCn(wn−λnAwn).

If yn=wn, then stop and yn is the solution of the VIP (1). Otherwise, go to Step 2.

Step 2. Compute

xn+1=βnf(xn)+(1−βn)zn

and

λn+1=minμ∥wn−yn∥∥Awn−Ayn∥, λn,ifAwn−Ayn≠0,λn,otherwise,

where zn=yn+λn(Awn−Ayn). Set n≔n+1 and return to Step 1.

It is easy to extend Lemmas 3.2, 3.4, and 3.6 for Algorithms 3.1–3.8.

Lemma 3.9

The sequence{λn}generated by Algorithm 3.8 is monotonically decreasing with lower boundminμLλ0.

Lemma 3.10

Let{xn}be a sequence generated by Algorithm 3.8. Then, the sequence{xn}is bounded.

Lemma 3.11

Let{xn}, {wn}, {yn}, {λn}, {βn}, andμbe as defined in Algorithm 3.8 andM5>0be some constant. Then, the following inequality holds:

(34)(1−βn)1−λn2μ2λn+12∥yn−wn∥2≤∥xn−x∗∥2−∥xn+1−x∗∥2+βnM5,

wherex∗∈VI(C,A).

Lemma 3.12

Assume that{wn}and{yn}are sequences generated by Algorithm (3.8) such thatlimn→∞∥wn−yn∥=0. If{wnj}converges weakly to somexˆ∈Hasj→∞, thenxˆ∈VI(C,A).

Proof

Since wnj⇀xˆ, it follows that ynj⇀xˆ as j→∞. Since ynj∈Cnj, by the definition of Cn, we get

h(wnj)+〈ξnj,ynj−wnj〉≤0.

Since {xn} is bounded by Lemma 3.10, then {wn} and {yn} are also bounded, and by condition (B3) there exists a constant M>0 such that ∥ξnj∥≤Mfor allj≥0. So h(wnj)≤M∥wnj−ynj∥→0asj→∞, and this in turn implies that liminfj→∞h(wnj)≤0. Using condition (B2), we have h(xˆ)≤liminfj→∞h(wnj)≤0. This means that xˆ∈C. From Lemma 2.3, we obtain

〈ynj−wnj+λnjAwnj, z−ynj〉≥0,∀ z∈C⊆Cnj.

Since A is monotone, we have

0≤〈ynj−wnj, z−ynj〉+λnj〈Awnj, z−ynj〉=〈ynj−wnj, z−ynj〉+λnj〈Awnj, z−wnj〉+λnj〈Awnj, wnj−ynj〉≤〈ynj−wnj, z−ynj〉+λnj〈Az, z−wnj〉+λnj〈Awnj, wnj−ynj〉.

Letting j→∞, and since limj→∞∥ynj−wnj∥=0, we have

〈Az, z−xˆ〉≥0,∀ z∈C.

Applying Lemma 2.6, we have that xˆ∈VI(C,A).□

Now, we prove the convergence theorem for Algorithm 3.8.

Theorem 3.13

Assume that (A1), (A2), (A3), (B1), and (B2) hold. Let the sequence{αn}be chosen such that it satisfies (8). Then, the sequence{xn}generated by Algorithm 3.8 converges strongly to an elementxˆ∈VI(C,A), wherexˆ=PVI(C,A)∘f(xˆ).

Proof

From (24), we have

(35)∥xn+1−x∗∥2≤(1−(1−k)βn)∥xn−x∗∥2+(1−k)βn21−k〈f(x∗)−x∗,xn+1−x∗〉+3M1−k⋅αnβn∥xn−xn−1∥,

where M≔supn∈N{∥xn−x∗∥,αn∥xn−xn−1∥}>0.

We claim that the sequence {∥xn−x∗∥} converges to zero. In order to establish this, by Lemma 2.5, it suffices to show that limsupk→∞〈f(x∗)−x∗,xnk+1−x∗〉≤0 for every subsequence {∥xnk−x∗∥} of {∥xn−x∗∥} satisfying

liminfk→∞(∥xnk+1−x∗∥−∥xnk−x∗∥)≥0.

Suppose that {∥xnk−x∗∥} is a subsequence of {∥xn−x∗∥} such that

liminfk→∞(∥xnk+1−x∗∥−∥xnk−x∗∥)≥0.

By applying Lemma 3.11 and following similar argument as in Theorem 3.7 we have

(36)limk→∞∥ynk−wnk∥=0,limk→∞∥znk−wnk∥=0,limk→∞∥xnk−wnk∥=0,limk→∞∥xnk+1−xnk∥=0.

Since {xnk} is bounded, there exists a subsequence {xnkj} that converges weakly to some z0∈H, such that

(37)limsupk→∞〈f(x∗)−x∗,xnk−x∗〉=limj→∞〈f(x∗)−x∗,xnkj−x∗〉=〈f(x∗)−x∗,z0−x∗〉.

By Lemma 3.12 and (36), we have z0∈VI(C,A). Since the solution set VI(C,A) is a closed, convex subset and f is a strict contraction, the mapping PVI(C,A)∘f is a contraction mapping. By the Banach contraction mapping principle, there exists a unique element x∗∈VI(C,A) such that x∗=PVI(C,A)∘f(x∗). Applying Lemma 2.3, we have

(38)〈f(x∗)−x∗, z−x∗〉≤0,∀ z∈VI(C,A).

Therefore, we have that

(39)limsupk→∞〈f(x∗)−x∗, xnk−x∗〉=〈f(x∗)−x∗, z0−x∗〉≤0.

From (36) and (39), we have

(40)limsupk→∞〈f(x∗)−x∗, xnk+1−x∗〉≤limsupk→∞〈f(x∗)−x∗, xnk+1−xnk〉+limsupk→∞〈f(x∗)−x∗, xnk−x∗〉=〈f(x∗)−x∗, z0−x∗〉≤0.

Hence, by (40), limn→∞αnβn∥xn−xn−1∥=0, (35), and applying Lemma 2.5, we have limn→∞∥xn−x∗∥=0, which is the required result.□

4 Numerical experiments

In this section, we present some numerical examples to demonstrate the efficiency of our algorithms in comparison with Algorithms 1.1 and 1.2 in the literature. All numerical computations were carried out using Matlab 2016(b) on an HP personal computer, 8-Gb RAM.

Table 1

Comparison between Algorithms 3.1, 1.1, and 1.2 for Problem 1

Dimension	Algorithm 3.1		Algorithm 1.1		Algorithm 1.2
Dimension	Iter.	CPU time (s)	Iter.	CPU time (s)	Iter.	CPU time (s)
m=4	23	6.0523	59	16.5546	29	7.6139
m=20	22	7.4086	58	18.7391	29	8.7219
m=100	24	15.9526	63	35.6063	31	18.4151

Figure 1

Problem 1, left: m=4; middle: m=20; and right: m=100.

Table 2

Comparison between Algorithms 3.1, 1.1, and 1.2 for Problem 2

	Algorithm 3.1		Algorithm 1.1		Algorithm 1.2
	Iter.	CPU time (s)	Iter.	CPU time (s)	Iter.	CPU time (s)
Case I	7	1.7756	6	1.9701	11	5.6289
Case II	7	5.0223	7	5.2426	11	7.0342
Case III	8	3.1035	8	3.1066	11	8.0202

Figure 2

Problem 1, left: Case I; middle: Case II; and right: Case III.

We choose λ0=0.9,βn=1n+1,f(x)=x5, and μ=0.6 and use ∥xn+1−xn∥∥x2−x1∥<ε as a stopping criterion to terminate the algorithm in each example. The projection onto the feasible set C is computed using the function “fmincon” in the optimization tool box. We take θ=0.6 and choose the sequence {αn} such that

αn=minθ,βn2∥xn−xn−1∥ifxn≠xn−1,θotherwise.

Problem 1

The first test (also considered in [34]) is a classical example for which the usual gradient method does not converge. The feasible set is C=Rm and A(x)=Mx, where M is a square m×m matrix given by

ai,j=−1,ifj=m+1−iandj>i,1ifj=m+1−iandj<1,0,otherwise.

For even m, the zero vector x∗=(0,…,0) is the solution of Problem 1. In this example, we take (Case I:) m=4, (Case II:) m=20, and (Case III:) m=100 and ε=10−4. The initial points are generated randomly using x0=rand(m,1) and x1=10×rand(m,1). The numerical results are summarized in Table 1 and Figure 1.

Problem 2

Suppose that H=L2([0,1]) with the inner product

〈x,y〉≔∫01x(t)y(t)dt,∀ x,y∈H

and the induced norm

∥x∥≔∫01|x(t)|2dt,∀ x∈H.

Let C≔{x∈H:∥x∥≤1} be the unit ball and define the operator A:C→H by

(Ax)(t)=max{0,x(t)}.

It can be easily verified that A is 1-Lipschitz continuous and monotone on C. With these given C and A, the solution set of the VIP (1) is given by

VI(C,A)={0}≠∅.

It is known that

PC(x)=x∥x∥L2if∥x∥L2>1,xif∥x∥L2≤1.

We test the algorithms for three different starting points and use ε=10−3 as a stopping criterion. The numerical results are summarized in Table 2 and Figure 2.

Case I: x0=t35, x1=t3+5t2−1;

Case II: x0=exp(−5t), x1=(2t2−1)5;

Case III: x0=5sin(2πt), x1=cos(2t)4.

Problem 3

Next, we consider the Kojima-Shindo nonlinear complementarity problem, where n=4 and the mapping A is defined by

A(x1,x2,x3,x4)=3x12+2x1x2+2x22+x3+3x4−62x12+x1+x22+10x3+2x4−23x12+x1x2+2x22+2x3+9x4−9x12+3x22+2x3+3x4−3.

It is known that A is Lipschitz continuous [45]. The feasible set is C={x∈R+4 | x1+x2+x3+x4=4}. We choose the starting points: Case I: x0=(1,2,0,1)′;x1=(1,1,1,1)′ and Case II: x0=(2,0,0,2)′;x1=(1,0,1,2)′. For all the starting points, we have two tests: with ε=10−3 and ε=10−6. The results are summarized in Tables 3–4 and Figures 3–4.

5 Conclusion

In this article, we introduce two modifications of the inertial TEGM with self-adaptive step size for solving monotone VIPs. The algorithms were constructed in such a way that only one projection onto the feasible set C was made in each iteration. The results obtained improve many known results in this direction in the literature.

Table 3

Comparison between Algorithms 3.1, 1.1, and 1.2 for Problem 3, ε=10−3

	Algorithm 3.1		Algorithm 1.1		Algorithm 1.2
	Iter.	CPU time (s)	Iter.	CPU time (s)	Iter.	CPU time (s)
Case I	15	4.4316	35	9.5863	18	4.8669
Case II	18	5.5598	36	10.1659	23	6.3572

Table 4

Comparison between Algorithms 3.1, 1.1, and 1.2 for Problem 3, ε=10−6

	Algorithm 3.1		Algorithm 1.1		Algorithm 1.2
	Iter.	CPU time (s)	Iter.	CPU time (s)	Iter.	CPU time (s)
Case I	31	9.5930	75	22.0722	41	10.6315
Case II	31	9.5526	75	21.7023	41	11.5232

Figure 3

Problem 1, left: Case I; right: Case II.

Figure 4

Problem 1, left: Case I; right: Case II.

Acknowledgments

The authors sincerely thank the anonymous reviewers for their careful reading, constructive comments, and fruitful suggestions that substantially improved the manuscript. Oluwatosin Temitope Mewomo is supported in part by the National Research Foundation (NRF) of South Africa Incentive Funding for Rated Researchers (Grant Number 119903). Opinions expressed and conclusions arrived are those of the authors and are not necessarily to be attributed to the NRF.

Conflict of interest: The authors declare that they have no competing interests.

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Received: 2019-07-12

Revised: 2020-05-07

Accepted: 2020-05-30

Published Online: 2020-09-15

This work is licensed under the Creative Commons Attribution 4.0 International License.

Two modifications of the inertial Tseng extragradient method with self-adaptive step size for solving monotone variational inequality problems

Abstract

1 Introduction

Algorithm 1.1

Algorithm 1.2

2 Preliminaries

Definition 2.1

Lemma 2.2

Lemma 2.3

Lemma 2.4

Lemma 2.5

Lemma 2.6

3 Main results

Algorithm 3.1

Lemma 3.2

Proof

Remark 3.3

Lemma 3.4

Proof

Lemma 3.5

Proof

Lemma 3.6

Proof

Theorem 3.7

Proof

Algorithm 3.8

Lemma 3.9

Lemma 3.10

Lemma 3.11

Lemma 3.12

Proof

Theorem 3.13

Proof

4 Numerical experiments

Problem 1

Problem 2

Problem 3

5 Conclusion

Acknowledgments

References

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