# Convergence Analysis of New Construction Explicit Methods for Solving Equilibrium Programming and Fixed Point Problems

Authors: Chainarong Khunpanuk; Nuttapol Pakkaranang; Bancha Panyanak
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this paper, we present improved iterative methods for evaluating the numerical solution of an equilibrium problem in a Hilbert space with a pseudomonotone and a Lipschitz-type bifunction. The method is built around two computing phases of a proximal-like mapping with inertial terms. Many such simpler step size rules that do not involve line search are examined, allowing the technique to be enforced more effectively without knowledge of the Lipschitz-type constant of the cost bifunction. When the control parameter conditions are properly defined, the iterative sequences converge weakly on a particular solution to the problem. We provide weak convergence theorems without knowing the Lipschitz-type bifunction constants. A few numerical tests were performed, and the results demonstrated the appropriateness and rapid convergence of the new methods over traditional ones.


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## Body

## 1. Introduction

LetΠ stand for a certain Hilbert space and Ξ stand for a nonempty closed convex subset of Π. The research is about an iterative technique for solving the equilibrium problem ((1), to make it short). Let Γ:Π×Π⟶ℝ be a bifunction with Γy1,y1=0, for each y1∈Ξ. An equilibrium problem for granted bifunction Γ on Ξ is interpreted this way: find ℏ∗∈Ξ such that
(1)Γℏ∗,y1≥0,∀y1∈Ξ.The numerical evaluation of the equilibrium problem under the following conditions is the focus of this study. We will assume that the following conditions have been satisfied:ForΓ1, the solution set of a problem (1) is denoted by solΓ,Ξ and it is nonempty.ForΓ2, a bifunction Γ is said to be pseudomonotone [1, 2], i.e.,
(2)Γy1,y2≥0⇒Γy2,y1≤0,∀y1,y2∈Ξ.ForΓ3, a bifunction Γ is said to be Lipschitz-type continuous [3] on Ξ if there exist two constants c1,c2>0, such that
(3)Γy1,y3≤Γy1,y2+Γy2,y3+c1y1−y22+c2y2−y32,∀y1,y2,y3∈Ξ.ForΓ4, for any sequence yk⊂Ξ satisfying yk⇀y∗, then, the following inequality holds:
(4)limsupk⟶+∞Γyk,y1≤Γy∗,y1,∀y1∈Ξ.For (Γ5), Γy1,· is convex and subdifferentiable on Π for each fixed y1∈Π.Let us represent a problem’s solution set assolΓ,Ξ, and we will assume in the following text that this solution set is not empty. Researchers are interested in the equilibrium problem because it connects many mathematical problems, including fixed point problems, vector and scalar minimization problems, variational inequalities, complementarity problems, saddle point problems, Nash equilibrium problems in noncooperative games, and inverse optimization problems (see for further information [2, 4–9]). It also has a variety of applications in economics [10], the dynamics of offer and demand [11], and it continues to use the theoretical framework of noncooperative games and Nash’s equilibrium models [12, 13]. The phrase “equilibrium problem” was first used in the literature in 1992 by Muu and Oettli [9] and was further investigated by Blum [2]. More precisely, we consider two applications for the problem (1). (i) A variational inequality problem for an operator I1:Ξ⟶Π is stated as follows: find ℏ∗∈Ξ such that
(5)I1ℏ∗y1−ℏ∗≥0,∀y1,y2∈Ξ.Let us define a bifunctionΓ as follows:
(6)Γy1,y2≔I1y1,y2−y1,∀y1,y2∈Ξ.Then, the equilibrium problem converts into the problem of variational inequalities defined in (5) and Lipschitz constants of the mapping I1 are L=2c1=2c2. (ii) Letting a mapping I2:Ξ⟶Ξ is said to κ-strict pseudocontraction [14] if there exists a constant κ∈0,1 such that
(7)I2y1−I2y22≤y1−y22+κy1−I2y1−y2−I2y22,∀y1,y2∈Ξ.A fixed point problem (FPP) forI2:Ξ⟶Ξ is to find ℏ∗∈Ξ such that I2ℏ∗=ℏ∗. Let us define a bifunction Γ as follows:
(8)Γy1,y2=y1−I2y1,y2−y1,∀y1,y2∈Ξ.It can be easily seen in [15] that expression (8) satisfies the conditions (Γ1)–(Γ5) as well as the values of Lipschitz constants are c1=c2=3−2κ/2−2κ.The extragradient method developed by Tran et al. [16] is one useful approach. Take an arbitrary starting point x0∈Π; and the next iteration as follows:
(9)x0∈Ξ,yk=argminy∈ΞℶΓxk,y+12xk−y2,xk+1=argminy∈ΞℶΓyk,y+12xk−y2,where 0<ℶ<min1/2c1,1/2c2 and c1,c2 are two Lipschitz-type constants.The main goal is to create an inertial-type technique in the case of [16] that will be designed to increase the convergence rate of the iterative sequence. Such techniques have already been established as a result of the oscillator equation with damping and conservative force restoration. This second-order dynamical system is known as a “heavy friction ball,” and it was first proposed by Polyak in [17]. The important feature of this method is that the next iteration is built on the previous two iterations. Numerical results show that inertial terms improve the performance of the approaches in terms of the number of iterations and elapsed time in this context. Inertial-type approaches have been extensively studied in recent years for certain classes of equilibrium problems [18–26] and others in [27–33].As a result, the following natural question arises: Is it possible to develop new inertial-type weakly convergent extragradient-type methods with monotone and nonmonotone step size rules to solve equilibrium problems?In our study, we provide a positive answer to this question, namely, that the gradient approach still generates a weak convergence sequence when solving equilibrium problems involving pseudomonotone bifunctions using a novel monotone and nonmonotone variable step size rule. Motivated by the work of Censor et al. [34] and Tran et al. [16], we will describe new inertial extragradient-type approaches to solving problem (1) in the context of an infinite-dimensional real Hilbert space. Our primary contributions to this work are as follows:
(i)
We build an inertial subgradient extragradient technique with a novel monotone variable step size rule to solve equilibrium problems in a real Hilbert space and show that the resulting sequence is weakly convergent(ii)
To solve equilibrium problems, we devise another inertial subgradient extragradient technique that leverages a novel variable nonmonotone step size rule that is independent of the Lipschitz constants(iii)
Some results are investigated in order to address different kinds of equilibrium problems in a real Hilbert space(iv)
We offer numerical demonstrations of the suggested methodologies for the verification of theoretical conclusions and compare them to earlier results [22, 35, 36]. Our numerical results indicate that the new approaches are useful and outperform the current onesThe paper is structured as follows: in Section2, preliminary results were presented. Section 3 gives all new approaches and their convergence analysis. Finally, Section 5 gives some numerical results to explain the practical efficiency of the proposed methods.


## 2. Preliminaries

In this part, we will go over several fundamental identities as well as crucial lemmas and definitions. Ametric projectionPΞy1 of y1∈Π is defined by
(10)PΞy1=argminy1−y2:y2∈Ξ.The following sections outline the key characteristics of projection mapping.Lemma 1 (see [37]).
LetPΞ:Π⟶Ξ be a metric projection. Then, there are the following features:
(11)y1−PΞy22+PΞy2−y22≤y1−y22,y1∈Ξ,y2∈Π,y3=PΞy1,if and only if
(12)y1−y3,y2−y3≤0,∀y2∈Ξ,y1−PΞy1≤y1−y2,y2∈Ξ,y1∈Π.Lemma 2 (see [37]).
For anyy1,y2∈Π and ℓ∈ℝ. Then, the following conditions were met:
(13)ℓy1+1−ℓy22=ℓy12+1−ℓy22−ℓ1−ℓy1−y22,y1+y22≤y12+2y2,y1+y2.
Anormal cone of Ξ at y1∈Ξ is defined by
(14)NΞy1=y3∈Π:y3,y2−y1≤0,∀y2∈Ξ.
Assume that℧:Ξ⟶ℝ is a convex function and subdifferential of ℧ at y1∈Ξ is defined by
(15)∂℧y1=y3∈Π:℧y2−℧y1≥y3,y2−y1,∀y2∈Ξ.Lemma 3 (see [38]).
Let℧:Ξ⟶ℝ be a subdifferentiable, convex, and lower semicontinuous function on Ξ. An element x∈Ξ is a minimizer of a function ℧ if and only if
(16)0∈∂℧x+NΞx,where ∂℧x stands for the subdifferential of ℧ at x∈Ξ and NΞx the normal cone of Ξ at x.Lemma 4 (see [39]).
LetΞ be a nonempty subset of Π and xk be a sequence in Π satisfying two conditions:
(i)
For eachx∈Ξ,limk⟶+∞xk−x exists(ii)
Each sequentially weak cluster point ofxk belongs to Ξ
Then, sequencexk weakly converges to an element in Ξ.Lemma 5 (see [40]).
Suppose thatak and tk are two sequences of nonnegative real numbers satisfying the inequality
(17)ak+1≤ak+tk,forallk∈ℕ.
If∑‍tk<+∞, then, limk⟶+∞ak exists.


## 3. Main Results

In this section, we present a numerical iterative method for accelerating the rate of convergence of an iterative sequence by combining two strong convex optimization problems with an inertial term. We propose the techniques listed below for solving equilibrium problems.Algorithm 1
STEP 0: Choose ℶ0>0,x−1,x0∈Π,ζ∈0,1,ϰ∈0,1, ϕ∈0,2−2 with a sequence ψk⊂0,+∞ such that∑k=0+∞‍ψk<+∞.Moreover, chooseζk such that 0≤ζk≤βk such thatβk=minζ,ψk/xk−xk−1ifxk≠xk−1,ζotherwise.STEP 1: Computeyk=argminy∈ΞℶkΓνk,y+1/2νk−y2wwhere νk=xk+ζkxk−xk−1.STEP 2: Given the current iterates xk−1,xk,yk. Firstly choose ωk∈∂2Γνk,yk satisfying νk−ℶkωk−yk∈NΞyk and generate a half-spaceΠk=z∈Π:νk−ℶkωk−yk,z−yk≤0.Computexk+1=argminy∈ΠkℶkΓyk,y+1/2νk−y2.STEP 3: Computeℶk+1=minℶk,2−2−ϕϰνk−yk2+2−2−ϕϰxk+1−yk2/2Γνk,xk+1−Γνk,yk−Γyk,xk+1ifΓνk,xk+1−Γνk,yk−Γyk,xk+1>0,ℶk,otherwise.STEP 4: If yk=νk, then complete the computation. Otherwise, set k≔k+1 and go back STEP 1.Remark 6.
(i) Ifζ=0 is used in the abovementioned method, then, it is equivalent to the default extragradient method [16] with the updated step size rule. (ii) From the expressions in Algorithm 1, we have
(18)∑k=1+∞ζkxk−xk−1≤∑k=1+∞βkxk−xk−1<+∞.
It further implies that(19)limk⟶+∞βkxk−xk−1=0.Lemma 7.
A sequenceℶk is converged to ℶ and
(20)minϰ2−2−ϕmax2c1,2c2,ℶ0≤ℶ≤ℶ0.Proof.
Assume thatΓνk,xk+1−Γνk,yk−Γyk,xk+1>0, such that
(21)ϰ2−2−ϕνk−yk2+xk+1−yk22Γνk,xk+1−Γνk,yk−Γyk,xk+1≥ϰ2−2−ϕνk−yk2+xk+1−yk22c1νk−yk2+c2xk+1−yk2≥ϰ2−2−ϕ2maxc1,c2.
Thus, we obtainlimk⟶+∞ℶk=ℶ This completes the proof of the lemma.Algorithm 2
STEP 0: Choose ℶ0>0,x−1,x0∈Π,ζ∈0,1,ϰ∈0,1, ϕ∈0,2−2 with a sequence ψk⊂0,+∞ such that∑k=0+∞ψk<+∞.Moreover, choose a non-negative real sequencepk such that ∑k=1+∞pk<+∞ and ζk such that 0≤ζk≤βk such thatβk=minζ,ψk/xk−xk−1ifxk≠xk−1,ζotherwise.STEP 1: Computeyk=argminy∈ΞℶkΓνk,y+1/2νk−y2where νk=xk+ζkxk−xk−1.STEP 2: Given the current iterates xk−1,xk,yk. Firstly choose ωk∈∂2Γνk,yk satisfying νk−ℶkωk−yk∈NΞyk and generate a half-spaceΠk=z∈Π:νk−ℶkωk−yk,z−yk≤0.Computexk+1=argminy∈ΠkℶkΓyk,y+1/2νk−y2.STEP 3: Computeℶk+1=minℶk+pk,2−2−ϕϰνk−yk2+2−2−ϕϰxk+1−yk2/2Γνk,xk+1−Γνk,yk−Γyk,xk+1ifΓνk,xk+1−Γνk,yk−Γyk,xk+1>0,ℶk+pk,otherwise.STEP 4: If yk=νk, then complete the computation. Otherwise, set k≔k+1 and go back STEP 1.Lemma 8.
A sequenceℶk is converged to ℶ and
(22)minϰ2−2−ϕmax2c1,2c2,ℶ0≤ℶ≤ℶ0,where P=∑k=1+∞pk.Proof.
Assume thatΓνk,xk+1−Γνk,yk−Γyk,xk+1>0 such that
(23)ϰ2−2−ϕνk−yk2+xk+1−yk22Γνk,xk+1−Γνk,yk−Γyk,xk+1≥ϰ2−2−ϕνk−yk2+xk+1−yk22c1νk−yk2+c2xk+1−yk2≥ϰ2−2−ϕ2maxc1,c2.
Applying mathematical induction on the concept ofℶk+1, we have
(24)minϰ2−2−ϕmax2c1,2c2,≤ℶk≤ℶ0+P.
Suppose thatℶk+1−ℶk+=max0,ℶk+1−ℶk and ℶk+1−ℶk−=max0,−ℶk+1−ℶk. Due to the definition of ℶk, we get
(25)∑k=1+∞ℶk+1−ℶk+=∑k=1+∞max0,ℶk+1−ℶk≤P<+∞.
That is, the series∑k=1+∞ℶk+1−ℶk+ is convergent. The convergence must now be proven of ∑k=1+∞ℶk+1−ℶk−. Let ∑k=1+∞ℶk+1−ℶk−=+∞. Due to the fact that ℶk+1−ℶk=ℶk+1−ℶk+−ℶk+1−ℶk−, we could get
(26)ℶk+1−ℶ0=∑k=0kℶk+1−ℶk=∑k=0kℶk+1−ℶk+−∑k=0kℶk+1−ℶk−.
Lettingk⟶+∞ in (26), we have ℶk⟶−∞ as k⟶+∞. This is an absurdity. As a result of the series convergence ∑k=0kℶk+1−ℶk+ and ∑k=0kℶk+1−ℶk− taking k⟶+∞ in expression (26), we obtain limk⟶+∞ℶk=ℶ. This brings the proof to a conclusion.Lemma 9.
The following useful inequality is derived in Algorithm3.
(27)ℶkΓyk,y−ℶkΓyk,xk+1≥νk−xk+1,y−xk+1,∀y∈Πk.Proof.
By use of Lemma3, we have
(28)0∈∂2ℶkΓyk,·+12νk−·2xk+1+NΠkxk+1.
Thus, forυ∈∂Γyk,xk+1, there exists a vector υ¯∈NΠkxk+1 such that
(29)ℶkυ+xk+1−νk+υ¯=0.
Thus, we have(30)νk−xk+1,y−xk+1=ℶkυ,y−xk+1+υ¯,y−xk+1,∀y∈Πk.
Sinceυ¯∈NΠkxk+1 implies that υ¯,y−xk+1≤0 for all y∈Πk, thus, we have
(31)νk−xk+1,y−xk+1≤ℶkυ,y−xk+1,∀y∈Πk.
Sinceυ∈∂Γyk,xk+1, we have
(32)Γyk,y−Γyk,xk+1≥υ,y−xk+1,∀y∈Π.
Combining expressions (31) and (32), we have
(33)ℶkΓyk,y−ℶkΓyk,xk+1≥νk−xk+1,y−xk+1,∀y∈Πk.Lemma 10.
In Algorithm3, we also have the following useful inequality:
(34)ℶkΓνk,y−ℶkΓνk,yk≥νk−yk,y−yk,∀y∈Ξ.Proof.
The proof is analogous to the proof of Lemma9. Next, substituting y=xk+1, we have
(35)ℶkΓνk,xk+1−Γνk,yk≥νk−yk,xk+1−yk.Theorem 11.
Letxk be a sequence generated by Algorithm 3, and the conditions (Γ1)–(Γ5) are satisfied. Then, the sequence xk converges weakly to ℏ∗.Proof.
By substitutingy=ℏ∗ into Lemma 9, we have
(36)ℶkΓyk,ℏ∗−ℶkΓyk,xk+1≥νk−xk+1,ℏ∗−xk+1.
By the use of conditionΓ2, we obtain
(37)νk−xk+1,xk+1−ℏ∗≥ℶkΓyk,xk+1.
From the expression in Algorithm1, we obtain
(38)Γνk,xk+1−Γνk,yk−Γyk,xk+1≤2−2−ϕϰνk−yk2+xk+1−yk22ℶk+1,which after multiplying both sides by ℶk>0 implies that
(39)ℶkΓyk,xk+1≥ℶkΓνk,xk+1−ℶkΓνk,yk−2−2−ϕℶkϰνk−yk2+xk+1−yk22ℶk+1.
Combining expressions (37) and (39), we obtain
(40)νk−xk+1,xk+1−ℏ∗≥ℶkΓνk,xk+1−Γνk,yk−2−2−ϕℶkϰνk−yk2+xk+1−yk22ℶk+1.
By using expression (35), we have
(41)ℶkΓvk,xk+1−Γvk,yk≥vk−yk,xk+1−yk.
Combining expressions (40) and (41), we have
(42)νk−xk+1,xk+1−ℏ∗≥νk−yk,xk+1−yk−2−2−ϕℶkϰνk−yk2+xk+1−yk22ℶk+1.
The following facts are available to us:(43)2vk−xk+1,xk+1−ℏ∗=vk−ℏ∗2−xk+1−vk2−xk+1−ℏ∗2,2yk−νk,yk−xk+1=νk−yk2+xk+1−yk2−νk−xk+12.
Thus, we have(44)xk+1−ℏ∗2≤νk−ℏ∗2−νk−yk2−xk+1−yk2+2−2−ϕℶkϰνk−yk2+xk+1−yk2ℶk+1.
Sinceℶk⟶ℶ, thus, there exists a fixed natural number k1∈ℕ such that
(45)limk⟶+∞ϰℶkℶk+1≤1.
Thus, we have(46)xk+1−ℏ∗2≤vk−ℏ∗2−vk−yk2−xk+1−yk2+2−2−ϕvk−yk2+xk+1−yk2.
Furthermore, it implies that(47)xk+1−ℏ∗2≤vk−ℏ∗2−2−1vk−yk2−2−1xk+1−yk2−ϕvk−yk2+xk+1−yk2.
From expression (47), we obtain
(48)xk+1−ℏ∗2≤vk−ℏ∗2,∀k≥k1.
It is possible to write as an expression for everyk≥k1 such that
(49)xk+1−ℏ∗≤xk+ζkxk−xk−1−ℏ∗≤xk−ℏ∗+ζkxk−xk−1.
Combining expressions (18) and (49) and Lemma 5 implies that
(50)limk⟶+∞xk−ℏ∗=l,forsomefinitel≥0.
By using the definition ofνk, we have
(51)vk−ℏ∗2=xk+ζkxk−xk−1−ℏ∗2=1+ζkxk−ℏ∗−ζkxk−1−ℏ∗2=1+ζkxk−ℏ∗2−ζkxk−1−ℏ∗2+ζk1+ζkxk−xk−12≤1+ζkxk−ℏ∗2−ζkxk−1−ℏ∗2+2ζkxk−xk−12.
By using expressions (50) and (19) in the abovementioned formula, we may deduce that
(52)limk⟶+∞vk−ℏ∗=l.
Thus, we have(53)xk+1−ℏ∗2≤νk−ℏ∗2−νk−yk2−xk+1−yk2+2−2−ϕℶkϰνk−yk2+xk+1−yk2ℶk+1.
By using expressions (51) and (53), we obtain
(54)xk+1−ℏ∗2≤1+ζkxk−ℏ∗2−ζkxk−1−ℏ∗2+2ζkxk−xk−12−1−2−2−ϕϰℶkℶk+1νk−yk2−1−2−2−ϕϰℶkℶk+1yk−xk+12.
Consequently, this implies that(55)1−2−2−ϕϰℶkℶk+1νk−yk2+1−2−2−ϕϰℶkℶk+1yk−xk+12≤xk−ℏ∗2−xk+1−ℏ∗2+ζkxk−ℏ∗2−xk−1−ℏ∗2+2ζkxk−xk−12.
By taking the limit ask⟶+∞ in expression (55), we obtain
(56)limk⟶+∞νk−yk=limk⟶+∞yk−xk+1=0.
Thus, expressions (52) and (56) give that
(57)limk⟶+∞yk−ℏ∗=l.
By using expressions (50), (52), and (57), so that the sequences xk, νk, and yk are bounded, therefore xk, νk, and yk exist. Thus, limk⟶+∞xk−ℏ∗2, limk⟶+∞yk−ℏ∗2, limk⟶+∞vk−ℏ∗2. Following that, we will show that the sequence xk weakly converges to ℏ∗. As a result, all sequences xk, νk, and yk are bounded. We now demonstrate that each sequential weak cluster point in the sequence xk is in solΓ,Ξ. Consider that z is a weak cluster point of xk, which means that there is a subsequence of xk that is weakly convergent to z. Then, z∈Ξ,ykm is also weakly convergent to z.Now let demonstrate thatz∈solΓ,Ξ. We have obtained the following by combining Lemma 9 with expressions (39) and (35):
(58)ℶkmΓykm,y≥ℶkmΓykm,xkm+1+νkm−xkm+1,y−xkm+1≥ℶkmΓνkm,xkm+1−ℶkmΓνkm,ykm−2−2−ϕϰℶkm2ℶkm+1ykm−νkm2−2−2−ϕϰℶkm2ℶkm+1ykm−xkm+12+νkm−xkm+1,y−xkm+1≥νkm−ykm,xkm+1−ykm−2−2−ϕϰℶkm2ℶkm+1ykm−νkm2−2−2−ϕϰℶkm2ℶkm+1ykm−xkm+12+νkm−xkm+1,y−xkm+1,where y is any member of Πk. The use of expression (56) and the boundedness of the sequence xk implies that the right-hand side of the last inequality is convergent to zero. By using the condition Γ4 and ykm⇀z, we have ℶkm≥ℶ>0 such as
(59)0≤limsupm⟶+∞ykm,y≤Γz,y,∀y∈Πk.
SinceΞ is a subset of half-space Πk, it follows that Γz,y≥0,∀y∈Ξ. This proves that z∈solΓ,Ξ. Thus, Lemma 4 assures that νk,xk, and yk converge weakly to ℏ∗ as k⟶+∞.We now present two iterative methods based on a monotone and nonmonotone variable step size rule and two strongly convex minimization problems without the need for subgradient methods. The following is a description of the second major result.Algorithm 3
STEP 0: Choose ℶ0>0,x−1,x0∈Π,ζ∈0,1,ϰ∈0,1, ϕ∈0,2−2 with a sequence ψk⊂0,+∞ such that∑k=0+∞ψk<+∞.Moreover, chooseζk such that 0≤ζk≤βk such thatβk=minζ,ψk/xk−xk−1ifxk≠xk−1,ζotherwise.STEP 1: Computeyk=argminy∈ΞℶkΓνk,y+1/2νk−y2,wwhere νk=xk+ζkxk−xk−1.STEP 2: Computexk+1=argminy∈ΞℶkΓyk,y+1/2νk−y2.STEP 3: Computeℶk+1=minℶk,2−2−ϕϰνk−yk2+2−2−ϕϰxk+1−yk2/2Γνk,xk+1−Γνk,yk−Γyk,xk+1ifΓνk,xk+1−Γνk,yk−Γyk,xk+1>0,ℶk,otherwise.STEP 4: If yk=νk, then complete the computation. Otherwise, set k≔k+1 and go back STEP 1.Algorithm 4
STEP 0: Choose ℶ0>0,x−1,x0∈Π,ζ∈0,1,ϰ∈0,1, ϕ∈0,2−2 with a sequence ψk⊂0,+∞ such that∑k=0+∞‍ψk<+∞.Moreover, choose a non-negative real sequencepk such that ∑k=1+∞pk<+∞ and ζk such that 0≤ζk≤βk such thatβk=minζ,ψk/xk−xk−1ifxk≠xk−1,ζotherwise.STEP 1: Computeyk=argminy∈ΞℶkΓνk,y+1/2νk−y2,wwhere νk=xk+ζkxk−xk−1.STEP 2: Computexk+1=argminy∈ΞℶkΓyk,y+1/2νk−y2.STEP 3: Computeℶk+1=minℶk+pk,2−2−ϕϰνk−yk2+2−2−ϕϰxk+1−yk2/2Γνk,xk+1−Γνk,yk−Γyk,xk+1ifΓνk,xk+1−Γνk,yk−Γyk,xk+1>0,ℶk+pk,otherwise.STEP 4: If yk=νk, then complete the computation. Otherwise, set k≔k+1 and go back STEP 1.


## 4. Results to Solve the Fixed Point Problem and Variational Inequalities

In this section, we solve fixed point problems and variational inequalities using the results from our main results. Expressions (6) and (8) are employed to obtain the following conclusions. All the methods are based on our main findings, which are interpreted as follows.Corollary 12.
Assume thatI1:Ξ⟶Π is a pseudomonotone, weakly continuous, and L-Lipschitz continuous operator and the solution set solI1,Ξ≠∅. Choose ℶ0>0, x−1,x0∈Π,ζ∈0,1,ϰ∈0,1,ϕ∈0,2−2 with a sequence ψk⊂0,+∞ such that
(60)∑k=0+∞ψk<+∞.Moreover, chooseζk such that 0≤ζk≤βk such that
(61)βk=minζ,ψkxk−xk−1,ifxk≠xk−1,ζ,otherwise.First, we have to compute(62)νk=xk+ζkxk−xk−1,yk=PΞνk−u^kI1νk.Havingxk−1,xk,yk with
(63)Πk=z∈Π:νk−ℶkI1νk−yk,z−yk≤0,foreachk≥0.Compute(64)xk+1=PΠkνk−ℶkI1yk.Update the step size in the following way:(65)ℶk+1=minℶk,2−2−ϕϰνk−yk2+2−2−ϕϰxk+1−yk22I1νk−I1yk,xk+1−yk,ifI1νk−I1yk,xk+1−yk>0,ℶk,otherwise.Then, the sequencesxk converge weakly to ℏ∗∈solI1,Ξ.Corollary 13.
Assume thatI1:Ξ⟶Π is a pseudomonotone, weakly continuous, and L-Lipschitz continuous operator and the solution set solI1,Ξ≠∅. Choose ℶ0>0, x−1,x0∈Π,ζ∈0,1,ϰ∈0,1,ϕ∈0,2−2 with a sequence ψk⊂0,+∞ such that
(66)∑k=0+∞ψk<+∞.Moreover, choose a non-negative real sequencepk such that ∑k=1+∞pk<+∞ and ζk such that 0≤ζk≤βk such that
(67)βk=minζ,ψkxk−xk−1,ifxk≠xk−1,ζ,otherwise.First, we have to compute(68)νk=xk+ζkxk−xk−1,yk=PΞνk−u^kI1νk.Havingxk−1,xk,yk with
(69)Πk=z∈Π:νk−ℶkI1νk−yk,z−yk≤0,foreachk≥0.Compute(70)xk+1=PΠkνk−ℶkI1yk.Update the step size in the following way:(71)ℶk+1=minℶk+pk,2−2−ϕϰνk−yk2+2−2−ϕϰxk+1−yk22I1νk−I1yk,xk+1−yk,ifI1νk−I1yk,xk+1−yk>0,ℶk+pk,otherwise.Then, the sequencesxk converge weakly to ℏ∗∈solI1,Ξ.Corollary 14.
Assume thatI1:Ξ⟶Π is a pseudomonotone, weakly continuous, and L-Lipschitz continuous operator and the solution set solI1,Ξ≠∅. Choose ℶ0>0, x−1,x0∈Π,ζ∈0,1,ϰ∈0,1,ϕ∈0,2−2 with a sequence ψk⊂0,+∞ such that
(72)∑k=0+∞ψk<+∞.Moreover, chooseζk such that 0≤ζk≤βk such that
(73)βk=minζ,ψkxk−xk−1,ifxk≠xk−1,ζ,otherwise.First, we have to compute(74)νk=xk+ζkxk−xk−1,yk=PΞνk−ℶkI1νk,xk+1=PΞνk−ℶkI1yk.Update the step size in the following way:(75)ℶk+1=minℶk,2−2−ϕϰνk−yk2+2−2−ϕϰxk+1−yk22I1νk−I1yk,xk+1−yk,ifI1νk−I1yk,xk+1−yk>0,ℶk,otherwise.Then, the sequencesxk converge weakly to ℏ∗∈solI1,Ξ.Corollary 15.
Assume thatI1:Ξ⟶Π is a pseudomonotone, weakly continuous, and L-Lipschitz continuous operator and the solution set solI1,Ξ≠∅. Choose ℶ0>0, x−1,x0∈Π,ζ∈0,1,ϰ∈0,1,ϕ∈0,2−2 with a sequence ψk⊂0,+∞ such that
(76)∑k=0+∞ψk<+∞.Moreover, choose a non-negative real sequencepk such that ∑k=1+∞pk<+∞ and ζk such that 0≤ζk≤βk such that
(77)βk=minζ,ψkxk−xk−1,ifxk≠xk−1,ζ,otherwise.First, we have to compute(78)νk=xk+ζkxk−xk−1,yk=PΞνk−ℶkI1νk,xk+1=PΞνk−ℶkI1yk.Update the step size in the following way:(79)ℶk+1=minℶk+pk,2−2−ϕϰνk−yk2+2−2−ϕϰxk+1−yk22I1νk−I1yk,xk+1−yk,ifI1νk−I1yk,xk+1−yk>0,ℶk+pk,otherwise.Then, the sequencesxk converge weakly to ℏ∗∈solI1,Ξ.Corollary 16.
Assume thatI2:Ξ⟶Π is a κ-strict pseudocontraction, weakly continuous, and L-Lipschitz continuous operator and the solution set solI2,Ξ≠∅. Choose ℶ0>0, x−1,x0∈Π,ζ∈0,1,ϰ∈0,1,ϕ∈0,2−2 with a sequence ψk⊂0,+∞ such that
(80)∑k=0+∞ψk<+∞.Moreover, chooseζk such that 0≤ζk≤βk such that
(81)βk=minζ,ψkxk−xk−1,ifxk≠xk−1,ζ,otherwise.Compute(82)νk=xk+ζkxk−xk−1,yk=PΞνk−ℶkνk−I2νk.Havingxk−1,xk,yk, with
(83)Πk=z∈E:1−ℶkνk+ℶkI2νk−yk,z−yk≤0.Compute(84)xk+1=PΠkνk−ℶkyk−I2yk.The step size rule for the next iteration is evaluated as follows:(85)ℶk+1=minℶk,2−2−ϕϰνk−yk2+2−2−ϕϰxk+1−yk22νk−yk−I2νk−I2yk,xk+1−yk,ifνk−yk−I2νk−I2yk,xk+1−yk>0,ℶk,otherwise.Then, the sequencexk converges weakly to ℏ∗∈solI1,Ξ.Corollary 17.
Assume thatI2:Ξ⟶Π is a κ-strict pseudocontraction, weakly continuous, and L-Lipschitz continuous operator and the solution set solI2,Ξ≠∅. Choose ℶ0>0, x−1,x0∈Π,ζ∈0,1,ϰ∈0,1, ϕ∈0,2−2 with a sequence ψk⊂0,+∞ such that
(86)∑k=0+∞ψk<+∞.Moreover, choose a non-negative real sequencepk such that ∑k=1+∞pk<+∞ and ζk such that 0≤ζk≤βk such that
(87)βk=minζ,ψkxk−xk−1,ifxk≠xk−1,ζ,otherwise.Compute(88)νk=xk+ζkxk−xk−1,yk=PΞνk−ℶkνk−I2νk.Havingxk−1,xk,yk, with
(89)Πk=z∈E:1−ℶkνk+ℶkI2νk−yk,z−yk≤0.Compute(90)xk+1=PΠkνk−ℶkyk−I2yk.The step size rule for the next iteration is evaluated as follows:(91)ℶk+1=minℶk+pk,2−2−ϕϰνk−yk2+2−2−ϕϰxk+1−yk22νk−yk−I2νk−I2yk,xk+1−yk,ifνk−yk−I2νk−I2yk,xk+1−yk>0,ℶk+pk,otherwise.Then, the sequencexk converges weakly to ℏ∗∈solI2,Ξ.Corollary 18.
Assume thatI2:Ξ⟶Π is a κ-strict pseudocontraction, weakly continuous, and L-Lipschitz continuous operator and the solution set solI2,Ξ≠∅. Choose ℶ0>0, x−1,x0∈Π,ζ∈0,1, ϰ∈0,1, ϕ∈0,2−2 with a sequence ψk⊂0,+∞ such that
(92)∑k=0+∞ψk<+∞.Moreover, chooseζk such that 0≤ζk≤βk such that
(93)βk=minζ,ψkxk−xk−1,ifxk≠xk−1,ζ,otherwise.Compute(94)νk=xk+ζkxk−xk−1,yk=PΞνk−ℶkνk−I2νk,xk+1=PΞνk−ℶkyk−I2yk.The step size rule for the next iteration is evaluated as follows:(95)ℶk+1=minℶk,2−2−ϕϰνk−yk2+2−2−ϕϰxk+1−yk22νk−yk−I2νk−I2yk,xk+1−yk,ifνk−yk−I2νk−I2yk,xk+1−yk>0,ℶk,otherwise.Then, the sequencexk converges weakly to ℏ∗∈solI2,Ξ.Corollary 19.
Assume thatI2:Ξ⟶Π is a κ-strict pseudocontraction, weakly continuous, and L-Lipschitz continuous operator and the solution set solI2,Ξ≠∅. Choose ℶ0>0, x−1,x0∈Π,ζ∈0,1,ϰ∈0,1, ϕ∈0,2−2 with a sequence ψk⊂0,+∞ such that
(96)∑k=0+∞ψk<+∞.Moreover, choose a non-negative real sequencepk such that ∑k=1+∞pk<+∞ and ζk such that 0≤ζk≤βk such that
(97)βk=minζ,ψkxk−xk−1,ifxk≠xk−1,ζ,otherwise.Compute(98)νk=xk+ζkxk−xk−1,yk=PΞνk−ℶkνk−I2νk,xk+1=PΞνk−ℶkyk−I2yk.The step size rule for the next iteration is evaluated as follows:(99)ℶk+1=minℶk+pk,2−2−ϕϰνk−yk2+2−2−ϕϰxk+1−yk22νk−yk−I2νk−I2yk,xk+1−yk,ifνk−yk−I2νk−I2yk,xk+1−yk>0,ℶk+pk,otherwise.Then, the sequencexk converges weakly to ℏ∗∈solI2,Ξ.


## 5. Numerical Illustrations

This section describes a number of numerical experiments conducted to demonstrate the validity of the proposed methods. Some of these numerical experiments provide a thorough understanding of how to select effective control parameters. Some of them demonstrate the advantages of the proposed methods over existing ones in the literature. All MATLAB codes were run in MATLAB 9.5 (R2018b) on an Intel(R) Core(TM) i5-6200 Processor CPU @ 2.30 GHz 2.40 GHz, with 8.00 GB RAM.Example 20.
The first sample problem here is drawn from the Nash-Cournot oligopolistic equilibrium model in [16]. In this example, the bifunction Γ can be formulated as having
(100)Γx,y=Px+Qy+c,y−x,where P,Q, and vector c are defined by
(101)P=3.1200023.6000003.5200023.3000003,Q=1.6100011.6000001.5100011.5000002,c=1−2−12−1.The eigenvalues of the matrixQ−P are as follows: −2.9050,−2.7808,−1.0000,−0.8950,−0.7192. As a result, the matrices Q−P and Q are symmetrically negative semidefinite and symmetrically positive semidefinite, respectively. Furthermore, the values for Lipschitz-like parameters are c1=c2=1/2P−Q=1.4525. The constraint set Ξ⊂ℝM is regarded as
(102)Ξ≔x∈ℝM:−2≤xi≤5.The beginning points for these numerical investigations vary, as does the error termDk=xk+1−xk. Figures 1–8 and Tables 1 and 2 show several results for the error term 10−5. Consider the following information regarding control settings:
(1)
For Algorithm 1 in [22] (in short, Itr.Method1), we use(103)ϕ=0.45,ℶ=12c2+8c1(2)
For Algorithm 2 in [41] (in short, Itr.Method2), we use(104)ζk=0.12,ϰ=0.11,ℶ0=1(3)
For Algorithm1 (in short, Itr.Method3), we use(105)ℶ0=0.50,ζ=0.50,ϰ=0.55,ϕ=0.05,ψk=1k2(4)
For Algorithm2 (in short, Itr.Method4), we use(106)ℶ0=0.50,ζ=0.50,ϰ=0.55,ϕ=0.05,ψk=1k2,pk=1001+k2.Example 21.
Consider that the possible setΞ⊂ℝN is defined as follows:
(107)Ξ=u∈ℝN:Au≤b,where matrix A has an order 100×N. Consider that Γ:Ξ×Ξ⟶ℝ is expressed by
(108)Γu,y=Lu,y−u,∀u,y∈Ξ,where L:ℝN⟶ℝN is an operator evaluated as Lu=Pu+r with r∈ℝN and P=QQT+R+S, where Q is an N×N matrix, R is an N×N skew-symmetric matrix, and S is an N×N positive definite diagonal matrix. It is simple to demonstrate that Γ is monotone and that the Lipschitz constants are 2c1=2c2=M (for more information, see [42, 43]). The beginning points for these numerical investigations vary, as does the error term Dk=xk+1−xk. Figures 9–14 and Tables 3 and 4 show several results for the error term 10−5. Consider the following information regarding control settings:
(1)
For Algorithm 1 in [22] (in short, Itr.Method1), we use(109)ϕ=0.45,ℶ=12c2+8c1(2)
For Algorithm 2 in [41] (in short, Itr.Method2), we use(110)ζk=0.12,ϰ=0.11,ℶ0=1(3)
For Algorithm1 (in short, Itr.Method3), we use(111)ℶ0=0.50,ζ=0.50,ϰ=0.55,ϕ=0.05,ψk=1k2(4)
For Algorithm2 (in short, Itr.Method4), we use(112)ℶ0=0.50,ζ=0.50,ϰ=0.55,ϕ=0.05,ψk=1k2,pk=1001+k2Figure 1
All methods are compared computationally whilex0=0,0,0,0,0T.Figure 2
All methods are compared computationally whilex0=0,0,0,0,0T.Figure 3
All methods are compared computationally whilex0=1,2,1,2,1T.Figure 4
All methods are compared computationally whilex0=1,2,1,2,1T.Figure 5
All methods are compared computationally whilex0=1,2,3,−4,5T.Figure 6
All methods are compared computationally whilex0=1,2,3,−4,5T.Figure 7
All methods are compared computationally whilex0=2,−1,3,−4,5T.Figure 8
All methods are compared computationally whilex0=2,−1,3,−4,5T.Table 1
All methods’ numerical values for Figures1–8.
x0Number of iterationsExecution time in secondsAlgorithm1Algorithm2Algorithm1Algorithm20,0,0,0,0T22140.1802602000000000.1276095000000001,2,1,2,1T23160.2261622000000000.1522214000000001,2,3,−4,5T25160.2266679000000000.1542963000000002,−1,3,−4,5T25160.2750091000000000.144512100000000Table 2
All methods’ numerical values for Figures1–8.
x0Number of iterationsExecution time in secondsAlgorithm 1 n [22]Algorithm 2 in [35]Algorithm 1 in [22]Algorithm 2 in [35]0,0,0,0,0T44330.3408147000000000.3129066000000001,2,1,2,1T54350.6523779000000000.3518180000000001,2,3,−4,5T56350.5266949000000000.3325744000000002,−1,3,−4,5T57400.4948373000000000.359039600000000Figure 9
All methods are compared computationally whilex0=2,3,2,5,2T.Figure 10
All methods are compared computationally whilex0=2,3,2,5,2T.Figure 11
All methods are compared computationally whilex0=1,3,5,4,7T.Figure 12
All methods are compared computationally whilex0=1,3,5,4,7T.Figure 13
All methods are compared computationally whilex0=2,−3,5,9,−5T.Figure 14
All methods are compared computationally whilex0=2,−3,5,9,−5T.Table 3
All methods’ numerical values for Figures9–14.
x0Number of iterationsExecution time in secondsAlgorithm 1 in [22]Algorithm 2 in [35]Algorithm 1 in [22]Algorithm 2 in [35]2,3,2,5,2T22170.93052020000.8089937001,3,5,4,7T30231.84773040000.9452039002,−3,5,9,−5T33251.31130050000.816565900Table 4
All methods’ numerical values for Figures9–14.
x0Number of iterationsExecution time in secondsAlgorithm1Algorithm2Algorithm1Algorithm22,3,2,5,2T09050.3661678000000000.2027593000000001,3,5,4,7T12070.4467526000000000.3411427000000002,−3,5,9,−5T13070.4457636000000000.257909300000000


## 6. Conclusion

The research proposed four explicit extragradient-like strategies for dealing with an equilibrium problem in a real Hilbert space involving a pseudomonotone and a Lipschitz-type bifunction. A novel step size rule that does not rely on Lipschitz-type constant information has been proposed. The convergence theorems and applications of the main results have been demonstrated. Several experiments are given to show the numerical behavior of our two algorithms and to compare them to other well-known algorithms in the literature.



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