# Novel Algorithms for Solving a System of Absolute Value Variational Inequalities

Authors: Safeera Batool; Muhammad Aslam Noor; Riaz Ahmad; Ilyas Khan; Mulugeta Andualem
Journal: Journal of Function Spaces
 (2022)

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## Abstract

The goal of this paper is to study a new system of a class of variational inequalities termed as absolute value variational inequalities. Absolute value variational inequalities present a rational, pragmatic, and novel framework for investigating a wide range of equilibrium problems that arise in a variety of disciplines. We first develop a system of absolute value auxiliary variational inequalities to calculate the approximate solution of the system of absolute variational inequalities, and then by employing the projection technique, we prove the existence of solutions of the system of absolute value auxiliary variational inequalities. By utilizing an auxiliary principle and the existence result, we propose several new iterative algorithms for solving the system of absolute value auxiliary variational inequalities in the frame of four different operators. Furthermore, the convergence of the proposed algorithms is investigated in a thorough manner. The efficiency and supremacy of the proposed schemes is exhibited through some special cases of the system of absolute value variational inequalities and an illustrative example. The results presented in this paper are more general and rehash a number of some previously published findings in this field.


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## Body

## 1. Introduction

The theory of variational inequalities, which was presented in the 1960s, exhibits an exceptional evolution as a fascinating and stimulating branch of applied mathematics that assumes a significant role in economics, finance, industry, transportation, optimization, and network analysis. Stampacchia [1] was the first to demonstrate the existence and uniqueness of variational inequality solutions. Variational inequalities have been utilized to examine problems that occurred in a variety of basic and applied sciences since their origin (see [2–6]). These significant applications prompted researchers to develop and broaden variational inequalities and associated optimization problems in various formations employing advanced and innovative methodologies, which include auxiliary principal technique, Wiener-Hopf equations, projection methods, and dynamical systems (see [7–10] and the references therein). It is noted that the operator must be Lipschitz continuous and strongly monotone for projection schemes to converge which is a very difficult set of requirements to verify. This fact led researchers to modify the projection method or to establish new ones. Extragradient-type methods address this difficulty as their convergence requires only the existence of solution and the Lipschitz continuity of the monotone operator. Various modified projection and extragradient-type algorithms have been proposed for finding the solution of variational inequalities. We would like to point out that the projection technique is not appropriate for some variational inequality classes that include nonlinear functions which fail to be differentiable. These factors prompted us to employ the auxiliary principle technique, presented by Glowinski et al. [11]. They employed this method to investigate the existence of a mixed variational inequality solution. Adopting the fixed point approach, this strategy finds the auxiliary variational inequality and proves that the solution obtained from the auxiliary problem is the same as the solution of the underlying problem.The system of variational inequalities is a natural and useful generalization of variational inequalities because it can be used to describe a variety of equilibrium problems, including traffic equilibrium, spatial equilibrium, the Nash equilibrium, and general equilibrium problems (see [12–16]). The emergence of this approach can be noticed as the simultaneous acquisition of two distinct figurations of research; that is, it validates the qualitative features of the solution of major types of problems, while also empowering us to build useful and effective new problem-solving strategies. Various iterative algorithms have been suggested to solve the different systems of variational inequalities. Agarwal et al. [17] considered a system of generalized nonlinear mixed quasi-variational inclusions and proved its associated sensitivity analysis. However, Pang [18] has shown that several equilibrium-type problems other than the Nash equilibrium problem may also be stated as a variational inequality problem that is equivalent to a system of variational inequalities. Thus, the variational inequality theory gives a natural, comprehensive, ordered, and effective framework for analyzing many linear and nonlinear problems.In recent years, another remarkable extension of the variational inequalities known as absolute value variational inequalities is introduced and studied by Batool et al. [9]. They have shown that the absolute value variational inequalities can be transformed into a system of absolute value equations if the underlying domain is the entire space. The system of absolute value equations was proposed and analyzed by Mangasarian [19]. In fact, the system of absolute value equations has become an appealing direction for researchers as various mathematical and engineering problems including linear programs, quadratic programs, and bimatrix games can be reduced into an absolute system of equations (see [20–22] and the references therein). Inspired by the significant boost in this field, in this investigation, the approaches for constructing a novel system of absolute value variational inequalities in connection with the fixed point formulation were proposed with the help of the projection method. By equivalency, several new projection algorithms have been developed that are useful for solving the system of absolute value variational inequalities. Moreover, we examine the convergence of these algorithms under suitable constraints. Various special cases are also considered. A test example illustrates the graphical view of our proposed results. The suggested methods associate a variety of iterative algorithms in this direction. The findings in this study are more invigorating and can be viewed as an improvement and extension of the previously known results.


## 2. Results and Discussion

LetH be a real Hilbert space, whose norm and inner product are denoted by ⋅ and ⋅,⋅, respectively. Let K1 and K2 be two closed and convex sets in H. For given operators T1,T2,B1,B2:H⟶H, consider the problem of finding y∈K1 and x∈K2 such that
(1)T1x+B1x−f1,v−y≥0,∀v∈K1,T2y+B2y−f2,v−x≥0,∀v∈K2,where f1 and f2 are the continuous functionals defined on H and . contains the absolute values of components of x,y∈H. The system (Equation (1)) is called a system of absolute value variational inequalities with four operators.We will now discuss some special cases of the system of absolute value variational inequalities (Equation (1)).
(1)
IfB1=B2=B, then system (Equation (1)) reduces to find y∈K1 and x∈K2 such that(2)T1x+Bx−f1,v−y≥0,∀v∈K1,T2y+By−f2,v−x≥0,∀v∈K2,which is called a system of absolute value variational inequalities with three operators.
(2)
IfK1=K2=K, then system (Equation (2)) reduces to find x,y∈K such that(3)T1x+Bx−f1,v−y≥0,∀v∈K,T2y+By−f2,v−x≥0,∀v∈K,which is a system of absolute value variational inequalities.
(3)
IfB1x=B2y=0,∀x,y∈H, then system (Equation (1)) is equivalent to find y∈K1 and x∈K2 such that(4)T1x−f1,v−y≥0,∀v∈K1,T2y−f2,v−x≥0,∀v∈K2,which is called the system of variational inequalities.
(4)
IfT1=T2=T,∀x,y∈H, then system (Equation (1)) is equivalent to find y∈K1 and x∈K2 such that(5)Tx+B1x−f1,v−y≥0,∀v∈K1,Ty+B2y−f2,v−x≥0,∀v∈K2,which is a system of absolute value variational inequalities with three operators.
(5)
IfT1=T2=T and K1=K2=K, then system (Equation (2)) reduces to find x∈K such that(6)Tx+Bx−f,y−x≥0,∀y∈Kis called an absolute value variational inequality.
(6)
IfBx=0,∀x∈H, then Equation (4) collapses to find x∈K such that(7)Tx−f,y−x≥0,∀y∈K,which are well-known classical variational inequalities, introduced by Lions and Stampacchia [23, 24] and have been studied extensively in many directions. Variational inequalities are useful to formulate various equilibrium problems.
(7)
IfK∗=x∈K:x,y≥0,y∈K is the polar cone of the closed and convex cone K in H, then Equation (4) is equivalent to find x∈K such that(8)x∈K,Tx+Bx−f∈K∗,Tx−Bx,x=0,which is an absolute value complementarity problem. The absolute value complementarity problem was introduced and studied by Noor et al. [25].
(8)
IfBx=0,∀x∈H, then Equation (6) reduces to find x∈K such that(9)x∈K,Tx−f∈K∗,Tx−f,x=0,which is called a complementarity problem. The complementarity problem was introduced and studied by Lemke [5] and has also been investigated by Cottle and Dantzig [26].
(9)
IfT1=T2=T, B1=B2=B, and K1=K2=H, then Equation (1) is equivalent to find x∈H such that(10)Tx+Bx=f,which is known as the system of absolute value equations and is addressed in Reference [27]. The system of absolute value equations is widely applied in various branches of engineering and mathematics. Hence, the proper choice of operators and spaces may generate several known and new types of absolute value variational inequalities and its variant forms.In order to obtain the main results of this paper, some basic definitions and results are needed which are essential for the further analysis.Definition 1.
T:H⟶H is said to be strongly monotone, if there exists a constant α>0 such that
(11)Tx−Ty,x−y≥αx−y2,∀x,y∈H.Definition 2.
An operatorT:H⟶H is said to be Lipschitz continuous, if there exists a constant β>0 such that
(12)Tx−Ty≤βx−y,∀x,y∈H.Definition 3.
An operatorT:H⟶H is said to be monotone if
(13)Tx−Ty,x−y≥0,∀x,y∈H.Definition 4.
An operatorT:H⟶H is said to be pseudomonotone if
(14)Tx,y−x≥0implies
(15)Ty,y−x≥0,∀x,y∈H.We now consider the well-known projection lemma which is due to Reference [4]. The projection lemma transforms the variational inequalities into a fixed point problem.Lemma 5 (see [4]).
LetK be a closed and convex set in H. Then, for a given z∈H,x∈K satisfies
(16)x−z,y−x≥0,∀y∈K,if and only if
(17)x=PKz,where PK is the projection of H onto a closed and convex set K in H.It is notable that the projection operatorPK is a nonexpansive operator, that is
(18)PKx−PKy≤x−y,∀x,y∈H.The above lemma is important to obtain the main results of this paper.Lemma 6 (see [28]).
Ifδnn=0∞ is a nonnegative sequence satisfying the following inequality
(19)δn+1≤1−λnδn+σn,∀n≥0,
with0≤λn≤1, ∑n=0∞λn=∞, and σn=0λn, then limn⟶∞δn=0.Since the projection-type techniques could not be used to suggest iterative algorithms for mixed variational inequalities, Glowinski et al. [11] suggested a new technique for solving the variational inequalities. It is called auxiliary principle technique which proved to be useful as it does not depend on the projection. Also, it is worth mentioning that unified descent algorithms for variational inequalities can be suggested by using an auxiliary principle technique.Hence, Equation (1) can easily be written in an equivalent form by using the auxiliary principle technique, that is to find y∈K1 and x∈K2 such that
(20)γ1T1x+γ1B1x−γ1f1+y−x,v−y≥0,∀v∈K1,γ2T2y+γ2B2y−γ2f2+x−y,v−x≥0,∀v∈K2,where γ1>0 and γ2>0 are the constants.We use this equivalent system to suggest some new iterative algorithms for solving the system of absolute value variational inequalities and its alternative systems.


### 2.1. Main Results

In this section, we establish the equivalence between system of absolute value Equation (20) and the fixed point problems. We use this equivalent formulation to suggest some iterative algorithms for solving the system of absolute value equations. The convergence analysis of the proposed methods is also demonstrated.Lemma 7.
The system of absolute value variational inequalities (Equation (20)) has a solution y∈K1 and x∈K2 if and only if y∈K1 and x∈K2 satisfy the relations:
(21)y=PK1x−γ1T1x−γ1B1x+γ1f1,(22)x=PK2y−γ2T2y−γ2B2y+γ2f2,where γ1>0 and γ2>0 are constants.It is clear from Lemma7 that the system (Equation (20)) is equivalent to the fixed point problems (Equations (21) and (22)). This equivalent formulation is very important from theoretical as well as from the numerical point of view (see [29]). We propose and analyze some iterative schemes by using the composition (Equations (21) and (22)).Equations (21) and (22) can be rewritten in the following equivalent forms:
(23)y=1−ηny+ηnPK1x−γ1T1x−γ1B1x+γ1f1,(24)x=1−ζnx+ζnPK2y−γ2T2y−γ2B2y+γ2f2,where 0≤ζn,ηn≤1 for all n≥0.We use this equivalent formulation to suggest the following iterative algorithms for solving the system of absolute value variational inequalities (Equation (20)) and its related formations.Algorithm 1.
For giveny0∈K2 and x0∈K1, compute xn+1 and yn+1 by the iterative schemes:
(25)yn+1=1−ηnyn+ηnPK1xn−γ1T1xn−γ1B1xn+γ1f1,(26)xn+1=1−ζnxn+ζnPK2yn−γ2T2yn−γ2B2yn+γ2f2,where 0≤ζn,ηn≤1 for all n≥0.Algorithm1 is known as a parallel algorithm which can be considered as the Jacobi method for solving the system of absolute value equations. It is proved that parallel algorithms outperform the sequential schemes.We now discuss some of the special cases of Algorithm1.(1)
IfB1=B2=B, then Algorithm 1 reduces to the following parallel algorithm to find the solution of the system (Equation (2))Algorithm 2.
For giveny0∈K2 and x0∈K1, compute xn+1 and yn+1 by the iterative schemes:
(27)yn+1=1−ηnyn+ηnPK1xn−γ1T1xn−γ1Bxn+γ1f1,xn+1=1−ζnxn+ζnPK2yn−γ2T2yn−γ2Byn+γ2f2,where 0≤ζn,ηn≤1 for all n≥0.(2)
IfK1=K2=K, then Algorithm 2 reduces to the following projection algorithm to solve the system of absolute value variational inequalities (Equation (3))Algorithm 3.
For givenx0,y0∈K, compute xn+1 and yn+1 by the iterative schemes:
(28)yn+1=1−ηnyn+ηnPK1xn−γ1T1xn−γ1Bxn+γ1f1,xn+1=1−ζnxn+ζnPK2yn−γ2T2yn−γ2Byn+γ2f2,where 0≤ζn,ηn≤1 for all n≥0.(3)
IfB1x=B2y=0,∀x,y∈H, then Algorithm 2 reduces to the following projection algorithm to solve the system of variational inequalities (Equation (4))Algorithm 4.
For giveny0∈K2 and x0∈K1, compute xn+1 and yn+1 by the iterative schemes:
(29)yn+1=1−ηnyn+ηnPK1xn−γ1T1xn+γ1f1,xn+1=1−ζnxn+ζnPK2yn−γ2T2yn+γ2f2,where 0≤ζn,ηn≤1 for all n≥0.(4)
IfT1=T2=T, then Algorithm 1 reduces to the following algorithmAlgorithm 5.
For giveny0∈K2 and x0∈K1, compute xn+1 and yn+1 by the iterative schemes:
(30)yn+1=1−ηnyn+ηnPK1xn−γ1Txn−γ1B1xn+γ1f1,xn+1=1−ζnxn+ζnPK2yn−γ2Tyn−γ2B2yn+γ2f2,where 0≤ζn,ηn≤1 for all n≥0.(5)
IfT1=T2=T and B1=B2=B, then Algorithm 2 reduces to the following parallel algorithmAlgorithm 6.
For giveny0∈K2 and x0∈K1, compute xn+1 and yn+1 by the iterative schemes:
(31)yn+1=1−ηnyn+ηnPK1xn−γ1Txn−γ1Bxn+γ1f1,xn+1=1−ζnxn+ζnPK2yn−γ2Tyn−γ2Byn+γ2f2,where 0≤ζn,ηn≤1 for all n≥0.Several new and known iterative schemes can be suggested for solving absolute value variational inequalities and the associated problems by making proper and appropriate choice for operators and spaces.We now examine the convergence analysis of Algorithm1 which is the key motivation of the next result.Theorem 8.
Let the operatorsT1,T2:H⟶H be strongly monotone with constants αT1>0,αT2>0 and Lipschitz continuous with constants βT1>0,βT2>0 and the operators B1,B2:H⟶H be Lipschitz continuous with constants βB1>0,βB2>0, respectively. If the following conditions
(1)
ϕT1=1−2γ1αT1+γ12βT12<1(2)
ϕT2=1−2γ2αT2+γ22βT22<1(3)
0≤ζn,ηn≤1∀n≥0,(32)ζn−ηnϕT1+ϕB1≥0,ηn−ζnϕT2+ϕB2≥0,such that
(33)∑n=0∞ζn−ηnϕT1+ϕB1=∞,∑n=0∞ηn−ζnϕT2+ϕB2=∞,
where(34)ϕT1=1−2γ1αT1+γ12βT12,ϕT2=1−2γ2αT2+γ22βT22,ϕB1=γ2βB2,ϕB2=γ1βB1,hold, then sequences xn and yn obtained from Algorithm 1 converge to x and y, respectively.Proof.
Letx,y∈H such that y∈K1 and x∈K2 be a solution of the system (Equation (20)). Then, from Equation (23) and Equation (26), we have the following:
(35)xn+1−x=1−ζnxn+ζnPK2yn−γ2T2yn−γ2B2yn+γ2f2−1−ζnx−ζnPK2y−γ2T2y−γ2B2y+γ2f2≤1−ζnxn−x+ζnPK2yn−γ2T2yn−γ2B2yn+γ2f2−PK2y−γ2T2y−γ2B2y+γ2f2≤1−ζnxn−x+ζnyn−y−γ2T2yn−T2y+ζnγ2B2yn−B2y.Since the operatorT2 is strongly monotone and Lipschitz continuous with constants αT2>0 and βT2>0, respectively, then we have the following:
(36)yn−y−γ2T2yn−T2y≤yn−y2−2γ2T2yn−T2y,yn−y+T2yn−T2y2≤1−2γ2αT2+γ22βT22yn−y2.Also, using the Lipschitz continuity of the operatorB2 with constant βB2>0, we have the following:
(37)B2yn−B2y≤βB2yn−y.Combining Equations (35), (36), and (37), we obtain the following:
(38)xn+1−x≤1−ζnxn−x+ζn1−2γ2αT2+γ22βT22+γ2βB2yn−y=1−ζnxn−x+ζnϕT2+ϕB2yn−y.In a similar way, from Equation (23) and Equation (25), we have the following:
(39)yn+1−y=1−ηnyn+ηnPK1xn−γ1T1xn−γ1B1xn+γ1f1−1−ηny−ηnPK1x−γ1T1x−γ1B1x+γ1f1≤1−ηnyn−y+ηnPK1xn−γ1T1xn−γ1B1xn+γ1f1−PK1x−γ1T1x−γ1B1x+γ1f1≤1−ηnyn−y+ηnxn−x−γ1T1xn−T1x+ηnγ1B1xn−B1x≤1−ηnyn−y+ηn1−2γ1αT1+γ12βT12xn−x+ηnγ1βB1xn−x=1−ηnyn−y+ηn1−2γ1αT1+γ12βT12+γ1βB1xn−x=1−ηnyn−y+ηnϕT1+ϕB1xn−x,where we have used the strong monotonicity ofT1 with constant αT1>0 and Lipschitz continuity of the operators T1 and B1 with constants βT1>0 and βB1>0, respectively.Adding Equations (38) and (39), we have the following:
(40)xn+1−x+yn+1−y≤1−ζn+ηnϕT1+ϕB1xn−x+1−ηn+ζnϕT2+ϕB2yn−y≤max1−ζn+ηnϕT1+ϕB1,1−ηn+ζnϕT2+ϕB2xn−x+yn−y=maxξ1,ξ2xn−x+yn−y=ϕxn−x+yn−y,where
(41)ϕ=maxξ1,ξ2,ξ1=1−ζn+ηnϕT1+ϕB1,ξ2=1−ηn+ζnϕT2+ϕB2.From assumption (iii), it follows thatϕ<1. Hence, using Lemma 6, we obtain from Equation (44) the following:
(42)limn⟶∞xn−x+yn−y=0.This further implies that(43)limn⟶∞xn−x=0,limn⟶∞yn−y=0,which is the required result.We now propose and examine some new iterative schemes for solving system of absolute value variational inequalities, by employing a useful substitution.It can easily be shown, by using Lemma5, that x,y∈H such that y∈K1 and x∈K2 is a solution of the system of absolute value variational inequalities (Equation (1)), if and only if x,y∈H:y∈K1,x∈K2 satisfies the following:
(44)y=PK1z,(45)x=PK2w,(46)z=x−γ1T1x−γ1B1x+γ1f1,(47)w=y−γ2T2y−γ2B2y+γ2f2.By using this alternative formation, we can propose and examine the following iterative schemes to solve the system (Equation (1)).Algorithm 7.
For giveny0∈K1 and x0∈K2, find xn+1 and yn+1 by the iterative schemes:
(48)yn+1=1−ηnyn+ηnPK1zn,(49)xn+1=1−ζnxn+ζnPK2wn,(50)zn=xn−γ1T1xn−γ1B1xn+γ1f1,(51)wn=yn−γ2T2yn−γ2B2yn+γ2f2,where 0≤ζn,ηn≤1 for all n≥0.By choosing the useful operators and proper spaces, one can have various new as well as known iterative schemes for solving the system of absolute value variational inequalities and its variant forms. Now, we examine the convergence analysis of Algorithm7 by employing the approach of Theorem 8.Theorem 9.
Let the operatorsT1,T2:H⟶H be strongly monotone with constants αT1>0,αT2>0 and Lipschitz continuous with constants βT1>0,βT2>0 and the operators B1,B2:H⟶H be Lipschitz continuous with constants βB1>0,βB2>0, respectively. If the following conditions
(i)
ϕT1=1−2γ1αT1+γ12βT12<1(ii)
ϕT2=1−2γ2αT2+γ22βT22<1(iii)
0≤ζn,ηn≤1∀n≥0(52)ζn−ηnϕT1+ϕB1≥0,ηn−ζnϕT2+ϕB2≥0,such that
(53)∑n=0∞ζn−ηnϕT1+ϕB1=∞,∑n=0∞ηn−ζnϕT2+ϕB2=∞,where
(54)ϕT1=1−2γ1αT1+γ12βT12,ϕT2=1−2γ2αT2+γ22βT22,ϕB1=γ2βB2,ϕB2=γ1βB1,hold, then sequences xn and yn obtained from Algorithm 7 converge to x and y, respectively.Proof.
Letx,y∈H such that y∈K1 and x∈K2 be a solution of the system (Equation (20)). Then, from Equation (45) and Equation (49), we have the following:
(55)xn+1−x=1−ζnxn+ζnPK2wn−1−ζnx−ζnPK2w≤1−ζnxn−x+ζnPK2wn−PK2w≤1−ζnxn−x+ζnwn−w.In a similar way, from Equation (44) and Equation (48), we have the following:
(56)yn+1−y=1−ηnyn+ηnPK1zn−1−ηny−ηnPK1z≤1−ηnyn−y+ηnPK1zn−PK1z≤1−ηnyn−y+ηnzn−z.From Equations (36), (37), (47), and (51), we have the following:
(57)wn−w=yn−y−γ2T2yn−T2y+B2yn−B2y≤ϕT2+ϕB2yn−y.Also, from Equations (39), (46), and (50), we have the following:
(58)zn−z=xn−x−γ1T1xn−T1x+B1xn−B1x≤ϕT1+ϕB1xn−x.Combining Equations (55), (56), (57), and (58), we have the following:
(59)xn+1−x≤1−ζnxn−x+ζnϕT2+ϕB2yn−y,(60)yn+1−y≤1−ηnyn−y+ηnϕT1+ϕB1xn−x.Addition of Equations (59) and (60) implies
(61)xn+1−x+yn+1−y≤1−ζn+ηnϕT1+ϕB1xn−x+1−ηn+ζnϕT2+ϕB2yn−y≤max1−ζn+ηnϕT1+ϕB1,1−ηn+ζnϕT2+ϕB2xn−x+yn−y=maxξ1,ξ2xn−x+yn−y=ϕxn−x+yn−y,where
(62)ϕ=maxξ1,ξ2,ξ1=1−ζn+ηnϕT1+ϕB1,ξ2=1−ηn+ζnϕT2+ϕB2.From assumption (iii), it follows thatϕ<1. Hence, using Lemma 6, we obtain from Equation (23) the following:
(63)limn⟶∞xn−x+yn−y=0.This further implies that(64)limn⟶∞xn−x=0,limn⟶∞yn−y=0,which is the required result.Example 10.
H=ℝ,K1=−∞,0 and K2=0,∞.. Let T1,T2:H⟶H be the single-valued mappings defined by the following:
(65)T1x=2x−13,T2x=5x−26,∀x∈H.Also, the mappingsB1,B2:H⟶H are defined by the following:
(66)B1x=3x−25,B2x=5x+17,∀x∈H.Then, it can easily be verified that for eachi=1,2, Ti is strongly monotone and Lipschitz continuous with αT1=2/3=βT1 and αT2=5/6=βT2, and Bi is strongly monotone and Lipschitz continuous with αB1=3/5=βB1 and αB2=5/7=βB2.Then, forγ1,γ2=1,(67)ϕT1=1−2γ1αT1+γ12βT12=1−223+232=23<1,ϕT2=1−2γ2αT2+γ22βT22=1−256+562=16<1.Also, for eachn=1,2,
(68)0<12=ζn,ηn=12<1,we have the following:
(69)Ω1=ζn−ηnϕT1+ϕB1=12−1213+35=130>0,Ω2=ηn−ζnϕT2+ϕB2=12−1216+57=584>0,where ϕB1=γ1βB1=3/5 and ϕB2=γ2βB2=5/7. Clearly, we see that all the assumptions of Theorem 8 and Theorem 9 are satisfied. Hence, by using Algorithm 1 and Algorithm 7, the conclusions of Theorem 8 and Theorem 9 follow.Figure1 is the graphical representation of the operators defined in Example 10. Figure 2 depicts the behaviour of Theorem 8 satisfying Example 10. Similarly, Figure 3 interprets the behaviour of Theorem 9 via a three-dimensional plot satisfying Example 10.Figure 1
Graphical behaviour ofT1x, T2x, B1x, and B2x.Figure 2
Behaviour of Theorem8 via a three-dimensional plot satisfies Example 10.Figure 3
Behaviour of Theorem9 via a three-dimensional plot satisfies Example 10 approach. The equivalence between the system of absolute value variational inequalities and the system of fixed point problems has been established. Further, we suggested several innovative iterative algorithms for solving our considered system of variational inequalities and the convergence of the proposed schemes has been analyzed under some suitable conditions. Finally, to demonstrate the existence and convergence results, a numerical example was given. We also discussed some special cases of the system of absolute value variational inequalities. The concept and technique of this paper may encourage researchers to analyze the innovative and unique applications of the system of absolute value variational inequalities and its associated optimization problems. In futuristic research, we extend this study to exponential absolute value variational inequalities and their variant forms.


## 2.1. Main Results

In this section, we establish the equivalence between system of absolute value Equation (20) and the fixed point problems. We use this equivalent formulation to suggest some iterative algorithms for solving the system of absolute value equations. The convergence analysis of the proposed methods is also demonstrated.Lemma 7.
The system of absolute value variational inequalities (Equation (20)) has a solution y∈K1 and x∈K2 if and only if y∈K1 and x∈K2 satisfy the relations:
(21)y=PK1x−γ1T1x−γ1B1x+γ1f1,(22)x=PK2y−γ2T2y−γ2B2y+γ2f2,where γ1>0 and γ2>0 are constants.It is clear from Lemma7 that the system (Equation (20)) is equivalent to the fixed point problems (Equations (21) and (22)). This equivalent formulation is very important from theoretical as well as from the numerical point of view (see [29]). We propose and analyze some iterative schemes by using the composition (Equations (21) and (22)).Equations (21) and (22) can be rewritten in the following equivalent forms:
(23)y=1−ηny+ηnPK1x−γ1T1x−γ1B1x+γ1f1,(24)x=1−ζnx+ζnPK2y−γ2T2y−γ2B2y+γ2f2,where 0≤ζn,ηn≤1 for all n≥0.We use this equivalent formulation to suggest the following iterative algorithms for solving the system of absolute value variational inequalities (Equation (20)) and its related formations.Algorithm 1.
For giveny0∈K2 and x0∈K1, compute xn+1 and yn+1 by the iterative schemes:
(25)yn+1=1−ηnyn+ηnPK1xn−γ1T1xn−γ1B1xn+γ1f1,(26)xn+1=1−ζnxn+ζnPK2yn−γ2T2yn−γ2B2yn+γ2f2,where 0≤ζn,ηn≤1 for all n≥0.Algorithm1 is known as a parallel algorithm which can be considered as the Jacobi method for solving the system of absolute value equations. It is proved that parallel algorithms outperform the sequential schemes.We now discuss some of the special cases of Algorithm1.(1)
IfB1=B2=B, then Algorithm 1 reduces to the following parallel algorithm to find the solution of the system (Equation (2))Algorithm 2.
For giveny0∈K2 and x0∈K1, compute xn+1 and yn+1 by the iterative schemes:
(27)yn+1=1−ηnyn+ηnPK1xn−γ1T1xn−γ1Bxn+γ1f1,xn+1=1−ζnxn+ζnPK2yn−γ2T2yn−γ2Byn+γ2f2,where 0≤ζn,ηn≤1 for all n≥0.(2)
IfK1=K2=K, then Algorithm 2 reduces to the following projection algorithm to solve the system of absolute value variational inequalities (Equation (3))Algorithm 3.
For givenx0,y0∈K, compute xn+1 and yn+1 by the iterative schemes:
(28)yn+1=1−ηnyn+ηnPK1xn−γ1T1xn−γ1Bxn+γ1f1,xn+1=1−ζnxn+ζnPK2yn−γ2T2yn−γ2Byn+γ2f2,where 0≤ζn,ηn≤1 for all n≥0.(3)
IfB1x=B2y=0,∀x,y∈H, then Algorithm 2 reduces to the following projection algorithm to solve the system of variational inequalities (Equation (4))Algorithm 4.
For giveny0∈K2 and x0∈K1, compute xn+1 and yn+1 by the iterative schemes:
(29)yn+1=1−ηnyn+ηnPK1xn−γ1T1xn+γ1f1,xn+1=1−ζnxn+ζnPK2yn−γ2T2yn+γ2f2,where 0≤ζn,ηn≤1 for all n≥0.(4)
IfT1=T2=T, then Algorithm 1 reduces to the following algorithmAlgorithm 5.
For giveny0∈K2 and x0∈K1, compute xn+1 and yn+1 by the iterative schemes:
(30)yn+1=1−ηnyn+ηnPK1xn−γ1Txn−γ1B1xn+γ1f1,xn+1=1−ζnxn+ζnPK2yn−γ2Tyn−γ2B2yn+γ2f2,where 0≤ζn,ηn≤1 for all n≥0.(5)
IfT1=T2=T and B1=B2=B, then Algorithm 2 reduces to the following parallel algorithmAlgorithm 6.
For giveny0∈K2 and x0∈K1, compute xn+1 and yn+1 by the iterative schemes:
(31)yn+1=1−ηnyn+ηnPK1xn−γ1Txn−γ1Bxn+γ1f1,xn+1=1−ζnxn+ζnPK2yn−γ2Tyn−γ2Byn+γ2f2,where 0≤ζn,ηn≤1 for all n≥0.Several new and known iterative schemes can be suggested for solving absolute value variational inequalities and the associated problems by making proper and appropriate choice for operators and spaces.We now examine the convergence analysis of Algorithm1 which is the key motivation of the next result.Theorem 8.
Let the operatorsT1,T2:H⟶H be strongly monotone with constants αT1>0,αT2>0 and Lipschitz continuous with constants βT1>0,βT2>0 and the operators B1,B2:H⟶H be Lipschitz continuous with constants βB1>0,βB2>0, respectively. If the following conditions
(1)
ϕT1=1−2γ1αT1+γ12βT12<1(2)
ϕT2=1−2γ2αT2+γ22βT22<1(3)
0≤ζn,ηn≤1∀n≥0,(32)ζn−ηnϕT1+ϕB1≥0,ηn−ζnϕT2+ϕB2≥0,such that
(33)∑n=0∞ζn−ηnϕT1+ϕB1=∞,∑n=0∞ηn−ζnϕT2+ϕB2=∞,
where(34)ϕT1=1−2γ1αT1+γ12βT12,ϕT2=1−2γ2αT2+γ22βT22,ϕB1=γ2βB2,ϕB2=γ1βB1,hold, then sequences xn and yn obtained from Algorithm 1 converge to x and y, respectively.Proof.
Letx,y∈H such that y∈K1 and x∈K2 be a solution of the system (Equation (20)). Then, from Equation (23) and Equation (26), we have the following:
(35)xn+1−x=1−ζnxn+ζnPK2yn−γ2T2yn−γ2B2yn+γ2f2−1−ζnx−ζnPK2y−γ2T2y−γ2B2y+γ2f2≤1−ζnxn−x+ζnPK2yn−γ2T2yn−γ2B2yn+γ2f2−PK2y−γ2T2y−γ2B2y+γ2f2≤1−ζnxn−x+ζnyn−y−γ2T2yn−T2y+ζnγ2B2yn−B2y.Since the operatorT2 is strongly monotone and Lipschitz continuous with constants αT2>0 and βT2>0, respectively, then we have the following:
(36)yn−y−γ2T2yn−T2y≤yn−y2−2γ2T2yn−T2y,yn−y+T2yn−T2y2≤1−2γ2αT2+γ22βT22yn−y2.Also, using the Lipschitz continuity of the operatorB2 with constant βB2>0, we have the following:
(37)B2yn−B2y≤βB2yn−y.Combining Equations (35), (36), and (37), we obtain the following:
(38)xn+1−x≤1−ζnxn−x+ζn1−2γ2αT2+γ22βT22+γ2βB2yn−y=1−ζnxn−x+ζnϕT2+ϕB2yn−y.In a similar way, from Equation (23) and Equation (25), we have the following:
(39)yn+1−y=1−ηnyn+ηnPK1xn−γ1T1xn−γ1B1xn+γ1f1−1−ηny−ηnPK1x−γ1T1x−γ1B1x+γ1f1≤1−ηnyn−y+ηnPK1xn−γ1T1xn−γ1B1xn+γ1f1−PK1x−γ1T1x−γ1B1x+γ1f1≤1−ηnyn−y+ηnxn−x−γ1T1xn−T1x+ηnγ1B1xn−B1x≤1−ηnyn−y+ηn1−2γ1αT1+γ12βT12xn−x+ηnγ1βB1xn−x=1−ηnyn−y+ηn1−2γ1αT1+γ12βT12+γ1βB1xn−x=1−ηnyn−y+ηnϕT1+ϕB1xn−x,where we have used the strong monotonicity ofT1 with constant αT1>0 and Lipschitz continuity of the operators T1 and B1 with constants βT1>0 and βB1>0, respectively.Adding Equations (38) and (39), we have the following:
(40)xn+1−x+yn+1−y≤1−ζn+ηnϕT1+ϕB1xn−x+1−ηn+ζnϕT2+ϕB2yn−y≤max1−ζn+ηnϕT1+ϕB1,1−ηn+ζnϕT2+ϕB2xn−x+yn−y=maxξ1,ξ2xn−x+yn−y=ϕxn−x+yn−y,where
(41)ϕ=maxξ1,ξ2,ξ1=1−ζn+ηnϕT1+ϕB1,ξ2=1−ηn+ζnϕT2+ϕB2.From assumption (iii), it follows thatϕ<1. Hence, using Lemma 6, we obtain from Equation (44) the following:
(42)limn⟶∞xn−x+yn−y=0.This further implies that(43)limn⟶∞xn−x=0,limn⟶∞yn−y=0,which is the required result.We now propose and examine some new iterative schemes for solving system of absolute value variational inequalities, by employing a useful substitution.It can easily be shown, by using Lemma5, that x,y∈H such that y∈K1 and x∈K2 is a solution of the system of absolute value variational inequalities (Equation (1)), if and only if x,y∈H:y∈K1,x∈K2 satisfies the following:
(44)y=PK1z,(45)x=PK2w,(46)z=x−γ1T1x−γ1B1x+γ1f1,(47)w=y−γ2T2y−γ2B2y+γ2f2.By using this alternative formation, we can propose and examine the following iterative schemes to solve the system (Equation (1)).Algorithm 7.
For giveny0∈K1 and x0∈K2, find xn+1 and yn+1 by the iterative schemes:
(48)yn+1=1−ηnyn+ηnPK1zn,(49)xn+1=1−ζnxn+ζnPK2wn,(50)zn=xn−γ1T1xn−γ1B1xn+γ1f1,(51)wn=yn−γ2T2yn−γ2B2yn+γ2f2,where 0≤ζn,ηn≤1 for all n≥0.By choosing the useful operators and proper spaces, one can have various new as well as known iterative schemes for solving the system of absolute value variational inequalities and its variant forms. Now, we examine the convergence analysis of Algorithm7 by employing the approach of Theorem 8.Theorem 9.
Let the operatorsT1,T2:H⟶H be strongly monotone with constants αT1>0,αT2>0 and Lipschitz continuous with constants βT1>0,βT2>0 and the operators B1,B2:H⟶H be Lipschitz continuous with constants βB1>0,βB2>0, respectively. If the following conditions
(i)
ϕT1=1−2γ1αT1+γ12βT12<1(ii)
ϕT2=1−2γ2αT2+γ22βT22<1(iii)
0≤ζn,ηn≤1∀n≥0(52)ζn−ηnϕT1+ϕB1≥0,ηn−ζnϕT2+ϕB2≥0,such that
(53)∑n=0∞ζn−ηnϕT1+ϕB1=∞,∑n=0∞ηn−ζnϕT2+ϕB2=∞,where
(54)ϕT1=1−2γ1αT1+γ12βT12,ϕT2=1−2γ2αT2+γ22βT22,ϕB1=γ2βB2,ϕB2=γ1βB1,hold, then sequences xn and yn obtained from Algorithm 7 converge to x and y, respectively.Proof.
Letx,y∈H such that y∈K1 and x∈K2 be a solution of the system (Equation (20)). Then, from Equation (45) and Equation (49), we have the following:
(55)xn+1−x=1−ζnxn+ζnPK2wn−1−ζnx−ζnPK2w≤1−ζnxn−x+ζnPK2wn−PK2w≤1−ζnxn−x+ζnwn−w.In a similar way, from Equation (44) and Equation (48), we have the following:
(56)yn+1−y=1−ηnyn+ηnPK1zn−1−ηny−ηnPK1z≤1−ηnyn−y+ηnPK1zn−PK1z≤1−ηnyn−y+ηnzn−z.From Equations (36), (37), (47), and (51), we have the following:
(57)wn−w=yn−y−γ2T2yn−T2y+B2yn−B2y≤ϕT2+ϕB2yn−y.Also, from Equations (39), (46), and (50), we have the following:
(58)zn−z=xn−x−γ1T1xn−T1x+B1xn−B1x≤ϕT1+ϕB1xn−x.Combining Equations (55), (56), (57), and (58), we have the following:
(59)xn+1−x≤1−ζnxn−x+ζnϕT2+ϕB2yn−y,(60)yn+1−y≤1−ηnyn−y+ηnϕT1+ϕB1xn−x.Addition of Equations (59) and (60) implies
(61)xn+1−x+yn+1−y≤1−ζn+ηnϕT1+ϕB1xn−x+1−ηn+ζnϕT2+ϕB2yn−y≤max1−ζn+ηnϕT1+ϕB1,1−ηn+ζnϕT2+ϕB2xn−x+yn−y=maxξ1,ξ2xn−x+yn−y=ϕxn−x+yn−y,where
(62)ϕ=maxξ1,ξ2,ξ1=1−ζn+ηnϕT1+ϕB1,ξ2=1−ηn+ζnϕT2+ϕB2.From assumption (iii), it follows thatϕ<1. Hence, using Lemma 6, we obtain from Equation (23) the following:
(63)limn⟶∞xn−x+yn−y=0.This further implies that(64)limn⟶∞xn−x=0,limn⟶∞yn−y=0,which is the required result.Example 10.
H=ℝ,K1=−∞,0 and K2=0,∞.. Let T1,T2:H⟶H be the single-valued mappings defined by the following:
(65)T1x=2x−13,T2x=5x−26,∀x∈H.Also, the mappingsB1,B2:H⟶H are defined by the following:
(66)B1x=3x−25,B2x=5x+17,∀x∈H.Then, it can easily be verified that for eachi=1,2, Ti is strongly monotone and Lipschitz continuous with αT1=2/3=βT1 and αT2=5/6=βT2, and Bi is strongly monotone and Lipschitz continuous with αB1=3/5=βB1 and αB2=5/7=βB2.Then, forγ1,γ2=1,(67)ϕT1=1−2γ1αT1+γ12βT12=1−223+232=23<1,ϕT2=1−2γ2αT2+γ22βT22=1−256+562=16<1.Also, for eachn=1,2,
(68)0<12=ζn,ηn=12<1,we have the following:
(69)Ω1=ζn−ηnϕT1+ϕB1=12−1213+35=130>0,Ω2=ηn−ζnϕT2+ϕB2=12−1216+57=584>0,where ϕB1=γ1βB1=3/5 and ϕB2=γ2βB2=5/7. Clearly, we see that all the assumptions of Theorem 8 and Theorem 9 are satisfied. Hence, by using Algorithm 1 and Algorithm 7, the conclusions of Theorem 8 and Theorem 9 follow.Figure1 is the graphical representation of the operators defined in Example 10. Figure 2 depicts the behaviour of Theorem 8 satisfying Example 10. Similarly, Figure 3 interprets the behaviour of Theorem 9 via a three-dimensional plot satisfying Example 10.Figure 1
Graphical behaviour ofT1x, T2x, B1x, and B2x.Figure 2
Behaviour of Theorem8 via a three-dimensional plot satisfies Example 10.Figure 3
Behaviour of Theorem9 via a three-dimensional plot satisfies Example 10 approach. The equivalence between the system of absolute value variational inequalities and the system of fixed point problems has been established. Further, we suggested several innovative iterative algorithms for solving our considered system of variational inequalities and the convergence of the proposed schemes has been analyzed under some suitable conditions. Finally, to demonstrate the existence and convergence results, a numerical example was given. We also discussed some special cases of the system of absolute value variational inequalities. The concept and technique of this paper may encourage researchers to analyze the innovative and unique applications of the system of absolute value variational inequalities and its associated optimization problems. In futuristic research, we extend this study to exponential absolute value variational inequalities and their variant forms.


## 3. Conclusion

In this paper, we have introduced a new system of variational inequalities, called the system of absolute value variational inequalities. To determine the approximate solution of the system of absolute value variational inequalities, we first built a system of absolute value auxiliary variational inequalities. We demonstrate the existence of a solution to the system of absolute value auxiliary variational inequalities using the projection.



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