ABSOLUTE VALUE VARIATIONAL INEQUALITIES AND DYNAMICAL SYSTEMS

In this paper, we consider the absolute value variational inequalities. We propose and analyze the projected dynamical system associated with absolute value variational inequalities by using the projection method. We suggest different iterative algorithms for solving absolute value variational inequalities by discretizing the corresponding projected dynamical system. The convergence of the suggested methods is proved under suitable constraints. Numerical examples are given to illustrate the efficiency and implementation of the methods. Results proved in this paper continue to hold for previously known classes of absolute value variational inequalities.


Introduction
Variational inequalities theory was introduced earlier by Stampacchia [45] and now it is developed as a well-established branch of nonlinear analysis and optimization. Variational inequalities theory is widely applied in industry, economics, social, pure and applied sciences, see [17, 20, 27-29, 31, 32, 35-38].
In fact, variational inequalities theory provides us the direct, natural, unified and dynamic framework for the natural analysis of a wide range of unrelated linear and nonlinear problems, see [5,7,[11][12][13]23].
Since the discovery of variational inequalities theory, a number of numerical methods including projection method, Wiener-Hopf equations, auxiliary principle and dynamical systems has been developed for solving the variational inequalities and the related optimization problems, see the references therein .
The relationship of variational inequalities and complementarity problems was proved by Karamardian [15], who proved that, if the involved set is a convex cone, then variational inequalities are equivalent to complementarity problem. Complementarity problem was introduced by Lemke [19]. The equivalence between variational inequalities and complementarity problem has been used in suggesting many iterative algorithms for solving complementarity problems, see [17,26,27,37]. Absolute value complementarity problem is an important and useful generalization of the complementarity problem. Absolute value complementarity problem was introduced by Noor et al. [41] and he also shown that absolute value complementarity problem is equivalent to absolute value variational inequalities. Absolute value variational inequalities include the variational inequalities as a special case. It is also proved that if the underlying set is the whole space, then absolute value variational inequalities transform into absolute system of equations, which were introduced and studied by Mangasarian [23], Rohn [43] and Hu and Huang [13]. For the applications of absolute system of equations, see [13,14,24,25,41,42].
Fixed point theory played an important role in developing various types of algorithms for finding the solution of variational inequalities. The equivalence between variational inequalities and fixed point problem can be established by using the projection technique, see [17]. This alternative formulation has been used to suggest the projected dynamical systems related to variational inequalities. Projected dynamical systems were introduced by Dupuis and Nagurney, see [7]. The projected dynamical systems are identified by the discontinuous right hand side. An innovative aspect of the projected dynamical system is that the set of its stationary points corresponds to the solution set of the associated variational inequality problems. Hence the equilibrium problems in different branches of pure and applied sciences which are studied in the framework of variational inequalities, can now be considered in the more general framework of projected dynamical systems. Projected dynamical systems are effective in the development of many efficient numerical techniques for approximating the solutions of variational inequalities and related nonlinear problems, see [3,6,10,11]. The global asymptotic stability of the projected dynamical systems has also been studied by Noor [33] and Xia and Wang [46].
In this paper, we consider the absolute value variational inequalities. We propose and analyze the projected dynamical system associated with absolute value variational inequalities by using projection method. It is important to mention that such type of the dynamical systems for absolute value variational inequalities have not been investigated earlier. We suggest different iterative algorithms for the solution of absolute value variational inequalities by dicretizing the corresponding projected dynamical system. The convergence of the suggested methods is proved under suitable conditions. Two examples are given to illustrate the implementation and efficiency of the proposed iterative methods. Since the absolute value variational inequalities include the classical variational inequalities, complementarity problems and absolute value equations as special cases, the results obtained in this paper continue to hold for these problems.

Formulations and Basic Results
Let H be a real Hilbert space, whose norm and inner product are denoted by . and ., . respectively.
Let K be a closed and convex set in H. For given operators T, B : H → H, consider the problem of finding where f is a continuous functional defined on H and |u| contains the absolute values of components of u ∈ H. The inequality (1) is called absolute value variational inequality.
We will now discuss some special cases of the problem (1).
(II). If K * = {u ∈ K : u, v ≥ 0, v ∈ K} is the polar cone of the closed and convex cone K in H, then problem (1) is equivalent to find u ∈ K such that which is called an absolute value complementarity problem. If B|u| = 0, then problem (3) is known as the complementarity problem, the origin of which can be traced back to Lemke [19] and have been studied by Cottle and Dantzig [4] who restated the linear and quadratic programming problem as a complementarity problem.
(III). If K = H, then problem (1) is equivalent to find u ∈ H such that which is called the system of absolute value equations. Absolute system of equations was introduced by Mangasarian and Meyer [24] and was investigated by Hu and Huang [13] in a more general context. A generalized Newton method was also suggested to find the solution of absolute system of equations, see [14].
An algorithm to compute all the solutions of (4) was proposed by Rohn [44] and equivalent formulations of (4) are presented by Prokopyev [43]. For the applications of absolute system of equations in various areas of engineering and mathematics, see [11,12,[23][24][25].
Hence, it is clear that the absolute value variational inequality (1) is more general and includes previously known classes of variational inequalities and the system of absolute value equations as special cases.
For the recent applications of absolute value variational inequalities, see [40,41].
In order to derive the main results of this paper, we recall some standard definitions and results.
We now consider the well-known projection lemma [17]. This result is useful to reformulate the variational inequalities as a fixed point problem.
Lemma 2.1. [17] Let K be a closed and convex set in H. Then for a given z ∈ H, u ∈ K satisfies if and only if where P K is the projection of H onto a closed and convex set K in H.
The above lemma plays an important role in obtaining the main results of this paper.
The projection operator P K has following properties: (1) The projection operator is non-expansive, that is, (2) The projection operator P K is co-coercive map, that is,

Existence Theory
In this section, we consider a new type of projected dynamical system associated with absolute value variational inequalities and discuss the existence of the solution, stability and convergence of that dynamical system to the solution of absolute value variational inequality (1). Using Lemma 2.1, we can now state the following result which shows that problem (1) is equivalent to a fixed point problem.
Lemma 3.1. Let K be a closed convex set in H. Then u ∈ K is a solution of absolute value variational inequality (1), if and only if, u ∈ K satisfies the relation where ρ > 0 is a constant.
This alternative equivalent formulation is very important from the theoretical as well as from the numerical point of view. This equivalent provides important tools to approximate the solution of variational inequalities. This equivalent formulation played an important part in establishing the various explicit and implicit iterative methods. Koperlevich [18] developed the the extragradient method for solving the variational inequalities, the convergence of which requires only the Lipschitz continuity.
We now define the residue vector R(u) by the following relation Using Lemma 3.1, one can conclude that u ∈ K is a solution of the absolute value variational inequality (1), if and only if, u ∈ K is a zero of the equation Using (7), we suggest a new projected dynamical system related to absolute value variational inequality (1) as follows: where γ > 0 is a parameter. The expression on the right hand side of (9) is discontinuous on the boundary of K. It can be observed from the definition that the solution of (9) always belongs to K. Hence, it is meaningful to discuss the existence, uniqueness and continuous dependence of the solution of (9). These type of projected dynamical systems, corresponding to variational inequalities, have been investigated widely, see [3, 6, 7, 9-11, 33, 46-48].
The dynamical system is said to be stable in the Lyapunov sense, if the small initial perturbation does not grow in time for a phase trajectory and asymptotically stable, if the small perturbation vanishes as the time goes by. If the dynamical system is asymptotically stable, then it is Lyapunov stable but not conversely, in general.
Definition 3.1. The dynamical system is said to converge globally to the solution set K of (1), if and only if, irrespective of the initial point, the trajectory of the dynamical system satisfies where It can be observed that if the set K contains a unique point u, then (10) implies that lim n→∞ u(t) = u.
Here, the Lyapunov stability of the dynamical system at u implies the global exponential stability of the dynamical system at u.
Definition 3.2. The dynamical system is said to be globally exponentially stable with degree λ > 0, at u ∈ K if and only if, irrespective of the initial point, the trajectory of the system u(t) satisfies where η 1 > 0 and λ > 0, are independent of the initial point. It can also be noted that global exponential stability always implies the global asymptotic stability and hence the dynamical system converges arbitrarily fast.
Now onwards, we assume that the nonempty set K is bounded, unless otherwise specified.
We now discuss the existence of the solution of absolute value variational inequality (1) via dynamical system (9), mainly using the technique of Noor [33] and Xia and Wang [46]. Proof. Let where β 1 > 0 and β 2 > 0 are Lipchitz constants of the operators T and B respectively. This shows that the operator H(u) is Lipchitz continuous in K. So, for each u 0 ∈ K, there exists a continuous and unique solution u(t) of the dynamical system (9), which is defined in an interval t 0 ≤ t ≤ T with initial condition u(t 0 ) = u 0 . Let its maximal interval of existence be [t 0 , T ). For any u ∈ H, consider where the Lipchitz continuity of the operators T and B with constants β 1 > 0, β 2 > 0, is used.
Hence, by using Gronwall's lemma 3.2, we have This proves that the solution u(t) is bounded on the interval [t 0 , S). Hence, S = ∞.
We now study the stability of dynamical system (9) by using the technique of Noor [33] and Xia and Wang [46]. Proof. Since the operators T and B are Lipchitz continuous with constants β 1 > 0, β 2 > 0. It follows from Theorem 3.1, that the dynamical system (9) has a continuous and unique solution u(t) over the interval [t 0 , S) for any fixed u 0 ∈ K. Let u(t) = u(t, t 0 ; u 0 ) be the solution of the initial value problem (9). For a given u ∈ K, consider the following Lyapunov function It can be observed that lim n−→∞ L(u n ) = +∞ whenever the sequence u n ⊂ K and lim n→∞ u n = +∞.
Therefore, it can be concluded that the level sets of L are bounded.
Let u ∈ K be a solution of (1). Then Using the pseudomonotonicity of the operators T and B in (14), we obtain Taking (15), we have Setting v = u, in (6), we get Adding (16) and (17) and using (8), we get which shows that Hence from (9), (13) and (19), we get This proves that L(u) is global Lyapunov function for the dynamical system (9) and hence the dynamical system (9) is Lyapunov stable. Since, we have is continuously differentiable function on the closed and bounded set K, LaSalle's invariance principle shows the convergence of the trajectory to the largest subset ϕ of the following subset: then, from (20), we have Using (21) in (9), we get du dt = 0, which implies that u is an equilibrium point of the system (9).
Thus, we conclude that where the nonempty set X is convex and invariant set which is contained in the solution set K. So, Using definition 7 and (22), we obtain the global convergence of the dynamical system (9) to the solution set of (1). In particular, if X = { u}, then lim n→∞ u(t) = u, which proves that the dynamical system (9) is globally asymptotically stable.
Theorem 3.3. Let the operators T and B be Lipchitz continuous with constants β 1 > 0 and β 2 > 0, respectively. If γ < 0, then the projected dynamical system (9) globally exponentially converges to the unique solution of the absolute value variational inequality (1).
Proof. It can be seen from Theorem 3.1 that a unique and continuously differentiable solution of the dynamical system (9) exists over the interval [t 0 , ∞). So, where u ∈ K is the solution of the absolute value variational inequality (1). So, we have From the nonexpansivity of P K and the Lipchitz continuity of the operators T and B, we have From (23) and (24), we obtain where θ = 1 + ρ(β 1 + β 2 ).
Thus, for γ = −γ 1 , where γ 1 is a positive constant, we get Hence, using definition 8, it is proved that the trajectory of the solution of the system (9) will converge globally exponentially to the unique solution of the absolute value variational inequality (1).

Iterative Methods
In this section, we use the projected dynamical system (9) associated with absolute variational inequality (1) to suggest and analyze some iterative schemes, which will be used to obtain the solution of absolute value variational inequalities. For the numerical computations of the suggested algorithms, the projection operator, P K , is defined by Consider the dynamical system (9) with γ = 1, For a given η ∈ [0, 1], using forward difference schemes, we discretize (24) and suggest the following iterative algorithms to find the solution of absolute value variational inequalities.
Algorithm 4.1. For a given u 0 ∈ K, compute u n+1 by the iterative scheme which is an implicit method.
For the implementation of Algorithm 4.1, we suggest the following two-step, iterative method.
Algorithm 4.2. For a given u 0 ∈ K, compute u n+1 by the iterative schemes Algorithm 4.2 is a new two-step iterative method for finding the solution of absolute value variational inequality (1).
We also consider some special cases of Algorithm 4.2.
(1) For η = 0, Algorithm 4.2 collapses to the following iterative method. Algorithm 4.3. For a given u 0 ∈ K, compute u n+1 by the iterative scheme Algorithm 4.3 is extragradient type methods in the sense of Korpelevich [18].
Algorithm 4.4. For a given u 0 ∈ K, compute u n+1 by the iterative scheme Algorithm 4.4 is modified extragradient type iterative method in the sense of Noor [35,36].
Algorithm 4.5. For a given u 0 ∈ K, compute u n+1 by the iterative scheme Clearly, Algorithm 4.2 contains the known iterative methods such as extragradient method by Korpelevich [18], modified extragradient method by Noor [35,36] as special cases. This shows that Algorithm 4.2 is quite general and unifying one.
We now discuss the convergence of Algorithm 4.2 which is the main motivation of the next result.
Theorem 4.1. Let u ∈ K be a solution of absolute value variational inequality (1). Let u n+1 be the approximate solution obtained from (26). If T and B are monotone operators, then Proof. Let u ∈ K be a solution of absolute value variational inequality (1) i.e; Using monotonicity of the operators T and B, we get Taking v = u n+1 in (28), we obtain Using lemma 2.1, Algorithm 4.1 can be rewritten in the following equivalent form, that is Subtituting v=u in (30) to have, From (29) and (31), we obtain From (32) and using inequality which is (27), the required result. Proof. Let T and B be a monotone operators. Then, from (27), it follows that sequence which shows that lim n→∞ ||u n+1 − u n || 2 = 0.
Since {u j } ∞ j=1 is bounded, so there exists a limit point u ∈ K such that subsequence {u j l } ∞ l=j converges to. Replacing u n by u nj in (32) and taking the limit u i −→ ∞ , we obtain which implies that u is the solution of (1)and It follows from the above inequality that the limit point u n is unique and which completes the proof.

Computational Results
In this section, we consider two numerical examples to examine the efficiency and implementation of the with random initial guess.
The comparison between Algorithm 16, the Yong method [49] and Algorithm 2.1 [8] is presented in Table   1. The average times taken by CPU for every order of n are presented by TOC, in Table 1. Note that for any size of dimension n, Algorithm 16 converges faster than both the Yong method [49] and Algorithm 2.1 [8]. We randomly select a matrix A according to the following structure: A = round(100 * (eye(n; n) − .02 * (2 * rand(n; n) − 1)); B = eye(n); b(i) = (−i) i , i = 1, 2, ..., n. From Table 2, all the iteration steps (linter number) and the elapsed CPU times show that Algorithm 16 is more efficient than the P-SSOR, P-JACOBI and P-CG iteration methods.

Conclusion
In this paper, we have considered and analyzed dynamical system associate with absolute value variational inequalities. We have discussed the existence of a solution of the absolute value variational inequalities using only the Lipschitz continuity of the underlying operators. Asymptotic stability of the solution is investigated using the Lyapunov functions. Dynamical system is used to consider some extrgradient type methods for solving the absolute value equations. Some numerical examples are given, which shows that the proposed methods perform better then the previous ones. Ideas and techniques of this paper may inspire further research in this dynamical field.