New classes of exponentially general nonconvex variational inequalities

ABSTRACT


Introduction
Variational inequalities theory, which was introduced by Stampacchia [1], can be viewed as a natural generalization and extension of the variational principles. It is tool of great power that can be applied to a wide variety of problems, which arise in almost all branches of pure, applied, physical, regional and engineering sciences. During this period, variational inequalities have played an important, fundamental and significant part as a unifying influence and as a guide in the mathematical interpretation of many physical phenomena. In fact, it has been shown that the variational inequalities provide the most natural, direct, simple and efficient framework for the general treatment of wide range of problems. Variational inequalities have been extended and generalized in several directions for studying a wide class of equilibrium problems arising in financial, economics, transportation, elasticity, optimization, pure and applied sciences, see [2], [3], [4]- [7], [8], [9]- [22], [23]- [36], [37], [38], [1], [39], [40] and the references therein. An im-portant and useful generalization of variational inequalities is called the general(Noor) variational inequality introduced by Noor [17] in 1988, which enables us to study the odd-order and nonsymmetric problems arising in physical oceanography, engineering and mathematical sciences a unified framework. See also [15], [11], [18]- [22], [13], [35]- [37] for the applications of general variational inequalities and related optimization problems. It is worth mentioning that almost all the results regarding the existence and iterative schemes for variational inequalities, which have been investigated and considered in the classical convexity. This is because all the techniques are based on the properties of the projection operator over convex sets, which may not hold in general, when the sets are nonconvex. Bounkhel at al. [2] and Noor [11]- [14] introduced and considered some new classes of variational inequalities, which are called the nonconvex variational inequality in A R T I C L E I N F O A B S T R A C T It is known that the accurate inequalities can be derived using the algorithmically convex functions. Exponentially convex(concave) functions are closely related to the log-convex(concave) function. The origin of exponentially convex functions can be traced back to Bernstein [41]. Avriel [42] introduced and studied the concept of r-convex functions. Exponentially convex functions have important applications in information theory, big data analysis, machine learning and statistic, see [43], [44], [42], [41], [37], [45] and the references therein. Noor et al [46]- [49], [31], [32], [35] considered the concept of exponentially convex functions and discussed the basic their properties. It is worth mentioning that these exponentially convex functions considered by Noor et al [17] are distinctly differ-ent from the exponentially convex functions considered and studied by Bernstein [41]. It have been shown that the exponentially functions enjoy the same interesting properties which convex functions have. It have shown that the minimum of the differentiable exponentially convex functions can be characterized by the exponentially variational inequalities. Noor et al. [46], [48], [31], [32] introduced and studied some new classes of variational inequalities, which are called exponentially variational inequalities.
It is worth mentioning that the general variational inequalities, nonconvex varia-tional inequalities and exponentially variational inequalities are quite distinct classes of variational inequalities along with different applications. It is natural to develop a unified framework for these classes of variational inequalities. These facts and obser-vations motivated us to consider some new classes of exponentially general nonconvex variational inequalities(EGNVI).
In Section 2, we formulate the problem and discuss its special cases along with pre-liminaries results. Equivalence between EGNVI and fixed point problem is establish in Scetion3. This alternative equivalent formulation is used to discuss the existence of the solution. We shown that this alternative formulation played an important role is suggesting inertial type iterative methods for solving EGNVI. Convergence analy-sis is analyzed under suitable conditions. Wiener-Hopf technique is used to discuss some iterative for solving EGNVI in Section 4. In Section 5, we consider the second order initial value problem associated with EGNVI. Using the forwardbackward finite difference scheme, we suggest and investigate some iterative methods for solving the EGNVI along with convergence criteria. Some special cases are also pointed as poten-tial applications of the obtained results. We have only considered theoretical aspects of the suggested methods. It is an interesting problem to implement these methods and to illustrate the efficiency. Comparison with other methods need further research efforts. The ideas and techniques of this paper may be extended for other classes of quasi variational inequalities and related optimization problems

Basic Concepts and Formulation
Let H be a real Hilbert space whose inner product and norm are denoted by (.,.) and ||.|| respectively. Let K be a nonempty closed convex set in H.
For given nonlinear operators , : → consider the problem of finding ∪ ∈ K, where K is a convex set in H, such that: Which is called the exponentially general variational inequality, see [32]. We now show that the problem (2.1) arises as an optimality condition of the differen-tiable exponentially general convex functions. To be more precise, we recall the well known concepts and results for the sake of completeness and to convey the main ideas for the readers. which are mainly due to Noor and Noor [13], [14], [46], [32].
It is important to emphasize that, if g = I, the identity operator, then the general convex set = K, the classical convex set. Every convex set is a general convex set, but the converse is not true. [46] A function F is said to be exponentially general convex function with to respect to an arbitrary function g, if

Definition (2.2)
We remark that Definition 2.6 can be rewritten in the following equivalent way, Definition (2.3) [13], [46] A function F is said to be exponentially general convex func-tion with respect to an arbitrary function g, if A function is called the exponentially general concave function F , if-F is exponentially general convex function.
If g = I, then Definition 2.7 reduces to Definition (2.4) [50], [11] A function F is said to be exponentially convex function, if which can be rewritten in the equivalent form Definition (2.5) A function F is said to be exponentially convex function, if It is obvious that two concepts are equivalent. This equivalent have been used to discuss various aspects of the exponentially general convex functions. It is worth men-tioning that one can also deduce the concept of exponentially convex functions from r-convex functions, which were considered by Avriel [42] and Bernstein [41].
For the applications of the exponentially convex functions in the mathematical pro-gramming and information theory, see Antczak [44], Alirezaei et al. [43], Zhao et al. [45] and Pal et al [37]. For the applications of the exponentially concave function in the communication and information theory, we have the following example.

Example [1]:
The error function Becomes an exponentially concave function in the form erf (√x), ≥ 0, which de-scribes the bit/symbol error probability of communication systems depending on the square root of the underlying signal-to-noise ratio. This shows that the exponentially concave functions can play important part in communication theory and information theory.
Proof. Let u ∈ be a minimum of the function F. Then ( ( )) ≤ ( ( )), ∀ ∈ From which, we have Since K is a general convex set, so, ∀ , ∈ , ∈ [0,1], which is the required (2.4) Since F is differentiable exponentially general convex function, so This shows that u Kg is the minimum of the differentiable exponentially general convex function, the required result.
The inequality of the type ((2.4)) is called the exponentially general variational in-equality, which is a special case of (2.1).
The basic concepts and definitions used in this paper are exactly the same as in Noor [11], [13], [14]. Poliquin et al. [51] and Clarke et al. [3] have introduced and studied a new class of nonconvex sets, which are called uniformly prox-regular sets. Definition 2.6 [3], [51] The proximal normal cone of K at u ∈ H is given by The proximal normal cone ( ) has the following charcterization.
Where ̅̅̅ means the closure of the convex hull.
Definition 2.8. For a given r ∈ (0,∞], a subset is said to be normalized uniformly r-prox-regular, if and only if, every nonzero proximal normal cone to can be realized by an r-ball, that is, ∀ ∈ , 0 ≠ ( ), one has It is clear that the class of normalized uniformly prox-regular sets is sufficiently large to include the class of convex sets, p-convex sets, 1,1 submanifolds (possibly with boundary) of H, the images under a 1,1 diffeomorphism of convex sets and many other nonconvex sets; see [3], [11], [12], [14], [31]. Obviously, for r = ∞, the uniformly prox-regularity of is is equivalent to the convexity of K. This class of uniformly prox-regular sets have played an important part in many nonconvex applications such as optimization, dynamic systems and differential inclusions. It is known that if is a uniformly proxregular set, then the proximal normal cone ( ) is closed as a set-valued mapping.
We now recall the well known proposition which summarizes some important properties of the uniformly prox-regular sets . We now consider the exponentially general nonconvex variational inequality(EGNVI). To be more precise, for given nonlinear operators T, g : H → R, we consider the problem of finding u ∈ such that.
Which is called the .
Special caes. Some important special cases of the problem (2.7) are discussed.
(III). We note that, if = ( ), where N is a nonlinear operator, then problem (2.7) is equivalent to finding u ∈ H : g(u) ∈ K such that. which is known as the general variational inequality, introduced and studied by Noor [17] in 1988. It have shown that nonsymmterics, nonpositive and odd order obstacle boundary value problems can be studied in the unified framework of the general variational inequalities. For the applications, numerical methods, formulation and other aspects of the general variational inequalities (2.10), see [19], [21], [22], [13], [35] and the references therein.
(IV). If g ≡ I, the identity operator, then problem (2.7) is equivalent to finding which is called the exponentially variational inequality.
(V). If ≡ , the convex set in H, and ≡ , the identity operator, then problem (2.7) is equivalent to finding u ∈ K such that. () Inequality of type (2.12) is called the exponentially variational inequality.
If is a nonconvex (uniformly prox-regular) set, then problem (2.7) is equivalent to finding u ∈ such that where ( ( )) (g(u)) denotes the normal cone of Kr at g(u) in the sense of nonconvex analysis. Problem (2.14) is called the general nonconvex variational inclusion problem associated with general nonconvex variational inequality (2.7). This equivalent formulation plays a crucial and basic part in this paper. We would like to point out this equivalent formulation allows us to use the projection operator technique for solvin the exponentially general nonconvex variational inequalities of the type (2.7). (ii).
It is well known that exponentially general monotonicity implies exponentially general pseudomonotonicity, but, the converse is not tru

Methods
In this section, we prove that the EGNVI are equivalent to the fixed point problems. This alternative equivalent formulation is used to discuss the existence results and propose several iterative methods for solving the EGNVI. Convergence analysis is also analyzed under suitable conditions. First of all, we establish the equivalence between ENGVI (2.7) and fixed point formulation. This is the main motivation of nest result.
is a solution of the exponentially general nonconvex variational inequality (2.7), if and only if, u ∈ satisfies the relation where is the projection of H onto the uniformly prox-regular set .
Proof. Let u ∈ . be a solution of (2.7). Then, for a constant ρ > 0, where we have used the well-known fact ≡ ( + ) −1 , the required results.
Lemma 3.1 implies that the exponentially general nonconvex variational inequality (2.7) is equivalent to the fixed point problem (3.1). This alternative equivalent formulation is very useful from the numerical and theoretical points of view. We rewrite the relation (3.1) in the following form which is used to study the existence of a solution of the exponentially general nonconvex variational inequality (refeq1.1).
We now study those conditions under which the exponentially general nonconvex variational inequality (2.7) has a solution and this is the main motivation of our next result.
then there exists a solution of the exponentially general nonconvex variational inequality (2.7).
Thus it is enough to show that the map F(u), defined by (3.2), has a fixed point. For all u 6= v ∈ , we have where we have used the fact that the operator is a Lipschitz continuous operator with constant δ.
Since the operator T is strongly exponentially general monotone with constant α > 0 and Lipschitz continuous with constant β > 0, it follows that In a similar way, we have where σ > 0 is the strongly monotonicity constant and δ > 0 is the Lipschitz continuity constant of the operator g respectively.  .4), it follows that θ < 1, which implies that the map F(u) defined by (3.2), has a fixed point, which is a unique solution of (2.7).
This fixed point formulation (3.1) is used to suggest the following iterative method for solving the exponentially general nonconvex variational inequality (2.7). Using the fixed point formulation (3.1), we suggest and analyze the several iterative methods for solving the exponentially general nonconvex variational inequality (2.7). The equation (3.1)can be rewritten equivalently as where α, ξ ∈ [0, 1] are constants. This equivalent formulation is used to suggest and analyzed the following iterative methods for solving the problem (2.7).
For α = 0, ξ = 0, Algorithm 3.1 reduces to Algorithm 3.2. For a given u0 ∈ H, find the approximate solution un+1 by the iterative schemes Which is called the explicit iterative method.
Algorithm 3.3 is an implicit iterative method for solving the exponentially general nonconvex variational inequalities (2.7).
To implement the Algorithm 3.3, we use the predictor-corrector technique. We use Algorithm 3.2 as predictor and Algorithm 3.3 as a corrector to obtain the following predictor-corrector method for solving the exponentially general nonconvex variational inequality (2.7). Algorithm 3.4 is known as the extragradient method in the sense of Korpelevich [8]. It is obvious that the implicit method (Algorithm 3.3 ) and the extragradient method (Algorithm 3.4 ) are equivalent. We use this equivalent formulation to study the convergence analysis of Algorithm 3.3 and this is the motivation of our nest result.
We now consider the convergence analysis of Algorithm 3.3 and this is the main motivation of our next result.
∈ be a solution of (2.7) and let +1 be the approximate solution obtained from Algorithm 3.3. If the operator T is exponentially general pseudomonotone, then be solution of (2.7). Then, using the exponentially general pseudomonotonicity of T, we have Taking v = u in (5.11), we have From (5.13) and (5.11), we have The required results (3.12).   To implement the Algorithm 3.5, we use the predictor-corrector technique. We use Algorithm 3.2 as predictor and Algorithm 3.5 as a corrector to obtain the following predictor-corrector method for solving the exponentially general nonconvex variational inequality (2.7). Algorithm 3.6. For a given 0 ∈ , find the approximate solution +1 by the iterative schemes Algorithm 3.6 is known as the modified extragradient method in the sense of Noor [18], [19]. We would like to remark that this modified extragradient method is quite different than the extragradient method, which was suggested by Korpelevich [8]. Here we would like to point out that the implicit method (Algorithm 3.5) and the extragradient method (Algorithm 3.6 ) are equivalent. We use this equivalent to prove the convergence of the implicit projection method (Algorithm 3.5), which requires only the partially relaxed strongly monotonicity. We now consider the convergence analysis of Algorithm 3.5. Proof. Let u ∈ H : g(u) ∈ be solution of (2.7). Then Proof. Let ̅ ∈ ∶ (̅) ∈ be a solution of (2.7). Then, the sequences {|| ( ) − (̅)||} is nonincreasing and bounded and.  which are called midpoint implicit methods for solving the exponentially general nonconvex variational inequalities. Using the above techniques and ideas of this paper, one can consider the convergence analysis of these algorithms. We again rewrite the equation (3.1) in the equivalently as: This equivalent fixed point formulations is used to suggest the following iterative method Algorithm 3.9. For given , 1 ∈ , find the approximate solution +1 by the iterative schemes.
which is known as the inertial implicit iterative method for solving the problem (3.1). In a similar way, we can propose the following method. This method is called the extragradient method of Korpelevich [8]. In this paper, we suggest and analyze the following two-step iterative method for solving the exponentially general nonconvex variational inequalities (2.7). Clearly for = = 1, Algorithm 3.11 reduces to Algorithm 3.6. It is worth mentioning that, if r = ∞, then the nonconvex set reduces to a convex set K. Consequently all the Algorithms collapse to the following algorithms for solving the general variational inequalities.
We now consider the convergence analysis of Algorithm 3.11 and this is the main motivation of our next result. In a similar way, one can consider the convergence criteria of other Algorithms. which is another fixed point formulation. This fixed-point formulation is used to suggest the following three-step iterative method for solving the nonconvex variational inequality (2.7). We would like to mention that three-step iterative methods are also known as Noor iteration for solving the variational inequalities and equilibrium problems. One can easily consider the convergence criteria of Algorithm 3.12 using the technique of this paper.
In a similar way, we can suggest the following inertial four step Noor iterations: It is worth mentioning that for different and suitable choice of the constants one can easily show that the Noor iterations three-step and Noor inertial four step iterations include the Mann and Ishikawa iterations as special cases. Thus we conclude that Noor iterations are more general and unifying ones.

Wiener-Hopf Equations Technique
We now consider the problem of solving the nonlinear Wiener-Hopf equations. To be more precise, let = − , where is the projection operator, and I is the identity operator. For given nonlinear operators T, g, consider the problem of finding z ∈ H such that where −1 is the inverse of the operator g. Equations of the type (4.1) are called the exponentially general nonconvex Wiener-Hopf equations. Note that, if r = ∞, then the Wiener-Hopf (4.1) equations are exactly the same Wiener-Hopf equations associated with the general variational inequalities (2.7). For g ≡ I, the identity operator and r = ∞, one can obtain the original Wiener-Hopf equations which were introduced and studied by Shi [39] in conjunction with the variational inequalities. This shows that the original Wiener-Hopf equations are the special case of the exponentially general nonconvex Wiener-Hopf equations (4.1). The Wiener-Hopf equations technique has been used to study and develop several iterative methods for solving variational inequalities and related optimization problems.
We first establish the equivalence between the EGNVI (2.7) and the Wiener-Hopf (4.1) using essentially the projection method. This equivalence is used to suggest and analyze some iterative methods for solving the problem (2.7). This fixed point formulation enables us to suggest the following iterative method for solving the exponentially general nonconvex variational inequality (2.7). Using this fixed point formulation, we suggest the following iterative method. We would like to point out that one can obtain a number of iterative methods for solving the exponentially general nonconvex variational inequality (2.7) for suitable and appropriate choices of the operators T, g and the space H. This shows that iterative methods suggested in this paper are more general and unifying ones. We now study the convergence analysis of Algorithm 4.1. In a similar way, one can analyze the convergence analysis of other iterative methods. Proof. Let u ∈ H be a solution of (2.7). Then, using Lemma 4.2, we have where 0 ≤ ≤ 1, ∑ = ∞.

Dynamical Systems
In this section, we consider the projected dynamical system associated with the general variational inequalities. Using the fixed-point formulation of the variational inequalities, Dupuis et al. [5] introduced and considered the projected dynamical systems, which the right hand side of the ordinary differential equation is a projected operator associated with variational inequalities. The innovative and novel feature of a projected dynamical system is that its set of stationary points corresponds to the set of solutions of the corresponding variational inequality problem. Hence, equilibrium and nonlinear problems arising in various branches in pure and applied sciences, which can be formulated in the setting of the variational inequalities, can now be studied in the more general setting of dynamical systems. It have been shown [10], [5], [29], [35] that the dynamical systems are useful in developing some efficient numerical techniques for solving variational inequalities and related optimization problems. In recent years, much attention has been given to study the globally asymptotic stability of these projected dynamical systems. In Section 3, we have shown that the exponentially general nonconvex variational inequalities are equivalent to the fixedpoint. We use this equivalent formulation to suggest and analyze the projected dynamical system associated with the exponentially general nonconvex variational inequalities (2.7).
where λ is a parameter. The system of type (5.1) is called the exponentially projected general dynamical system. Here the right hand side is related to the projection operator and is discontinuous on the boundary. It is clear from the definition that the solution to (5.1) always stays in the constraint set. This implies that the qualitative results such as the existence, uniqueness and continuous dependence of the solution on the given data can be studied.
The equilibrium points of the dynamical system (5.1) are naturally defined as follows. Thus it is clear that ∈ , ( ) ∈ is a solution of the general variational inequality (2.7), if and only if, ∈ , ( ) ∈ is an equilibrium point. In a similar way, one can define the concept of equilibrium points for other dynamical systems.

Definition 5.2
The dynamical system is said to converge to the solution set * * of (2.7), if , irrespective of the initial point, the trajectory of the dynamical system satisfies. If the dynamical system is still stable at * in the Lyapunov sense, then the dynamical system is globally asymptotically stable at * . Definition 5.3. The dynamical system is said to be globally exponentially stable with degree η at * , if, irrespective of the initial point, the trajectory of the system satisfies. This shows that the solution is bounded on [ 0 , 1 ). 1  Proof. Since the operators T, g are both Lipschitz continuo, it follows from Theorem 5.1 that the dynamical system (5.1) has unique solution u(t) over [ 0 , 1 ) for any fixed 0 ∈ . Let u(t) be a solution of the initial value problem (5.1). For a given * ∈ satisfying (2.7), consider the Lyapunov function. Where * ∈ is a solution of (2.7). Thus Using the Lipschitz continuity of the operators T, g, we have which shows that the trajectory of the solution of the dynamical system (5.1) converges globally exponentially to the unique solution of the exponentially general nonconvex variational inequality (2.7).
We now use the fixed point formulation to suggest and consider a new second order projection dynamical system associated with exponentially general nonconvex variational inequalities (2.7). We use this dynamical system to suggest and investigate some inertial proximal methods for solving the problem (2.7). These inertial implicit methods are constructed using the central finite difference schemes and its variant forms. To be more precise, we consider the problem of finding u ∈ H such that where γ > 0 and ρ > 0 are constants. Problem (5.6) is called second order dynamical system.
If γ = 0, then dynamical system (5.6) reduces to dynamical system (5.1). We discretize the second-order dynamical systems (5.6) using central finite difference and backward difference schemes to have. which is the extragradient method of Korpelevich [8] for solving the exponentially general nonconvex variational inequalities. Algorithm 5.1 is an implicit method. To implement the implicit method, we use the predictor-corrector technique to suggest the two-step inertial method.
Algorithm 5.2. For given 0 , 1 ∈ H, compute +1 by the iterative scheme Where is a constant. Similarly, we suggest the following iterative method. which is known as the double projection method, introduced and studied by Noor [13] and can be written as Where > 0, > 0 .  From which, for = 0, ℎ = 1, we have Algorithm 5.5. For a given 0 , 1 ∈ , compute +1 by the iterative scheme Or equivalenty Algorithm 5.6. For a given 0 , 1 ∈ , compute +1 by the iterative scheme which is called the new inertial iterative method for solving the exponentially general nonconvex variational inequality.
We discretize the second-order dynamical systems (5.6) using central finite difference and backward difference schemes to have Where h is the step size Using this discrete form, we can suggest the following an iterative method for solving the variational inequalities (2.7). Algorithm 5.7 is called the inertial proximal method for solving the exponentially general nonconvex variational inequalities and related optimization problems. This is a new proposed method.
We can rewrite the Algorithm 5.7 in the equivalent form as follows: We note that, for γ = 0, Algorithm 5.8 reduces to the following iterative method for solving the exponentially general nonconvex variational inequalities (2.7). We again discretize the second-order dynamical systems (5.6) using central difference scheme and forward difference scheme to suggest the following inertial proximal method for solving (2.7). Algorithm 5.10 is an proximal method for solving the nonconvex variational inequalities. Such type of proximal methods were suggested by Noor [48] using the fixed point problems. In brief, by suitable descritization of the second-order dynamical systems (5.6), one can construct a wide class of explicit and implicit method for solving inequalities. We now consider the convergence criteria of the Algorithm 5.8 using the technique of Alvarez [50], Noor [21] and Noor et al. [35]. Taking v = u in (5.9), we have be a solution of exponentially general nonconvex variational inequality (2.7). Let +1 be the approximate solution obtained from (5.9). If the operator T is monotone and −1 exists, then un+1 converges to u ∈ satisfying (2.7).
Proof. Let u ∈ be a solution of (2.7). From (5.10), it follows that the sequence {||u-||} is nonincreasing and consequently,{ } is bounded. Also from(5.10), we have. where we have used the fact that −1 exist. Since sequence { } =1 ∞ is bounded, so there exists a cluster point ̂ to which the subsequence { } = ∞ converges. Replacing by in (3.2), taking the limit as → ∞ and using (5.14),we have In this section, we have applied the dynamical systems associated with exponentially general nonconvex variational inequalities to suggest some iterative schemes for solving the variational inequalities. We have only considered the theoretical aspects of the proposed iterative methods. It is an interesting open problem to consider the implementable of these numerical methods. Comparison with other methods need further research efforts.