Implicit Schemes for Solving Extended General Nonconvex Variational Inequalities

We suggest and analyze some implicit iterative methods for solving the extended general nonconvex variational inequalities using the projection technique. We show that the convergence of these iterative methods requires only the gh-pseudomonotonicity, which is a weaker condition than gh-monotonicity. We also discuss several special cases. Our method of proof is very simple as compared with other techniques.


Introduction
Variational inequalities, which were introduced and studied in early sixties, contain wealth of new ideas. Variational inequalities can be considered as a natural extension of the variational principles. It is well known that the variational inequalities characterize the optimality conditions of the differentiable convex functions on the convex sets in normed spaces. In recent years, Noor 1-6 has introduced and studied a new class of variational inequalities involving three different operators, which is called the extended general variational inequalities. Noor 1-6 has shown that the minimum of a differentiable nonconvex gh-convex function on the nonconvex set gh-convex can be characterized by the class of extended general variational inequalities. The class of extended general variational include the general variational inequalities 1-33 and variational inequalities as special cases. This clearly shows that the extended general variational inequalities are more general and unifying ones. For applications, physical formulation, numerical methods, and other aspects of variational inequalities, see 1-35 and the references therein. However, all the work carried out in this direction assumes that the underlying set is a convex set. In many practical situations, a choice set may not be a convex set so that the existing results may not be applicable. To handle such situations,  has introduced and considered a new class of variational inequalities, called the general nonconvex variational inequality on the uniformly prox-regular sets. It is well known that uniformly prox-regular sets are nonconvex and include the convex sets as special cases, see 8,9,32 . Using the projection operator, Noor 27 proved a new characterization of the projection operator for the prox-regular sets. Using this characterization, one can easily show that nonconvex projection operator is Lipschitz continuous, which is a new result. Using this new characterization of the projection of the prox-regular sets, one can establish the equivalence between the nonconvex variational inequalities and the fixed point problems. This equivalence is useful to study various concepts for the nonconvex variational inequalities.
Motivated and inspired by the recent activities in this dynamic field, we consider the extended general noncomvex variational inequalities on the prox-regular sets. We use the projection technique to establish the equivalence between the extended general nonconvex variational inequalities and the fixed point problems. We use this alternative formulation to some unified implicit and extragradient methods for solving the extended general nonconvex variational inequalities. These new methods include the modified projection method of Noor 27 and the extragradient method of Korpelevič 11 as special cases. The main motivation of this paper is to improve the convergence criteria. We show that the convergence of the implicit iterative methods requires only the gh-pseudomonotonicity, which is weaker condition that gh-monotonicity. It is worth mentioning that we do not need the Lipschitz continuity of the operator. In this sense, our result represents an improvement and refinement of the known results. Our method of proof is very simple.

Basic Concepts
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. The basic concepts and definitions used in this paper are exactly the same as in Noor 20,22 . We now recall some basic concepts and results from nonsmooth analysis 9, 32 . Definition 2.1 see 9, 32 . The proximal normal cone of K at u ∈ H is given by where α > 0 is a constant and Here d K · is the usual distance function to the subset K, that is, The proximal normal cone N P K u has the following characterization.
Definition 2.3. The Clarke normal cone, denoted by N C K u , is defined as where co means the closure of the convex hull. Clearly N P K u ⊂ N C K u , but the converse is not true. Note that N P K u is always closed and convex, whereas N C K u is convex, but may not be closed 32 .

Definition 2.4 see 29 .
For a given r ∈ 0, ∞ , a subset K r is said to be normalized uniformly r-prox-regular if and only if every nonzero proximal normal to K r can be realized by an r-ball, that is, for all u ∈ K r and 0 / ξ ∈ N P K r u , 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, C 1,1 submanifolds possibly with boundary of H, the images under a C 1,1 diffeomorphism of convex sets, and many other nonconvex sets, see 11, 29 . It is well known 9, 32 that the union of two disjoint intervals a, b and c, d is a prox regular set with r c − b /2. Obviously, for r ∞, the uniformly prox-regularity of K r 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 K r is a uniformly prox-regular set, then the proximal normal cone N P K r u 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 K r .
For given nonlinear operators T, g, h, we consider the problem of finding u ∈ H : h u ∈ K r such that which is called the extended general nonconvex variational inequality. Here γ > 0 and ρ > 0 are constants.

Journal of Applied Mathematics
We remark that if g h, then problem 2.7 is equivalent to finding u ∈ H : g u ∈ K r such that which is called the general nonconvex variational inequality, introduced and studied by Noor 27 . We note that, if K r ≡ K, the convex set in H, then problem 2.7 is equivalent to finding u ∈ H : h u ∈ K such that which is known as the extended general variational inequality, introduced and studied by Noor 1-6 . For the applications, numerical methods, formulation, and other aspects of the extended general variational inequalities 2.8 , see 1-6, 30, 31 and the references therein.
If g ≡ h ≡ I, the identity operator, then both problems 2.7 and 2.8 are equivalent to finding u ∈ K r such that which is called the nonconvex variational inequality. For the formulation and numerical methods for the nonconvex variational inequalities, see 3, 8, 18-27 . We note that if K r ≡ K, the convex set in H, then problem 2.8 is equivalent to finding u ∈ H : g u ∈ K such that Tu, g v − g u ≥ 0, ∀v ∈ H : g v ∈ K. 2.11 Inequality of type 2.11 is called the general variational inequality involving two operators, which was introduced and studied by Noor 14 in 1988. It has been shown that the minimum of the differentiable g-convex function on a g-convex set can be characterized by the general variational inequality of type 3.4 . It has been shown that a wide class of odd-order and nonsymmetric problems can be studied via the general variational inequalities. For the numerical method, sensitivity analysis, dynamical systems, and other aspects of general variational inequalities, see 7, 15, 16 . For g I, the identity operator, one can obtain the original variational inequality, which was introduced and studied by Stampacchia 33 in 1964. It turned out that a number of unrelated obstacle, free, moving, unilateral, and equilibrium problems arising in various branches of pure and applied sciences can be studied via variational inequalities, see 1-35 and the references therein.
If K r is a nonconvex uniformly prox-regular set, then problem 2.7 is equivalent to finding u ∈ H : g u ∈ K r such that 0 ∈ ρT u h u − g u N P K r h u , 2.12 which is called the extended general nonconvex variational inclusion problem associated with general nonconvex variational inequality 2.7 . Here N P K r h u denotes the normal Journal of Applied Mathematics 5 cone of K r at h u in the sense of nonconvex analysis. This equivalent formulation plays a crucial and basic part in this paper. We would like to point out that this equivalent formulation allows us to use the projection operator technique for solving the general nonconvex variational inequalities of the type 2.7 . We now prove that the projection operator P K r has the following characterization for the prox-regular sets. This result is due to Noor 27 . We include its proof for the sake of completeness and to convey an idea of the technique. Lemma 2.6 see 27 . Let K r be a prox-regular and closed set in H. Then, for a given z ∈ H, u ∈ K r satisfies the inequality if and only if, u P K r z , 2.14 where P K r is the projection of H onto the prox-regular set K r .
Proof. Let u ∈ K r . Then, for given z ∈ H, we have

2.15
where P K r I N P K r −1 is the projection operator.
We note that, if K r ≡ K, the closed convex set, then Lemma 2.6 is a well-known result, see 10 . Using Lemma 2.6, one can easily prove that the nonconvex projection operator P K r is Lipschitz continuous.

Main Results
It is known that the extended general nonconvex variational inequalities 2.7 are equivalent to the fixed point problem. One can also prove this result using Lemma 2.6. where P K r is the projection of H onto the uniformly prox-regular set K r .
Lemma 3.1 implies that 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. Using the fixed point formulation 3.1 , we suggest and analyze the following iterative methods for solving the extended general nonconvex variational inequality 2.7 . We again use the fixed point formulation is used to suggest and analyze the following iterative method for solving 2.7 .

3.4
To implement 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 predictorcorrector method for solving the extended general nonconvex variational inequality 2.7 .
Algorithm 3.5. For a given u 0 ∈ H, find the approximate solution u n 1 by the iterative schemes g w n P K r g u n − ρT u n , h u n 1 P K r g u n − ρT w n , n 0, 1, . . . .

3.5
Algorithm 3.5 is known as the extended extragradient method. This method includes the extragradient method of Korpelevič 11 for h g I. Here we would like to point out Journal of Applied Mathematics 7 that the implicit method Algorithm 3.3 and the extragradient method Algorithm 3.5 are equivalent.
We now consider the convergence analysis of Algorithm 3.3, and this is the main motivation of our next result. Theorem 3.6. Let u ∈ H : h u ∈ K r be a solution of 2.7 and let u n 1 be the approximate solution obtained from Algorithm 3.3. If the operator T is gh-pseudomonotone, then Proof. Let u ∈ H : h u ∈ K r be a solution of 2.7 . Then since the operator T is gh-pseudomonotone. Take v u n 1 in 3.7 , we have Taking v u in 3.4 , we have From 3.8 and 3.9 , we have h u n 1 − g u n , g u − h u n 1 ρT u n 1 , g u − h u n 1 g u − h u n 1 2 ≥ 0.

3.10
It is well known that Using 3.11 , from 3.10 , one can easily obtain g u − h u n 1 2 les g u − g u n 2 − g u n − h u n 1 2 , 3.12 the required result 3.6 . Let u be a cluster point of {u n }. Then, there exists a subsequence {u n i } such that {u n i } converges to u. Replacing u n 1 by u n i in 3.4 , taking the limits in 3.4 and using 3.14 , we have This shows that u ∈ H : h u ∈ K r solves the extended general nonconvex variational inequality 2.7 and h u n 1 − g u 2 ≤ g u n − g u 2 , 3.16 which implies that the sequence {u n } has a unique cluster point, and lim n → ∞ h u n 1 g u is the solution of 2.7 , the required result.
We again use the fixed point formulation 3.1 to suggest the following method for solving 2.7 .
Algorithm 3.8. For a given u 0 ∈ H, find the approximate solution u n 1 by the iterative schemes h u n 1 P K r g u n 1 − ρT u n 1 , n 0, 1, 2, . . . , 3.17 which is also known an implicit method. To implement this method, we use the predictioncorrection technique. We use Algorithm 3.2 as the predictor and Algorithm 3.8 as the corrector. Consequently, we obtain the following iterative method.
Algorithm 3.9. For a given u 0 ∈ H, find the approximate solution u n 1 by the iterative schemes: h y n P K r g u n − ρT u n , h u n 1 P K r g y n − ρT y n , n 0, 1, 2, . . . .

3.18
Algorithm 3.9 is called the two-step or predictor-corrector method for solving the extended general nonconvex variational inequality 2.7 .
For a given step size η > 0, one can suggest and analyze the following two-step iterative method. h y n P K r g u n − ρT u n , h u n 1 P K r g u n − η g u n − g y n ρT y n , n 0, 1, 2, . . . .

3.19
Journal of Applied Mathematics 9 Note that for η 1, Algorithm 3.10 reduces to Algorithm 3.9. Using the technique of Noor 16 , one may study the convergence analysis of Algorithms 3.9 and 3.10.

Conclusion
In this paper, we have introduced and considered a new class of general variational inequalities, which is called the general nonconvex variational inequalities. Some new characterizations of the nonconvex projection operator are proved. We have established the equivalent between the general nonconvex variational inequalities and fixed point problem using the technique of the projection operator. This equivalence is used to suggest and analyze some iterative methods for solving the nonconvex general variational inequalities. Several special cases are also discussed. Results proved in this paper can be extended for multivalued and system of general nonconvex variational inequalities using the technique of this paper. The comparison of the iterative method for solving nonconvex general variational inequalities is an interesting problem for future research. We hope that the ideas and technique of this paper may stimulate further research in this field.