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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 657278, 8 pages
Proximal Point Methods for Solving Mixed Variational Inequalities on the Hadamard Manifolds
Mathematics Department, COMSATS Institute of Information Technology, Islamabad 44000, Pakistan
Received 21 March 2012; Accepted 28 March 2012
Academic Editor: Yonghong Yao
Copyright © 2012 Muhammad Aslam Noor and Khalida Inayat Noor. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We use the auxiliary principle technique to suggest and analyze a proximal point method for solving the mixed variational inequalities on the Hadamard manifold. It is shown that the convergence of this proximal point method needs only pseudomonotonicity, which is a weaker condition than monotonicity. Some special cases are also considered. Results can be viewed as refinement and improvement of previously known results.
In recent years, much attention has been given to study the variational inequalities and related problems on the Riemannian manifold and the Hadamard manifold. This framework is a useful for the developments of various fields. Several ideas and techniques from the Euclidean space have been extended and generalized to this nonlinear framework. The Hadamard manifolds are examples of hyperbolic spaces and geodesics; see [1–7] and the references therein. Németh , Tang et al. , and Colao et al.  have considered the variational inequalities and equilibrium problems on the Hadamard manifolds. They have studied the existence of a solution of the equilibrium problems under some suitable conditions. To the best of our knowledge, no one has considered the auxiliary principle technique for solving the mixed variational inequalities on the Hadamard manifolds. In this paper, we use the auxiliary principle technique to suggest and analyze a proximal iterative method for solving the mixed variational inequalities. If the nonlinearity in the mixed variational inequalities is an indicator function, then the mixed variational inequalities are equivalent to the variational inequality on the Hadamard manifold. This shows that the results obtained in this paper continue to hold for variational inequalities on the Hadamard manifold, which is due to Tang et al.  and Németh . We hope that the technique and idea of this paper may stimulate further research in this area.
Let be a simply connected -dimensional manifold. Given , the tangent space of at is denoted by and the tangent bundle of by , which is naturally a manifold. A vector field on is a mapping of into which associates to each point , a vector . We always assume that can be endowed with a Riemannian metric to become a Riemannian manifold. We denote by the scalar product on with the associated norm , where the subscript will be omitted. Given a piecewise smooth curve joining to (i.e., and ) by using the metric, we can define the length of as . Then, for any the Riemannian distance , which includes the original topology on , is defined by minimizing this length over the set of all such curves joining to .
Let be the Levi-Civita connection with . Let be a smooth curve in . A vector field is said to be parallel along if . If itself is parallel along , we say that is a geodesic and in this case is constant. When is said to be normalized. A geodesic joining to in is said to be minimal if its length equals .
A Riemannian manifold is complete, if for any all geodesics emanating from are defined for all . By the Hopf-Rinow theorem, we know that if is complete, then any pair of points in can be joined by a minimal geodesic. Moreover, is a complete metric space, and bounded closed subsets are compact.
Let be complete. Then the exponential map at is defined by for each , where is the geodesic starting at with velocity Then for each real number .
A complete simply connected Riemannian manifold of nonpositive sectional curvature is called a Hadamard manifold. Throughout the remainder of this paper, we always assume that is an -manifold Hadamard manifold.
We also recall the following well-known results, which are essential for our work.
Lemma 2.1 (see ). Let . Then is a diffeomorphism, and for any two points , there exists a unique normalized geodesic joining to , which is minimal.
So from now on, when referring to the geodesic joining two points, we mean the unique minimal normalized one. Lemma 2.1 says that is diffeomorphic to the Euclidean space . Thus has the same topology and differential structure as . It is also known that the Hadamard manifolds and Euclidean spaces have similar geometrical properties. Recall that a geodesic triangle of a Riemannian manifold is a set consisting of three points and three minimal geodesics joining these points.
Lemma 2.2 (see (comparison Theorem for Triangles [2, 3, 5])). Let be a geodesic triangle. Denote, for each , by the geodesic joining to , and , the angle between the vectors and , and . Then In terms of the distance and the exponential map, the inequality (2.2) can be rewritten as since
Lemma 2.3 (see ). Let be a geodesic triangle in a Hadamard manifold . Then, there exist such that The triangle is called the comparison triangle of the geodesic triangle , which is unique up to isometry of .
From the law of cosines in inequality (2.3), we have the following inequality, which is a general characteristic of the spaces with nonpositive curvature : From the properties of the exponential map, we have the following known result.
Lemma 2.4 (see ). Let and such that . Then the following assertions hold.(i)For any, (ii)If is a sequence such that and , then .(iii)Given the sequences and satisfying , if and , with , then
A subset is said to be convex if for any two points , the geodesic joining and is contained in ; that is, if is a geodesic such that and , then . From now on will denote a nonempty, closed and convex set, unless explicitly stated otherwise.
A real-valued function defined on is said to be convex if, for any geodesic of , the composition function is convex; that is,
The subdifferential of a function is the set-valued mapping defined as and its elements are called subgradients. The subdifferential at a point is a closed and convex (possibly empty) set. Let denote the domain of defined by
The existence of subgradients for convex functions is guaranteed by the following proposition; see .
For a given single-valued vector field and a real-valued function , we consider the problem of finding such that which is called the mixed variational inequality. This problem was considered by Colao et al. . They proved the existence of a solution of problem (2.12) using the KKM maps. For the applications, formulation, and other aspects of the mixed variational inequalities in the linear setting, see [8–16].
We remark that if the function is an indicator of a closed and convex set in , then problem (2.12) is equivalent to finding such that which is called the variational inequality on the Hadamard manifolds. Németh , Colao et al.  and Udrişte  studied variational inequalities on the Hadamard manifold from different point of views. In the linear setting, variational inequalities have been studied extensively; see [8–25] and the references therein.
Definition 2.6. An operator is said to be speudomonotone with respect a mapping , if and only if
3. Main Results
For a given satisfying (2.12), consider the problem of finding such that which is called the auxiliary mixed variational inequality on the Hadamard manifolds. We note that if , then is a solution of the mixed variational inequality (2.12). This observation enable to suggest and analyzes the following proximal point method for solving the mixed variational inequality (2.12).
Algorithm 3.1. For a given , compute the approximate solution by the iterative scheme: Algorithm 3.1 is called the implicit (proximal point) iterative method for solving the mixed variational inequality on the Hadamard manifold.
If , then Algorithm 3.1 collapses to the following algorithm:
Algorithm 3.2. For a given , find the approximate solution by the iterative scheme. which is known as the proximal pint method for solving the mixed variational inequalities. For the convergence analysis of Algorithm 3.2, see [11, 12].
Algorithm 3.3. For a given , compute the approximate solution by the iterative scheme
We would like to mention that Algorithm 3.1 can be rewritten in the following equivalent form.
Algorithm 3.4. For a given , compute the approximate solution by the iterative scheme: which is called the extraresolvent method for solving the mixed variational inequalities on the Hadamard manifolds.
In a similar way, one can obtain several iterative methods for solving the variational inequalities on the Hadamard manifold.
We now consider the convergence analysis of Algorithm 3.1, and this is the main motivation of our next result.
Theorem 3.5. Let be a pseudomonotone vector field. Let be the approximate solution of the mixed variational inequality (2.12) obtained from Algorithm 3.1; then where is the solution of the mixed variational inequality (2.12).
Proof. Let be a solution of the mixed variational inequality (). Then, by using the pseudomonotonicity of the vector filed, , we have Taking in (3.7), we have Taking in (3.2), we have From (3.8) and (3.9), we have For the geodesic triangle the inequality (3.10) can be written as, Thus, from (3.10) and (3.11), we obtained inequality (3.6), the required result.
Proof. Let be a solution of (2.12). Then, from (3.6), it follows that the sequence is bounded and it follows that Let be a cluster point of . Then there exists a subsequence such that converges to . Replacing by in (3.2), taking the limit, and using (3.13), we have This shows that solves (2.12) and which implies that the sequence has unique cluster point and is a solution of (2.12), the required result.
We have used the auxiliary principle technique to suggest and analyzed a proximal point iterative method for solving the mixed quasi-variational inequalities on the Hadamard manifolds. Some special cases are also discussed. Convergence analysis of the new proximal point method is proved under weaker conditions. Results obtained in this paper may stimulate further research in this area. The implementation of the new method and its comparison with other methods is an open problem. The ideas and techniques of this paper may be extended for other related optimization problems.
The authors would like to thank Dr. S. M. Junaid Zaidi, Rector, the COMSATS Institute of Information Technology, Islamabad, Pakistan, for providing excellent research facilities.
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