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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 691806, 8 pages
Explicit Iterative Method for Variational Inequalities on Hadamard Manifolds
Mathematics Department, COMSATS Institute of Information Technology, Islamabad, Pakistan
Received 2 April 2012; Revised 8 May 2012; Accepted 8 May 2012
Academic Editor: Yeong-Cheng Liou
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.
An explicit iterative method for solving the variational inequalities on Hadamard manifold is suggested and analyzed using the auxiliary principle technique. The convergence of this new method requires only the partially relaxed strongly monotonicity, which is a weaker condition than monotonicity. 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 Hadamard manifold. This framework is useful for the development of various fields on nonlinear setting. Several ideas and techniques from the Euclidean space have been extended and generalized to this nonlinear framework. Hadamard manifolds are examples of hyperbolic spaces and geodesics, see [1–7] and the references therein. Nemeth , Tang et al. , and Colao et al.  have considered the variational inequalities and equilibrium problems on Hadamard manifolds. They have studied the existence of a solution of the equilibrium problems under some suitable conditions. Several methods have been developed for solving the variational inequalities and related problems in the linear-normed spaces. The auxiliary principle technique is a powerful tool to suggest and analyze several implicit and explicit iterative methods for solving the equilibrium problems and variational inequalities. This technique is due to Glowinski et al. . M. A. Noor and K. I. Noor ; Noor et al.  have used the auxiliary principle technique to suggest some iterative methods for solving the variational inequalities and equilibrium problems on Hadamard manifolds. We again use the auxiliary principle technique to suggest and analyze an explicit iterative method for solving the variational inequalities, and this is the main motivation of this paper. We show that the convergence of this new method requires only the partially relaxed strongly monotonicity, which is a weaker condition than monotonicity. This represents the refinement of previously known results for the variational inequalities. 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 (that is, 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 and . 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 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 and three minimal geodesics joining these points.
Lemma 2.2 ([2, 3, 5] (comparison theorem for triangles)). 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), one has the following inequality, which is a general characteristic of the spaces with nonpositive curvature : From the properties of the exponential map, one has 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 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) Let denote the domain of defined by
The existence of subgradients for convex functions is guaranteed by the following proposition, see .
Lemma 2.5 (see [5, 7]). Let be a Hadamard manifold, and let be convex. Then, for any , the subdifferential of at is nonempty. That is, .
For a given single-valued vector field , one considers the problem of finding such that which is called the variational inequality. This problem was considered by Nemeth , Colao et al. , Tang et al. , and M. A. Noor and K. I. Noor . They proved the existence of a solution of Problem (2.12) using the KKM maps. In the linear setting, variational inequalities have been studied extensively, see [8–10, 12–26] and the references therein.
Definition 2.6. An operator is said to be partially relaxed strongly monotonicity if and only if there exists a constant such that We note that if , then partially relaxed strongly monotonicity reduces to monotonicity, but the converse is not true.
3. Main Results
For a given satisfying (2.12), consider the problem of finding such that which is called the auxiliary variational inequality on Hadamard manifolds. We note that if , then is a solution of the variational inequality (2.12). This observation enables as to suggest and analyze the following proximal point method for solving the variational inequality (2.12).
Algorithm 3.1. For a given , compute the approximate solution by the iterative scheme Algorithm 3.1 is called the explicit iterative method for solving the variational inequality on the Hadamard manifold.
If , then Algorithm 3.1 collapses to the following.
Algorithm 3.2. For a given , find the approximate solution by the iterative scheme which is known as the explicit method for solving the variational inequalities. For the convergence analysis of Algorithm 3.2, see [13, 14].
We now consider the convergence analysis of Algorithm 3.1, and this is the main motivation of our next result.
Theorem 3.3. Let be a partially relaxed strongly monotone vector field with a constant . Let be the approximate solution of the variational inequality (2.12) obtained from Algorithm 3.1. Then where is the solution of the variational inequality (2.12).
Proof . Let be a solution of the variational inequality (2.12). Then Taking in (3.5), we have Taking in (3.2), we have From (3.6) and (3.7), we have For the geodesic triangle , the inequality (3.2) can be written as Thus, from (3.8) and (3.9), we obtained the inequality (3.4), the required result.
Proof . Let be a solution of (2.12). Then, from (3.4), it follows that the sequence is bounded and It follows that Let be a cluster point of . Then there exits a subsequence such that converges to . Replacing by in (3.2), taking the limit, and using (3.10), we have This shows that solves (2.12) and which implies that the sequence has an unique cluster point and is a solution of (2.12), the required result.
We have used the auxiliary principle technique to suggest and analyze an explicit iterative method for solving the mixed quasivariational inequalities on 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, COMSATS Institute of Information Technology, Islamabad, Pakistan, for providing excellent research facilities. The authors are grateful to the referees for their very constructive comments and suggestions.
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