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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 691806, 8 pages
http://dx.doi.org/10.1155/2012/691806
Research Article

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.

Abstract

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.

1. Introduction

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 [17] and the references therein. Nemeth [8], Tang et al. [6], and Colao et al. [2] 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. [9]. M. A. Noor and K. I. Noor [10]; Noor et al. [11] 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.

2. Preliminaries

We now recall some fundamental and basic concept that need for a reading of this paper. These results and concepts can be found in the books on Riemannian geometry [2, 3, 5].

Let 𝑀 be a simply connected 𝑚-dimensional manifold. Given 𝑥𝑀, the tangent space of 𝑀 at 𝑥 is denoted by 𝑇𝑥𝑀 and the tangent bundle of 𝑀 by 𝑇𝑀=𝑥𝑀𝑇𝑥𝑀, which is naturally a manifold. A vector field 𝐴 on 𝑀 is a mapping of 𝑀 into 𝑇𝑀 which associates to each point 𝑥𝑀 a vector 𝐴(𝑥)𝑇𝑥𝑀. We always assume that 𝑀 can be endowed with a Riemannian metric to become a Riemannian manifold. We denote by ,, the scalar product on 𝑇𝑥𝑀 with the associated norm 𝑥, where the subscript 𝑥 will be omitted. Given a piecewise smooth curve 𝛾[𝑎,𝑏]𝑀 joining 𝑥 to 𝑦 (that is, 𝛾(𝑎)=𝑥 and𝛾(𝑏)=𝑦) by using the metric, we can define the length of 𝛾 as 𝐿(𝛾)=𝑏𝑎𝛾(𝑡)𝑑𝑡. Then, for any 𝑥,𝑦𝑀, the Riemannian distance 𝑑(𝑥,𝑦), which includes the original topology on 𝑀, is defined by minimizing this length over the set of all such curves joining 𝑥 to 𝑦.

Let Δ be the Levi-Civita connection with (𝑀,,). Let 𝛾 be a smooth curve in 𝑀. A vector field 𝐴 is said to be parallel along 𝛾 if Δ𝛾𝐴=0. If 𝛾 itself is parallel along 𝛾, we say that 𝛾 is a geodesic, and in this case, 𝛾 is constant. When 𝛾=1, 𝛾 is said to be normalized. A geodesic joining 𝑥 to 𝑦 in 𝑀 is said to be minimal if its length equals 𝑑(𝑥,𝑦).

A Riemannian manifold is complete, if for any 𝑥𝑀 all geodesics emanating from 𝑥 are defined for all 𝑡𝑅. By the Hopf-Rinow theorem, we know that if 𝑀 is complete, then any pair of points in 𝑀 can be joined by a minimal geodesic. Moreover, (𝑀,𝑑) is a complete metric space, and bounded closed subsets are compact.

Let 𝑀 be complete. Then the exponential map exp𝑥𝑇𝑥𝑀𝑀 at 𝑥 is defined by exp𝑥𝑣=𝛾𝑣(1,𝑥) for each 𝑣𝑇𝑥𝑀, where 𝛾()=𝛾𝑣(,𝑥) is the geodesic starting at 𝑥 with velocity 𝑣(i.e.,𝛾(0)=𝑥 and 𝛾(0)=𝑣). Then exp𝑥𝑡𝑣=𝛾𝑣(𝑡,𝑥) for each real number 𝑡.

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 𝑀 is an 𝑚-manifold Hadamard manifold.

We also recall the following well-known results, which are essential for our work.

Lemma 2.1 (see [5]). Let 𝑥𝑀. Then exp𝑥𝑇𝑥𝑀𝑀 is a diffeomorphism, and for any two points 𝑥,𝑦𝑀, there exists a unique normalized geodesic joining 𝑥 to 𝑦,𝛾𝑥,𝑦, 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 𝑀 is diffeomorphic to the Euclidean space 𝑅𝑚. Thus, 𝑀 has the same topology and differential structure as 𝑅𝑚. It is also known that Hadamard manifolds and euclidean spaces have similar geometrical properties. Recall that a geodesic triangle (𝑥1,𝑥2,𝑥3) of a Riemannian manifold is a set consisting of three points 𝑥1, 𝑥2, and  𝑥3 and three minimal geodesics joining these points.

Lemma 2.2 ([2, 3, 5] (comparison theorem for triangles)). Let (𝑥1,𝑥2,𝑥3) be a geodesic triangle. Denote, for each 𝑖=1,2,3(mod3), by 𝛾𝑖[0,𝑙𝑖]𝑀 the geodesic joining 𝑥𝑖 to 𝑥𝑖+1, and 𝛼𝑖;=𝐿(𝛾𝑖(0),𝛾𝑙(𝑖1)(𝑙𝑖1)), the angle between the vectors 𝛾𝑖(0) and 𝛾𝑖1(𝑙𝑖1), and 𝑙𝑖;=𝐿(𝛾𝑖). Then 𝛼1+𝛼2+𝛼3𝑙𝜋,(2.1)2𝑙+𝑙2𝑖+12𝐿𝑖𝑙𝑖+1cos𝛼𝑖+1𝑙2𝑖1.(2.2) In terms of the distance and the exponential map, the inequality (2.2) can be rewritten as 𝑑2𝑥𝑖,𝑥𝑖+1+𝑑2𝑥𝑖+1,𝑥𝑖+22exp𝑥1𝑖+1𝑥𝑖,exp𝑥1𝑖+1𝑥𝑖+2𝑑2𝑥𝑖1,𝑥𝑖,(2.3) since exp𝑥1𝑖+1𝑥𝑖,exp𝑥1𝑖+1𝑥𝑖+2𝑥=𝑑𝑖,𝑥𝑖+1𝑑𝑥𝑖+1,𝑥𝑖+2cos𝛼𝑖+1.(2.4)

Lemma 2.3 (see [5]). Let (𝑥,𝑦,𝑧) be a geodesic triangle in a Hadamard manifold 𝑀. Then there exist 𝑥,𝑦,𝑧𝑅2 such that 𝑑𝑥(𝑥,𝑦)=𝑦𝑦,𝑑(𝑦,𝑧)=𝑧𝑧,𝑑(𝑧,𝑥)=𝑥.(2.5) The triangle (𝑥,𝑦,𝑧) is called the comparison triangle of the geodesic triangle (𝑥,𝑦,𝑧), which is unique up to isometry of 𝑀.

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 [5]: exp𝑥1𝑦,exp𝑥1𝑧+exp𝑦1𝑥,exp𝑦1𝑧𝑑2(𝑥,𝑦).(2.6) From the properties of the exponential map, one has the following known result.

Lemma 2.4 (see [5]). Let 𝑥0𝑀 and {𝑥𝑛}𝑀 such that 𝑥𝑛𝑥0. Then the following assertions hold.(i)For any 𝑦𝑀, exp𝑥1𝑛𝑦exp𝑥1𝑜𝑦,exp𝑦1𝑥𝑛exp𝑦1𝑥𝑜.(2.7)(ii)If {𝑣𝑛} is a sequence such that 𝑣𝑛𝑇𝑥𝑛𝑀 and 𝑣𝑛𝑣0, then 𝑣0𝑇𝑥0𝑀.(iii)Given the sequences {𝑢𝑛} and {𝑣𝑛} satisfying 𝑢𝑛,𝑣𝑛𝑇𝑥𝑛𝑀, if 𝑢𝑛𝑢0 and 𝑣𝑛𝑣0, with 𝑢0,𝑣0𝑇𝑥0𝑀, then 𝑢𝑛,𝑣𝑛𝑢0,𝑣0.(2.8)

A subset 𝐾𝑀 is said to be convex if for any two points 𝑥,𝑦𝐾, the geodesic joining 𝑥 and 𝑦 is contained in 𝐾,𝐾 that is, if 𝛾[𝑎,𝑏]𝑀 is a geodesic such, that 𝑥=𝛾(𝑎) and 𝑦=𝛾(𝑏), then 𝛾((1𝑡)𝑎+𝑡𝑏)𝐾,forall𝑡[0,1]. From now on, 𝐾𝑀 will denote a nonempty, closed, and convex set, unless explicitly stated otherwise.

A real-valued function 𝑓 defined on 𝐾 is said to convex if for any geodesic 𝛾 of 𝑀, the composition function 𝑓𝛾𝑅𝑅 is convex, that is, [].(𝑓𝛾)(𝑡𝑎+(1𝑡)𝑏)𝑡(𝑓𝛾)(𝑎)+(1𝑡)(𝑓𝛾)(𝑏),𝑎,𝑏𝑅,𝑡0,1(2.9)

The subdifferential of a function 𝑓𝑀𝑅 is the set-valued mapping 𝜕𝑓𝑀2𝑇𝑀 defined as 𝜕𝑓(𝑥)=𝑢𝑇𝑥𝑀𝑢,exp𝑥1𝑦𝑓(𝑦)𝑓(𝑥),𝑦𝑀,𝑥𝑀,(2.10) and its elements are called subgradients. The subdifferential 𝜕𝑓(𝑥) at a point 𝑥𝑀 is a closed and convex (possibly empty) Let 𝐷(𝜕𝑓) denote the domain of 𝜕𝑓 defined by 𝐷(𝜕𝑓)={𝑥𝑀𝜕𝑓(𝑥)}.(2.11)

The existence of subgradients for convex functions is guaranteed by the following proposition, see [7].

Lemma 2.5 (see [5, 7]). Let 𝑀 be a Hadamard manifold, and let 𝑓𝑀𝑅 be convex. Then, for any 𝑥𝑀, the subdifferential 𝜕𝑓(𝑥) of 𝑓 at 𝑥 is nonempty. That is, 𝐷(𝜕𝑓)=𝑀.
For a given single-valued vector field 𝑇𝑀𝑇𝑀, one considers the problem of finding 𝑢𝐾 such that 𝑇𝑢,exp𝑢1𝑣0,𝑣𝐾,(2.12) which is called the variational inequality. This problem was considered by Nemeth [8], Colao et al. [2], Tang et al. [6], and M. A. Noor and K. I. Noor [10]. 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 [810, 1226] and the references therein.

Definition 2.6. An operator 𝑇 is said to be partially relaxed strongly monotonicity if and only if there exists a constant 𝛼>0 such that 𝑇𝑢,exp𝑣1𝑧+𝑇𝑣,exp𝑧1𝑣𝛼𝑑2(𝑧,𝑢),𝑢,𝑣,𝑧𝑀.(2.13) We note that if 𝑧=𝑢, then partially relaxed strongly monotonicity reduces to monotonicity, but the converse is not true.

3. Main Results

We now use the auxiliary principle technique of Glowinski et al. [9] to suggest and analyze an explicit iterative method for solving the variational inequality (2.12) on the Hadamard manifold.

For a given 𝑢𝐾 satisfying (2.12), consider the problem of finding 𝑤𝐾 such that 𝜌𝑇𝑢+exp𝑢1𝑤,exp𝑤1𝑣0,𝑣𝐾,(3.1) which is called the auxiliary variational inequality on Hadamard manifolds. We note that if 𝑤=𝑢, then 𝑤 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 𝑢0, compute the approximate solution by the iterative scheme 𝜌𝑇𝑢𝑛+exp𝑢1𝑛𝑢𝑛+1,exp𝑢1𝑛+1𝑣0,𝑣𝐾.(3.2) Algorithm 3.1 is called the explicit iterative method for solving the variational inequality on the Hadamard manifold.

If 𝑀=𝑅𝑛, then Algorithm 3.1 collapses to the following.

Algorithm 3.2. For a given 𝑢0𝐾, find the approximate solution 𝑢𝑛+1 by the iterative scheme 𝜌𝑇𝑢𝑛+𝑢𝑛+1𝑢𝑛,𝑣𝑢𝑛+10,𝑣𝐾,(3.3) 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 𝑇 be a partially relaxed strongly monotone vector field with a constant 𝛼>0. Let 𝑢𝑛+1 be the approximate solution of the variational inequality (2.12) obtained from Algorithm 3.1. Then 𝑑2𝑢𝑛+1,𝑢𝑑2𝑢𝑛,𝑢(12𝛼𝜌)𝑑2𝑢𝑛+1,𝑢𝑛,(3.4) where 𝑢𝑀 is the solution of the variational inequality (2.12).

Proof . Let 𝑢𝐾 be a solution of the variational inequality (2.12). Then 𝜌𝑇(𝑢),exp𝑢1𝑣0,𝑣K.(3.5) Taking 𝑣=𝑢𝑛+1 in (3.5), we have 𝜌𝑇(𝑢),exp𝑢1𝑢𝑛+10.(3.6) Taking 𝑣=𝑢 in (3.2), we have 𝜌𝑇𝑢𝑛+exp𝑢1𝑛𝑢𝑛+1,exp𝑢1𝑛+1𝑢0.(3.7) From (3.6) and (3.7), we have exp𝑢1𝑛+1𝑢𝑛,exp𝑢1𝑛+1𝑢𝜌𝑇(𝑢),exp𝑢1𝑣+𝑇𝑢𝑛,exp𝑢1𝑛+1𝑢.𝛼𝜌𝑑2𝑢𝑛+1,𝑢𝑛.(3.8) For the geodesic triangle (𝑢𝑛,𝑢𝑛+1,𝑢), the inequality (3.2) can be written as 𝑑2𝑢𝑛+1,𝑢+𝑑2𝑢𝑛+1,𝑢𝑛2exp𝑢1𝑛+1𝑢𝑛,exp𝑢1𝑛+1𝑢𝑑2𝑢𝑛,𝑢.(3.9) Thus, from (3.8) and (3.9), we obtained the inequality (3.4), the required result.

Theorem 3.4. Let 𝑢𝐾 be a solution of (2.12), and let 𝑢𝑛+1 be the approximate solution obtained from Algorithm 3.1. If 𝜌<1/2𝛼, then lim𝑛𝑢𝑛+1=𝑢

Proof . Let ̂𝑢𝐾 be a solution of (2.12). Then, from (3.4), it follows that the sequence {𝑢𝑛} is bounded and 𝑛=0(12𝛼𝜌)𝑑2𝑢𝑛+1,𝑢𝑛𝑑2𝑢0.,𝑢(3.10) It follows that lim𝑛𝑑𝑢𝑛+1,u𝑛=0.(3.11) Let ̂𝑢 be a cluster point of {𝑢𝑛}. Then there exits a subsequence {𝑢𝑛𝑖} such that {𝑢𝑢𝑖} converges to ̂𝑢. Replacing 𝑢𝑛+1 by 𝑢𝑛𝑖 in (3.2), taking the limit, and using (3.10), we have 𝑇̂𝑢,exp1𝑣̂𝑢0,𝑣𝐾.(3.12) This shows that ̂𝑢𝐾 solves (2.12) and 𝑑2𝑢𝑛+1,̂𝑢𝑑2𝑢𝑛,̂𝑢(3.13) which implies that the sequence {𝑢𝑛} has an unique cluster point and lim𝑛𝑢𝑛=̂𝑢 is a solution of (2.12), the required result.

4. Conclusion

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.

Acknowledgments

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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