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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 657278, 8 pages
http://dx.doi.org/10.1155/2012/657278
Research Article

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.

Abstract

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.

1. Introduction

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 [17] and the references therein. Németh [8], Tang et al. [6], and Colao et al. [2] 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. [6] and Németh [8]. 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 concepts needed for a reading of this paper. These results and concepts can be found in the books on the 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 (i.e., 𝛾(𝑎)=𝑥 and 𝛾(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)=𝑥and𝛾(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 the Hadamard manifolds and Euclidean spaces have similar geometrical properties. Recall that a geodesic triangle (𝑥1,𝑥2,and𝑥3) of a Riemannian manifold is a set consisting of three points 𝑥1,𝑥2,𝑥3 and three minimal geodesics joining these points.

Lemma 2.2 (see (comparison Theorem for Triangles [2, 3, 5])). 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), we have 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, we have the following known result.

Lemma 2.4 (see [5]). Let 𝑥0𝑀 and {𝑥𝑛}𝑀 such that 𝑥𝑛𝑥0. Then the following assertions hold.(i)For any𝑦𝑀, exp𝑥1𝑛𝑦exp𝑥10𝑦,exp𝑦1𝑥𝑛exp𝑦1𝑥0.(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 be 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) set. Let 𝐷(𝜕𝑓) denote the domain of 𝜕𝑓 defined by𝐷(𝜕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 𝑓𝑀𝑅 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 and a real-valued function 𝑓𝑀𝑅, we consider the problem of finding 𝑢𝑀 such that𝑇𝑢,exp𝑢1𝑣+𝑓(𝑣)𝑓(𝑢)0,𝑣𝑀,(2.12) which is called the mixed variational inequality. This problem was considered by Colao et al. [2]. 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 [816].

We remark that if the function 𝑓 is an indicator of a closed and convex set 𝐾 in 𝑀, then problem (2.12) is equivalent to finding 𝑢𝐾 such that𝑇𝑢,exp𝑥1𝑣0,𝑣𝐾,(2.13) which is called the variational inequality on the Hadamard manifolds. Németh [8], Colao et al. [2] and Udrişte [7] studied variational inequalities on the Hadamard manifold from different point of views. In the linear setting, variational inequalities have been studied extensively; see [825] and the references therein.

Definition 2.6. An operator 𝑇 is said to be speudomonotone with respect a mapping 𝑓, if and only if 𝑇(𝑢),exp𝑢1𝑣+𝑓(𝑣)𝑓(𝑢)0𝑇(𝑣),exp𝑣1𝑢+𝑓(𝑣)𝑓(𝑢)0,𝑢,𝑣𝑀.(2.14)

3. Main Results

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

For a given 𝑢𝑀 satisfying (2.12), consider the problem of finding 𝑤𝑀 such that𝜌𝑇𝑤+exp𝑢1𝑤,exp𝑤1𝑣+𝑓(𝑣)𝑓(𝑤)0,𝑣𝑀,(3.1) which is called the auxiliary mixed variational inequality on the Hadamard manifolds. We note that if 𝑤=𝑢, then 𝑤 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 𝑢0, compute the approximate solution by the iterative scheme: 𝜌𝑇𝑢𝑛+1+exp𝑢1𝑛𝑢𝑛+1,exp𝑢1𝑛+1𝑣𝑢+𝑓(𝑣)𝑓𝑛+10,𝑣𝑀,(3.2) Algorithm 3.1 is called the implicit (proximal point) iterative method for solving the mixed variational inequality on the Hadamard manifold.

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

Algorithm 3.2. For a given 𝑢0𝑅𝑛, find the approximate solution 𝑢𝑛+1 by the iterative scheme. 𝜌𝑇𝑢𝑛+1+𝑢𝑛+1𝑢𝑛,𝑣𝑢𝑛+1𝑢+𝜌𝑓(𝑣)𝑓𝑛+10,𝑣𝑅𝑛,(3.3) 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].

If 𝑓 is the indicator function of a closed and convex set 𝐾 in 𝑀, then Algorithm 3.1 reduces to the following method, which is due to Tang et al. [6].

Algorithm 3.3. For a given 𝑢0𝐾, compute the approximate solution by the iterative scheme 𝜌𝑇𝑢𝑛+1+exp𝑢1𝑛𝑢𝑛+1,exp𝑢1𝑛+1𝑣0,𝑣𝐾.(3.4)

We would like to mention that Algorithm 3.1 can be rewritten in the following equivalent form.

Algorithm 3.4. For a given 𝑢0𝑀, compute the approximate solution by the iterative scheme: 𝜌𝑇𝑢𝑛+exp𝑢1𝑛𝑦𝑛,exp𝑦1𝑛𝑣𝑦+𝜌𝑓(𝑣)𝜌𝑓𝑛0𝑣𝑀,𝑇𝑦𝑛+exp𝑢1𝑛𝑢𝑛+1,exp𝑢1𝑛+1𝑣𝑢+𝜌𝑓(𝑣)𝜌𝑓𝑛+1,𝑣𝑀,(3.5) 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 𝑇 be a pseudomonotone vector field. Let 𝑢𝑛 be the approximate solution of the mixed variational inequality (2.12) obtained from Algorithm 3.1; then 𝑑2𝑢𝑛+1,𝑢+𝑑2𝑢𝑛+1,𝑢𝑛𝑑2𝑢𝑛,,𝑢(3.6) where 𝑢𝑀 is the solution of the mixed variational inequality (2.12).

Proof. Let 𝑢𝑀 be a solution of the mixed variational inequality (). Then, by using the pseudomonotonicity of the vector filed, 𝑇(𝑢), we have 𝜌𝑇(𝑣),exp𝑢1𝑣+𝜌𝑓(𝑣)𝜌𝑓(𝑢)0,𝑣𝑀.(3.7) Taking 𝑣=𝑢𝑛+1 in (3.7), we have 𝑢𝜌𝑇𝑛+1,exp𝑢1𝑢𝑛+1𝑢+𝜌𝑓𝑛+1𝜌𝑓(𝑢)0.(3.8) Taking 𝑣=𝑢 in (3.2), we have 𝜌𝑇𝑢𝑛+1+exp𝑢1𝑛𝑢𝑛+1,exp𝑢1𝑛+1𝑢𝑢+𝑓(𝑢)𝑓𝑛+10.(3.9) From (3.8) and (3.9), we have exp𝑢1𝑛+1𝑢𝑛,exp𝑢1𝑛+1𝑢0.(3.10) For the geodesic triangle (𝑢𝑛,𝑢𝑛+1,𝑢) the inequality (3.10) can be written as, 𝑑2𝑢𝑛+1,𝑢+𝑑2𝑢𝑛+1,𝑢𝑛exp𝑢1𝑛+1𝑢𝑛,exp𝑢1𝑛+1𝑢𝑑2𝑢𝑛,𝑢.(3.11) Thus, from (3.10) and (3.11), we obtained inequality (3.6), the required result.

Theorem 3.6. Let 𝑢𝑀 be solution of (2.12), and let 𝑢𝑛+1 be the approximate solution obtained from Algorithm 3.1; then lim𝑛(𝑢𝑛+1)=𝑢.

Proof. Let ̂𝑢𝑀 be a solution of (2.12). Then, from (3.6), it follows that the sequence {𝑢𝑛} is bounded and 𝑛=0𝑑2𝑢𝑛+1,𝑢𝑛𝑑2𝑢0;,𝑢(3.12) it follows that lim𝑛𝑑𝑢𝑛+1,𝑢𝑛=0.(3.13) Let ̂𝑢 be a cluster point of {𝑢𝑛}. Then there exists a subsequence {𝑢𝑛𝑖} such that {𝑢𝑢𝑖} converges to ̂𝑢. Replacing 𝑢𝑛+1 by 𝑢𝑛𝑖 in (3.2), taking the limit, and using (3.13), we have 𝑇̂𝑢,exp1𝑣̂𝑢+𝑓(𝑣)𝑓(̂𝑢)0,𝑣𝑀.(3.14) This shows that ̂𝑢𝑀 solves (2.12) and 𝑑2𝑢𝑛+1̂𝑢𝑑2𝑢𝑛,̂𝑢,(3.15) which implies that the sequence {𝑢𝑛} has 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 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.

Acknowledgment

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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