# Implicit Methods for Equilibrium Problems on Hadamard Manifolds

Authors: Muhammad Aslam Noor; Saira Zainab; Yonghong Yao
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

We use the auxiliary principle technique to suggest and analyze an implicit method for solving the equilibrium problems on Hadamard manifolds. The convergence of this new implicit method requires only the speudomonotonicity, which is a weaker condition than monotonicity. Some special cases are also considered.


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## Body

## 1.  Introduction

Recently, much attention has been given to study the variational inequalities, equilibrium and related optimization problems on the Riemannian manifold and Hadamard manifold.  This framework is a useful for the developments of various fields. Several ideas and techniques form the Euclidean space have been extended and generalized to this nonlinear framework. Hadamard manifolds are examples of hyperbolic spaces and geodesics, see [1–8] and the references therein. Németh [9], Tang et al. [7], M. A. Noor and K. I. Noor [5], 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. To the best of our knowledge, no one has considered the auxiliary principle technique for solving the equilibrium problems on Hadamard manifolds. In this paper, we use the auxiliary principle technique to suggest and analyze an implicit method for solving the equilibrium problems on Hadamard manifold. As a special case, our result includes the recent result of Noor and Oettli [10] for variational inequalities on Hadamard manifold. This shows that the results obtained in this paper continue to hold for variational inequalities on Hadamard manifold, which are due to M. A. Noor and K. I. Noor [5], Tang et al. [7], and Németh [9]. 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  reading of this paper. These results and concepts can be found in the books on Riemannian geometry [1–3, 6].LetM be a simply connected m-dimensional manifold. Given x∈M, the tangent space of M at x is denoted by TxM and the tangent bundle of M by TM=∪x∈MTxM, which is naturally a manifold. A vector field A on M is a mapping of M into TM which associates to each point x∈M, 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)=∫ab∥γ′(t)∥dt. Then, for any x,y∈M, 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.LetΔ 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 Δγ′A=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 anyx∈M all geodesics emanating from x are defined for all t∈R. 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.LetM be complete. Then the exponential map expx:TxM→M at x is defined by expxv=γv(1,x) for each v∈TxM, 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 aHadamard 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 [6]).
Letx∈M. Then expx:TxM→M is a diffeomorphism, and, for any two points x,y∈M, 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. Lemma2.1 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, x3 and three minimal geodesics joining these points.Lemma 2.2 (See [2, 3, 6] (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′(i-1)(li-1)), the angle between the vectors γi′(0) and -γi-1′(li-1), and li;=L(γi). Then,
(2.1)α1+α2+α3≤π,(2.2)ll2+li+12-2Lili+1cos⁡αi+1≤li-12.In terms of the distance and the exponential map, the inequality (2.2) can be rewritten as(2.3)d2(xi,xi+1)+d2(xi+1,xi+2)-2〈exp⁡xi+1-1xi,exp⁡xi+1-1xi+2〉≤d2(xi-1,xi),
since(2.4)〈exp⁡xi+1-1xi,exp⁡xi+1-1xi+2〉=d(xi,xi+1)d(xi+1,xi+2)cos⁡αi+1.Lemma 2.3 (See [6]).
Let▵(x,y,z) be a geodesic triangle in a Hadamard manifold M. Then, there exist x′,y′,z′∈R2 such that
(2.5)d(x,y)=‖x′-y′‖,d(y,z)=‖y′-z′‖,d(z,x)=‖z′-x′‖.
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 [6]:(2.6)〈exp⁡x-1y,exp⁡x-1z〉+〈exp⁡y-1x,exp⁡y-1z〉≥d2(x,y).
From the properties of the exponential map, we have the following known result.Lemma 2.4 (See [6]).
Letx0∈M and {xn}⊂M such that xn→x0. Then the following assertions hold.(i)
For anyy∈M,
(2.7)exp⁡xn-1y⟶exp⁡x0-1y,exp⁡y-1xn⟶exp⁡y-1x0.(ii)
If{vn} is a sequence such that vn∈TxnM and vn→v0, then v0∈Tx0M.(iii)
Given the sequences{un} and {vn} satisfying un,vn∈TxnM, if un→u0 and vn→v0, with u0,v0∈Tx0M, then
(2.8)〈un,vn〉⟶〈u0,v0〉.A subsetK⊆M is said to be convex if for any two points x,y∈K, 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((1-t)a+tb)∈K,for all t∈[0,1]. From now on K⊆M will denote a nonempty, closed, and convex set, unless explicitly stated otherwise.A real-valued functionf defined on K is said to be convex, if, for any geodesic γ of M, the composition function f∘γ:R→R is convex, that is,(2.9)(f∘γ)(ta+(1-t)b)≤t(f∘γ)(a)+(1-t)(f∘γ)(b),∀a,b∈R,t∈[0,1].The subdifferential of a functionf:M→R is the set-valued mapping ∂f:M→2TM defined as(2.10)∂f(x)={u∈TxM:〈u,exp⁡x-1y〉≤f(y)-f(x),∀y∈M},∀x∈M,
and its elements are called subgradients. The subdifferential ∂f(x) at a point x∈M is a closed and convex (possibly empty) set. Let D(∂f) denote the domain of ∂f defined by(2.11)D(∂f)={x∈M:∂f(x)≠∅}.The existence of subgradients for convex functions is guaranteed by the following proposition, see [8].Lemma 2.5 (See [6, 8]).
LetM be a Hadamard manifold and let f:M→R be convex. Then, for any x∈M, the subdifferential ∂f(x) of f at x is nonempty. That is, D(∂f)=M.For a given bifunctionF(·,·):K×K→R, we consider the problem of finding u∈K such that(2.12)F(u,v)≥0,∀v∈K,
which is called the equilibrium problem on Hadamard manifolds. This problem was considered by Colao et al. [2]. They proved the existence of a solution of the problem (2.12) using the KKM maps. Colao et al. [2] have given an example of the equilibrium problem defined in a Euclidean space whose set K is not a convex set, so it cannot be solved using the technique of Blum and Oettli [11]. However, if one can reformulate the equilibrium problem on a Riemannian manifold, then it can be solved. This shows the importance of considering these problems on Hadamard manifolds. For the applications, formulation, and other aspects of the equilibrium problems in the linear setting, see [4, 9–22].IfF(u,v)=〈Tu,expu-1v〉, where T is a single valued vector filed T:K→TM, then problem (2.12) is equivalent to finding u∈K such that(2.13)〈Tu,exp⁡x-1v〉≥0,∀v∈K,
which is called the variational inequality on Hadamard manifolds. Németh [9], Colao et al. [2], Noor and Oettli [10], and M. A. Noor and K. I. Noor [5] studied variational inequalities on Hadamard manifold from different point of views. In the linear setting, variational inequalities have been studied extensively, see [5, 10, 11, 13–28] and the references therein.Definition 2.6.
A bifunctionF(·,·) is said to be speudomonotone, if and only if
(2.14)F(u,v)≥0,⟹F(v,u)≤0,∀u,v∈K.


## 3.  Main Results

We now use the auxiliary principle technique of Glowinski et al. [23] to suggest and analyze an implicit iterative method for solving the equilibrium problems (2.12).For a givenu∈K satisfying (2.12), consider the problem of finding w∈K such that(3.1)ρF(w,v)+〈exp⁡u-1w,exp⁡w-1v〉≥0,∀v∈K,
which is called the auxiliary equilibrium problem on Hadamard manifolds. We note that, if w=u, then w is a solution of (2.12). This observation enables to suggest and analyze the following implicit method for solving the equilibrium problems (2.12). This is the main motivation of this paper.Algorithm 3.1.
For a givenu0, compute the approximate solution by the iterative scheme
(3.2)ρF(un+1,v)+〈exp⁡un-1un+1,exp⁡un+1-1v〉≥0,∀v∈K.
Algorithm 3.1 is called the implicit (proximal point) iterative method for solving the equilibrium problem on the Hadamard manifold. Algorithm 3.1 can be written in the following equivalent form.Algorithm 3.2.
For a givenu0∈K, find the approximate solution un+1 by the iterative scheme:
(3.3)ρF(un,v)+〈exp⁡un-1yn,exp⁡yn-1v〉≥0,∀v∈K,ρF(yn,v)+〈exp⁡yn-1un+1,exp⁡un+1-1v〉≥0,∀v∈K.
Algorithm 3.2 is a two-step iterative method for solving the equilibrium problems on Hadamard manifolds. This method can be viewed as the extragradient method for solving the equilibrium problems.IfK is a convex set in Rn, then Algorithm 3.1 collapses to the following.Algorithm 3.3.
For a givenu0∈K, find the approximate solution un+1 by the iterative scheme:
(3.4)ρF(un+1,v)+〈un+1-un,v-un+1〉≥0,∀v∈K,
which is known as the implicit method for solving the equilibrium problem. For the convergence analysis of Algorithm 3.2, see [16, 19, 20].IfF(u,v)=〈Tu,expu-1v〉, where T is a single valued vector filed T:K→TM, then Algorithm 3.1 reduces to the following implicit method for solving the variational inequalities.Algorithm 3.4.
For a givenu0∈K, compute the approximate solution un+1 by the iterative scheme
(3.5)〈ρTun+1+(exp⁡un-1un+1),exp⁡un+1-1v〉≥0,∀v∈K.
Algorithm 3.4 is due according to Tang et al. [7] and M. A. Noor and K. I. Noor [5]. We can also rewrite Algorithm 3.4 in the following equivalent form.Algorithm 3.5.
For a givenu0∈K, compute the approximate solution un+1 by the iterative scheme
(3.6)〈ρTun+exp⁡un-1yn,exp⁡yn-1v〉≥0,∀v∈K,〈ρTyn+exp⁡un+1-1un,exp⁡un+1-1v〉≥0,∀v∈K,
which is the extragradient method for solving the variational inequalities on Hadamard manifolds and appears to be a new one.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 Algorithm3.1 and this is the motivation of our next result.Theorem 3.6.
LetF(·,·) be a speudomonotone bifunction. Let un be the approximate solution of the equilibrium problem (2.12) obtained from Algorithm 3.1, then
(3.7)d2(un+1,u)+d2(un+1,un)≤d2(un,u),
where u∈K is a solution of the equilibrium problem (2.12).Proof.
Letu∈K be a solution of the equilibrium problem (2.12). Then, by using the speudomonotonicity of the bifunction F(·,·), we have
(3.8)F(v,u)≤0,∀v∈K.
Taking v=un+1 in (3.9), we have
(3.9)F(un+1,u)≤0.
Taking v=u in (3.2), we have
(3.10)ρF(un+1,u)+〈exp⁡un-1un+1,exp⁡un+1-1u〉≥0.
From (3.10) and (3.9), we have
(3.11)〈exp⁡un+1-1un,exp⁡un+1-1u〉≤0.
For the geodesic triangle ▵(un,un+1,u), the inequality (2.3) can be written as
(3.12)d2(un+1,u)+d2(un+1,un)-〈exp⁡un+1-1un,exp⁡un+1-1u〉≤d2(un,u).
Thus, from (3.12) and (3.11), we obtained the inequality (3.8), the required result.Theorem 3.7.
Letu∈K be solution of (2.12) and let un+1 be the approximate solution obtained from Algorithm 3.1, then limn→un+1=u.Proof.
Letû be a solution of (2.12). Then, from (3.8), it follows that the sequence {un} is bounded and
(3.13)∑n=0∞d2(un+1,un)≤d2(u0,u),
then it follows that
(3.14)lim⁡n→∞d(un+1,un)=0.
Let û be a cluster point of {un}. Then there exists a subsequence {uni} such that {uui} converges to û. Replacing un+1 by uni in (3.2), taking the limit, and using (3.14), we have
(3.15)〈F(û,v)≥0,∀v∈K.
This shows that û∈K solves (2.12) and
(3.16)d2(un+1,û)≤d2(un,û)
which implies that the sequence {un} has unique cluster point and limn→∞un=û is a solution of (2.12), the required result.


## 4. Conclusion

In this paper, we have suggested and analyzed an implicit iterative method for solving the equilibrium problems on Hadamard manifold. It is shown that the convergence analysis of this methods requires only the speudomonotonicity, which is a weaker condition than monotonicity. Some special cases are also discussed. Results proved in this paper may stimulate research in this area.



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