Viscosity method for hierarchical variational inequalities and variational inclusions on Hadamard manifolds

This article aims to introduce and analyze the viscosity method for hierarchical variational inequalities involving a φ-contraction mapping defined over a common solution set of variational inclusion and fixed points of a nonexpansive mapping on Hadamard manifolds. Several consequences of the composed method and its convergence theorem are presented. The convergence results of this article generalize and extend some existing results from Hilbert/Banach spaces and from Hadamard manifolds. We also present an application to a nonsmooth optimization problem. Finally, we clarify the convergence analysis of the proposed method by some computational numerical experiments in Hadamard manifold.


Introduction
The variational inclusion in Hilbert space H can be stated as where K is nonempty closed, convex subset of H, M : K → H is an operator and F : H ⇒ H is a set-valued operator and (M + F) -1 (0) is the set of zeros of M + F. If M = 0, then the inclusion problem (1) reduces to Find x ∈ K such that x ∈ F -1 (0).
For a set-valued maximal monotone operator F : H ⇒ H in Hilbert spaces, problem (2) was studied by Rockafellar [20]. The iconic method to solve the inclusion problem (2) is the proximal point method which was first suggested by Martinet [15] and later generalized by Rockafellar [20]. Many mathematical problems arising in nonlinear analysis such as optimization, variational inequality problems, economics and partial differential equations are reduced to the inclusion problem (2). Therefore, in the recent past, many authors have extended and generalized the inclusion problem (2) in different directions; see, for example, [1, 3, 4, 8, 9, 11-14, 22, 24, 26] and the references therein. The fixed point problem of a nonexpansive self mapping T : K → K can be stated as The common solution of fixed point problem (3) of a nonexpansive self mapping T and variational inclusion problem (1) discussed by Takahashi et al. [24] in Hilbert spaces, which is defined by Later, Manaka and Takahashi [14] studied problem (4) with nonspreading mapping T in Hilbert spaces. Very recently, Al-Homidan et al. [1] extended the work of [14,24] to Hadamard manifolds settings. Moudafi [16] introduced the viscosity method to study the hierarchical variational inequality problem which consists of a contraction mapping f over a nonempty closed convex subset Fix(T) in Hilbert spaces, that is, If the set Fix(T) is a nonempty closed and convex subset of H, then problem (5) reduces to the following equivalent form: where P Fix(T) denotes the projection onto Fix(T). Xu [27] extended hierarchical variational inequality problem (6) to uniformly smooth Banach spaces. The advantage of this method is that it allows us to replace the fixed point set by some nonlinear problems which satisfy various variational inequalities. Very recently, Al-Homidan et al. [2] used this idea to extend the viscosity method for hierarchical variational inequality problem involving weakly contraction mapping and discussed its several cases on Hadamard manifolds. During the last ten years, many problems in nonlinear analysis such as fixed point problems, variational inequality problems, equilibrium problems and optimization problems have been transformed from the linear spaces, namely, Banach spaces, Hilbert spaces to nonlinear spaces because of their applications in many areas of sciences; see [1-3, 5, 8-13, 18, 25] and the references therein.
Inspired by the work discussed in [1,2,14,16], our motive is to present the viscosity method for the following hierarchical variational inequality problem (HVIP) involving φcontraction mapping in the framework of Hadamard manifold M: where 0 is a zero tangent vector, K is a nonempty, closed and convex subset of Hadamard manifold M, f : K → K is a φ-contraction mapping and T : K → K is a nonexpansive mapping with Fix(T) = ∅, M : K → TM is a single-valued and F : K ⇒ TM is a set-valued vector field such that (M +F) -1 (0) = ∅, (·, ·) is a Riemannian metric and exp -1 is an inverse exponential mapping. Equivalently, problem (7) can be written as: The rest of the paper is organized as follows: The next section consists of some preliminaries and auxiliary results of Riemannian manifolds. In Sect. 3, we propose a viscosity method to solve considered HVIP and establish a convergence result of the considered method. Some special cases and an application to nonsmooth optimization problem are also discussed in the subsequent sections that extend and improve some existing results in linear spaces and in Hadamard manifolds. In the last section, we analyze the convergence of the proposed viscosity method by some computational numerical experiments.

Preliminaries
Let M be a finite dimensional differentiable manifold. For any element q ∈ M, we denote the tangent space of M at q by T q M and the tangent bundle by TM = q∈M T q M. The tangent space T q M at q is a vector space and has the same dimension as M. An inner product q (·, ·) on T q M is the Riemannian metric on T q M. A tensor (·, ·) : q − → q (·, ·) is called a Riemannian metric on M, if for each q ∈ M, q (·, ·) is a Riemannian metric on T q M. We assume that M is endowed with the Riemannian metric q (·, ·) with the corresponding norm · q to be a Riemannian manifold. The angle between 0 = x, y ∈ T q M, denoted by ∠ q (x, y) is defined as cos ∠ q (x, y) = q (·,·) x y . For simplicity, we denote · q = · , q (·, ·) = (·, ·) and ∠ q (x, y) = ∠(x, y), where 0 is a zero tangent vector. For a piecewise smooth curve γ : [a, b] → M joining q to r (i.e. γ (a) = q and γ (b) = r), the length L of γ is defined as The Riemannian distance d(q, r) induces the original topology on M, minimize the length over the set of all such curves joining q to r.
Let ∇ be the Levi-Civita connection corresponding to Riemannian manifold M. A smooth mapping U : M → TM is said to be single-valued vector field, if for each q ∈ M, a tangent vector U(q) ∈ T q M is assigned. A vector field U is said to be parallel along a smooth curve γ if γ (s) U = 0. If γ is parallel along γ , i.e., ∇ γ (s) γ (s) = 0, then γ is called geodesic and in this case γ is constant and if γ = 1, then γ is said to be normalized geodesic. A geodesic joining q to r in M is called minimal geodesic if its length is equal to d(q, r).
A Riemannian manifold is said to be (geodesically) complete, if for any q ∈ M, all geodesics emanating from q, are defined for all s ∈ (-∞, ∞). We know by the Hopf-Rinow theorem [23] that, in a Riemannian manifold M, the following are equivalent: (1) M is complete, Assuming M is a complete Riemannian manifold, the exponential mapping exp q : is the geodesic starting at q with velocity ϑ (i.e., γ (0) = 0 and γ (0) = ϑ). We know that exp q (sϑ) = γ ϑ (s, q) for each real number s. One can easily see that exp q 0 = γ ϑ (0; q) = q. The exponential mapping exp q is differentiable on T q M for any q ∈ M.
A complete, simply connected Riemannian manifold of non-positive sectional curvature is called Hadamard manifold. From now on, we will suppose that M is a finite dimensional Hadamard manifold.

Proposition 1 ([23])
Let M be a Hadamard manifold. Then exp q : T q M → M is diffeomorphism for all q ∈ M and for any two points q, r ∈ M, there exists a unique normalized geodesic γ : A subset K ⊂ M is said to be convex if for any two points q, r ∈ K , then any geodesic joining q to r is contained in K , that is, if any γ : [a, b] → M geodesic such that q = γ (a) and r = γ (b), then γ ((1s)a + sb) ∈ K for all s ∈ [0, 1]. From now on, K ⊂ M will denote a nonempty, closed and convex subset of a Hadamard manifold M. The projection mapping onto K is defined by A function g : K → R is said to be convex if for any geodesic γ : In particular, for each q ∈ M, the function d(·, q) : M → R is a convex function.
If M is a finite dimensional manifold with dimension n, then Proposition 1 shows that M is diffeomorphism to the Euclidean space R n . Thus, we see that M has the same topology and differential structure as R n . Moreover, Hadamard manifolds and Euclidean spaces have some similar geometrical properties. We describe some of them in the following results.
Recall that a geodesic triangle (q 1 , q 2 , q 3 ) of Riemannian manifold is a set consisting of three points q 1 , q 2 and q 3 and the three minimal geodesics γ j joining q j to q j+1 , where j = 1, 2, 3 mod (3).
The points q 1 , q 2 , q 3 are called the comparison points to q 1 , q 2 , q 3 , respectively. The triangle (q 1 , q 2 , q 3 ) is called the comparison triangle of the geodesic triangle (q 1 , q 2 , q 3 ), which is unique up to isometry of M.
. Then the following inequality holds: (ii) Let p be a point on the geodesic joining q 1 to q 2 and p be its comparison point in the interval [q 1 , q 2 ]. Suppose that d(p, q 1 ) = pq 1 and d(p, q 2 ) = pq 2 . Then Proposition 3 (Comparison theorem for triangle, [23]) Let (q 1 , q 2 , q 3 ) be a geodesic triangle. Denote, for each j = 1, 2, 3 mod (3), by γ j : [0, l j ] → M the geodesic joining q j to q j+1 and set l j = L(γ j ), α 1 = ∠(γ j (0), -γ j-1 (l j-1 )). Then In terms of distance and exponential mapping, the above inequality can be rewritten as since Proposition 4 ([25]) Let K be a closed convex subset of a Hadamard manifold M. Then P K (q) is a singleton for each q ∈ M. Also, for any point q ∈ M, the following assertion holds: The set of all single-valued vector fields M : M → TM is denoted by (M) such that M(q) ∈ T q (M) for all q ∈ M. We denotes χ(M) the set of all set-valued vector fields F : is nonincreasing.
Firmly nonexpansive mappings are nonexpansive; see [12]. The following φ-contraction mapping was introduced by Boyd and Wong [6] in the setting of metric spaces.

Main results
We propose the following viscosity method for (HVIP) on Hadamard manifolds.
For the convergence of Algorithm 1, we impose the following conditions on the sequence {α n }: We make the following assumption on a single-valued vector field M : K → TM, which also appeared in [1] in the setting of Hadamard manifolds. Assumption 1 For any nonempty subset K of Hadamard manifold M. A single-valued vector field M : K → TM is said to satisfy the contraction type assumption if for any q, r ∈ K and any λ > 0, the following holds:
This completes the proof.

Consequences
If f is a contraction on K , then a corollary of Theorem 2, which can be seen as the extension of the work in [22] from Banach spaces to Hadamard manifolds, is mentioned now. If f = I, identity mapping in the Algorithm 1, then the following result is an extension from Hilbert spaces to Hadamard manifolds, discussed in [14,24]. Moreover, the following result is also appeared in [1] on a Hadamard manifold.

Nonsmooth optimization problem
In this section, we study composite minimization of a smooth and a nonsmooth realvalued functions defined on a Hadamard manifold M. Let Y, Z : M → R be real-valued functions such that Y is lower semicontinuous and convex, and Z is differentiable. We address the following minimization problem: to find Assume that S is the solution set of the problem (41). The directional derivative of a function Z : M → R at q in the direction u ∈ T q M is defined by s .

Conclusions
In this article, we have introduced the viscosity method for hierarchical variational inequalities involving a φ-contraction mapping defined over the common solution of variational inclusions and a fixed point problem. Some consequences of the proposed method are also provided. Furthermore, an application of the proposed viscosity method is presented to a nonsmooth optimization problem. Moreover, the convergence analysis of the proposed method is illustrated by some computational numerical experiments on Hadamard manifolds.