Iterative algorithm for singularities of inclusion problems in Hadamard manifolds

If A = 0, then the problem (1) becomes the inclusion problem introduced by Rockafellar [1]. Several nonlinear problems such as optimization problems, variational inequality problems, DEs [2–6] and economics can be formulated to find a singularity of the problem (1). The problem (1) is highly considered by many authors who performed dedicated work to theoretical results as well as iterative procedures; see, for instance [7–10] and the references therein. In 1979, Lions and Mercier [11] showed that the problem (1) is equivalent to find fixed points of the mapping JB λ (I – λA), that is, p∗ = JB λ (p∗ – λA(p∗)) ⇔ 0 ∈ (A + B)(p∗), where JB λ = (I + λB)–1. Owing to the fixed point formulation, Lions and Mercier [11] presented the following proximal point method: let p0 ∈ H be an initial point and


Introduction
Let H be a Hilbert space, A : H → H an operator and B : H → 2 H a multivalued operator. The inclusion problem is to find p * ∈ H such that 0 ∈ (A + B) p * . ( If A = 0, then the problem (1) becomes the inclusion problem introduced by Rockafellar [1]. Several nonlinear problems such as optimization problems, variational inequality problems, DEs [2][3][4][5][6] and economics can be formulated to find a singularity of the problem (1). The problem (1) is highly considered by many authors who performed dedicated work to theoretical results as well as iterative procedures; see, for instance [7][8][9][10] and the references therein. In 1979, Lions and Mercier [11] showed that the problem (1) is equivalent to find fixed points of the mapping J B λ (I -λA), that is, p * = J B λ (p * -λA(p * )) ⇔ 0 ∈ (A + B)(p * ), where J B λ = (I + λB) -1 . Owing to the fixed point formulation, Lions and Mercier [11] presented the following proximal point method: let p 0 ∈ H be an initial point and In 2012, Takahashi et al. [16] introduced an iterative scheme to solve the problem (1) by combining Mann-type and Halpern-type algorithms with the proximal point method. Recently, Lorenz and Pock [13] have defined an iterative algorithm related to the inertial extrapolation technique.
Over the past year, many significant techniques, concepts of nonlinear and analytical optimization that fit in Euclidean spaces have been extended to Riemannian manifolds. From the Riemannian geometry point of view, some non-convex constrained optimization problems can be viewed as convex unconstrained optimization problems by the introduction of a suitable Riemannian metric (see, e.g. [18][19][20][21][22][23][24][25][26] and the references therein).
In recent years, several researchers have extended the relevance of inclusion theory from linear spaces to the Riemannian context. For instance, Ferreira et al. [27] considered inclusion problem (1) in the setting of Hadamard manifolds. Later on, Ansari et al. [28] introduced Korpelevich's algorithm to solve the inclusion problem (1) and discussed its convergence. Moreover, they [28] obtained the relationship between the set of singularities of inclusion problems and fixed points of the resolvent of maximal monotone vector fields in Hadamard manifolds. In 2019, Al-Homidan et al. [29] presented Halpern-type and Mann-type iterative methods for approximating singularities of the inclusion problem (1) in the framework of Hadamard manifolds. Very recent, Ansari and Babu [30] presented the proximal point method for finding singularities of the inclusion problem (1) on Hadamard manifolds. The authors [30] also devoted their results to convex minimization problems and variational inequality problems.
Inspired by the work mentioned above, the purpose of this paper is to introduce a new class of inverse-strongly-monotone operators, and then develop a new class of iterative algorithms to solve the problem of finding singularities defined by the sum of an inverse-strongly-monotone vector field and a multivalued maximal monotone vector field in Hadamard manifolds.
The paper is organized as follows: In the next section, we give some fundamental concepts of geometry and nonlinear analysis in Riemannian manifolds. In Sect. 3, we construct the inclusion problem (1) in the setting of Hadamard manifolds and exhibit the concept of monotonicity for single-valued as well as for multivalued vector fields. Some fundamental realized results identified with the monotone vector fields are additionally mentioned. In Sect. 4, we present the Mann-type splitting method and establish convergence theorems of any sequence generated by the proposed algorithm converges to a solution of the proposed problem in Hadamard manifolds. In Sect. 5, an application of this results to solve the convex minimization problems and variational inequality in Hadamard manifolds were presented. In Sect. 6, we provide a numerical example to support the Mann-type splitting method.

Preliminaries
Let M be a connected finite-dimensional Riemannian manifold, ∇ a Levi-Civita connection, and χ a smooth curve on M. F is the unique vector field such that ∇ χ F = 0 for all t ∈ [a, b], where 0 is the zero section of the tangent bundle TM. Then the parallel transport P χ, If χ is a minimizing geodesic joining p to q, then we write P q,p instead of P χ,q,p . Note that, for every a, b, b 1 , b 2 ∈ R, we have ,χ(a) and P -1 , that is, the parallel transport preserves the inner product, for all υ, ν ∈ T χ(a) M.
A Riemannian manifold is complete if for any p ∈ M all geodesic emanating from p are defined for all t ∈ R.
Let M be a complete Riemannian manifold and p ∈ M. The exponential map exp p : T p M → M is defined as exp p ν = χ ν (1, p), then, for any value of t, we have exp p tν = χ ν (t, p). Note that the mapping exp p is differentiable on T p M for every p ∈ M. The exponential map has inverse exp -1 A complete simply connected Riemannian manifold of nonpositive sectional curvature is said to be an Hadamard manifold. Throughout, M always denotes a finite-dimensional Hadamard manifold. The following proposition is outstanding and will be helpful.

Proposition 1 ([31])
Let p ∈ M. The exp p : T p M → M is a diffeomorphism, and for any two points p, q ∈ M there exists a unique normalized geodesic joining p to q, which is can be expressed by the formula A geodesic triangle (p 1 , p 2 , p 3 ) of a Riemannian manifold M is a set consisting of three points p 1 , p 2 and p 3 , and three minimizing geodesics joining these points.

Proposition 2 ([31]
) Let (p 1 , p 2 , p 3 ) be a geodesic triangle. Then and Moreover, if θ is the angle at p 1 , then we have The following relation between geodesic triangles in Riemannian manifolds and triangles in R 2 can be found in [32].

Lemma 1 ([32]
) Let (p 1 , p 2 , p 3 ) be a geodesic triangle in a Hadamard manifold M. Then there exists a triangle (p 1 , p 2 , p 3 ) for (p 1 , p 2 , p 3 ) such that d(p i , p i+1 ) = p ip i+1 , with the indices taken modulo 3; it is unique up to an isometry of R 2 .
The triangle (p 1 , p 2 , p 3 ) in Lemma 1 is said to be a comparison triangle for (p 1 , p 2 , p 3 ). The points p 1 , p 2 , p 3 are called comparison points to the points p 1 , p 2 , p 3 , respectively. Lemma 2 Let (p 1 , p 2 , p 3 ) be a geodesic triangle in M and (p 1 , p 2 , p 3 ) be its comparison triangle.
(i) Let θ 1 , θ 2 , θ 3 (respectively, θ 1 , θ 2 , θ 3 ) be the angles of (p 1 , p 2 , p 3 ) (respectively, (p 1 , p 2 , p 3 )) at the vertices p 1 , p 2 , p 3 (respectively, p 1 , p 2 , p 3 ). Then (ii) Let q be a point on the geodesic joining p 1 to p 2 and q its comparison point in the interval [p 1 , Definition 1 A subset in a Hadamard manifold M is called geodesic convex if for all p and q in , and for any geodesic χ : Particularly, for all q ∈ M, the function d(·, q) : M → R is a geodesic convex function.
We now present the results of parallel transport which will be helpful in the sequel.
Remark 2 ([33]) Let p, q, r ∈ M and ν ∈ T p M, and using (5) and Remark 1, Let us end the preliminary section with the following results, which are important in establishing our convergence theorem.

Definition 3 ([19]) Let be a nonempty subset of M and {p n } be a sequence in M.
Then {p n } is said to be Fejér monotone with respect to if for all q ∈ and n ∈ N,

Lemma 3 ([19])
Let be a nonempty subset of M and {p n } ⊂ M be a sequence in M such that {p n } is a Fejér monotone with respect to . Then the following hold: (iii) assume that any cluster point of {p n } belongs to , then {p n } converges to a point in .

Problem formulations
Given is a nonempty subset of a Hadamard manifold M. Let ( ) denote the set of all single-valued vector fields A : → TM such that A(p) ∈ T p M, for each p ∈ . X( ) denote the set of all multivalued vector fields B : Let a vector field A ∈ ( ) and a vector field B ∈ X( ). In this paper, we consider the following inclusion problem: find p * ∈ such that We denote by (A + B) -1 (0) the set of singularities of the problem (8).
In this article we work mainly with specific classes of vector fields which are defined in the following.

Definition 4 ([34, 35]) A vector field
(ii) maximal monotone if it is monotone and for all p ∈ and υ ∈ T p , the condition implies that υ ∈ B(p).
The concept of the resolvent for multivalued vector fields and firmly nonexpansive mappings on Hadamard manifolds was introduce by Li et al. [24] and reads as follows.

Definition 6 ([37]) Let a vector field
Then T is said to be firmly nonexpansive if for any two points p, q ∈ , the function : is nonincreasing.
Let T : → be a nonexpansive mapping, i.e., d(T(p), T(q)) ≤ d(p, q) for all p, q ∈ . By Definition 7, it is clear that any firmly nonexpansive mapping T is nonexpansive. In particular, the monotonicity and nonexpansivity are firmly related.

Theorem 1 ([37]) Let a vector field B ∈ X( ) is monotone if and only if J B
λ is single-valued and firmly nonexpansive.
Let be a nonempty closed geodesic convex subset of M. The projection operator P (·) : M → is defined for any p ∈ M by P (p) := {r : d(p, r) ≤ d(p, q), ∀q ∈ }. The projection operator P is firmly nonexpansive as described in the following proposition [37].

Proposition 4 ([37]) Let be a nonempty closed geodesic convex subset of M. Then the following assertions holds:
(i) P is single-valued and firmly nonexpansive; Recently, Ansari et al. [28] obtained the relationship between a fixed point of T A,B λ (see Lemma 5) and a singularity of the inclusion problem (8) as follows.

Proposition 5 ([28])
For each p ∈ , the following assertions are equivalent: Moreover, they [28] also provided the following lemma which is useful in establishing the convergence result of the inclusion problem (8).
Next, let us introduce the concept of an inverse-strongly-monotone vector field in Hadamard manifolds.

Definition 8 A vector
In this case α-inverse-strongly-monotone. The reason for us to provide this definition is that β-strongly monotone and K -Lipschitz continuous vector field must be β K 2 -inversestrongly-monotone. (It is seen from the definition.) Moreover, we can see that if A is αinverse-strongly-monotone, then it is 1 α -Lipschitz continuous. Indeed, let A be α-inverse-strongly-monotone, then by the definition we have this implies that for all p, q ∈ , where α > 0. Thus, A is 1 α -Lipschitz continuous. Conversely, let A be 1 α -Lipschitz continuous, then by the definition we have for all p, q ∈ , where α > 0. Thus, A is α-inverse-strongly-monotone. Now, we provide some examples of inverse-strongly-monotone vector fields.
We have the following lemma.
Lemma 5 Let A ∈ ( ) be an α-inverse-strongly-monotone vector field, where α > 0, and B ∈ X( ) a maximal monotone vector field. Then the following properties hold: Proof Conclusion (ii) follows from Proposition 5 and the maximal monotonicity of B.

Mann-type splitting method
In this section, we present the conditions that guarantee the convergence of the Manntype splitting method in Hadamard manifolds and the proof. Theorem 2 Let be a nonempty, closed and geodesic convex subset of a Hadamard manifold M. Let A ∈ ( ) be an α-inverse-strongly-monotone vector field, where α > 0, and B ∈ X( ) a maximal monotone vector field with (A + B) -1 (0) = ∅. Choose p 0 ∈ and define {q n } and {p n } as follows: for all n ∈ N, where {γ n } is a sequence in (0, 1) and {λ n } is a real positive sequence satisfying the following conditions: (i) 0 < γ 1 ≤ γ n ≤ γ 2 < 1, ∀n ∈ N, (ii) 0 <λ ≤ λ n ≤ 2α < ∞, ∀n ∈ N. Then {p n } is convergent to a solution of the inclusion problem (8).
Proof Let x ∈ (A + B) -1 (0). From (ii) of Lemma 5, we have x = T A,B λ n (x) = J B λ (W λ (x)). By the nonexpansiveness of J B λ n and W λ n , gives Let χ : [0, 1] → M be geodesic joining p n to q n . Thus, (21) can be written as p n+1 = χ(1γ n ), respectively. By using the geodesic convexity of Riemannian distance, we have Fix n ∈ N and for x ∈ (A + B) -1 (0). Let (p n , q n , x) ⊆ M be a geodesic triangle with vertices p n , q n and x, and (p n , q n , x) ⊆ R 2 be the corresponding comparison triangle, one obtains d(p n , x) = p nx , d(q n , x) = q nx , and d(p n , q n ) = p nq n .
In the order to present an example in support of our main theorem, we need the following results.
Let M be a Riemannian manifold and φ : M → R a differentiable function. The direc- The gradient of φ at p ∈ M [39] is given by grad φ(p), ν := φ (p; ν) for all ν ∈ T p M. If φ : M → R is a twice differentiable function, then the Hessian of φ at p ∈ M [40], denoted by Hessφ, is defined by The set of all subgradients of ϕ, denoted by ∂ϕ(p) is said to be the subdifferential of ϕ at p, which is a closed geodesic convex (possibly empty) set. Next, we present an example in the cone of the positive semidefinite matrices with other metrics.
Example 3 Let S n be the set of symmetric matrices, S n + be the cone of the symmetric positive semidefinite matrices and S n ++ be the cone of the symmetric positive-definite matrices both n × n. X, Y ∈ S n + , X X (or X Y ) means that Y -X ∈ S n + and X Y (or X ≺ Y ) means that Y -X ∈ S n ++ . Following Rothaus [41], let M := (S n ++ , ·, · ) be the Riemannian manifold endowed with the Riemannian metric defined by where Tr(X) denotes the trace of matrix X ∈ S n and T X M ≈ S n , with the corresponding norm denoted by · . The gradient and the Hessian of a twice differentiable function φ : S n ++ → R are given by where V ∈ T X M, and φ (X) and φ (X) are the Euclidian gradient and Hessian of φ at X, respectively. In fact, M is a Hadamard manifold with curvature is not identically zero; see [42, Theorem 1.2. p. 325] for further details. The unique geodesic segment connecting any X, Y ∈ M is given by see, for example [43]. From the last equation Thus, for all X ∈ M, exp -1 X : M → T X M and exp X : T X M → M are defined, respectively, by Now, since the Riemannian distance d is given by d(X, Y ) = exp -1 X Y , from (27), along with first expression in (30), we have where η i (X -1/2 YX -1/2 ) denotes the ith eigenvalue of the symmetric matrix X -1/2 YX -1/2 . Following [22], let the function φ : S n ++ → R defined by where a > 0. The Euclidian gradient and Hessian of φ are given, respectively, by where X ∈ S n ++ and V ∈ S n . By combining (28), (29), (33) and (34), we obtain, after some calculations, for all X ∈ M and V ∈ T X M. We further have hess φ(X)V , V = 2a Tr(X -1 V ) 2 ≥ 0. Thus, φ is geodesic convex in M. Moreover, (27) together with (36) gives hess φ(X)V = 2a Tr(X -1 V ) for all X ∈ M and V ∈ T X M. If we assume that V 2 = Tr(VX -1 VX -1 ) = 1, then Tr(X -1 V ) ≤ √ n. Hence, Therefore, (25) and Proposition 6 imply that grad φ is Lipschitz with constant K ≤ 2a √ n. We also have grad φ is 1 K -inverse-strongly-monotone vector field with constant K ≤ 2a √ n. Let = {X ∈ S n ++ : η min (X) ≥ 1}, where η min (X) denotes the minimum eigenvalue of the matrix X, be a nonempty, closed and geodesic convex subset of M and ϕ : → R ∪ {+∞} be a proper, lower semicontinuous and geodesic convex defined by where η i (X -1 ) is the ith eigenvalue of the matrix X -1 . One can see that I is a minimizer of ϕ, where I denotes the identity matrix. By Definition 9, the subdifferential of ϕ at X is defined by The subdifferential ∂ϕ of ϕ is a maximal monotone vector field, according to Lemma 6. Moreover, we have Since the minimizer of ϕ is I, it is easy to see that 0 ∈ ∂ϕ(I).
Let A : → S n is a 1 K -inverse-strongly-monotone vector field defined by where K ≤ 2 √ n, and B : → 2 S n be a maximal monotone multivalued field defined by We see that (A + B) -1 (0) = {I}. Choose initial point X 0 ∈ , then Theorem 2 is applicable leading us to conclude that any sequence generated by Eq. (21) converges to a singularity of the inclusion problem (8).
Remark 3 It is worth noting that φ is non-convex with non-Lipschitz continuous gradient on S n ++ endowed with the Euclidean metric. Thus we cannot apply existence results, e.g. [8,16], to solve the corresponding inclusion problem in the Euclidean setting.

Applications
In this section, we shall utilize the Mann-type splitting method presented in the paper to study the convex minimization problems and variational inequality problems.

Convex minimization problems
Let φ, ϕ : → R ∪ {+∞} are proper, lower semicontinuous and geodesic convex functions such that φ is differentiable. We consider the problem of finding p * ∈ such that The problem is said to be a convex minimization problem. We denote S by the set of minimizers of the problem (37), that is, It is to see that the problem (37) is equivalent to the following inclusion problem: find p ∈ such that 0 ∈ grad f (p) + ∂g(p), that is, For further details see [30]. If φ : → R ∪ {+∞} is a proper, twice continuously differentiable and geodesic convex function such that Hess φ is bounded, then, by Proposition 6, grad φ is K -Lipschitz continuous vector field. Then grad φ is 1 K -inverse-strongly-monotone vector field. Moreover, from Lemma 6, ∂ϕ is a maximal monotone vector field. By replacing A and B by grad φ and ∂ϕ, respectively, in Theorem 2, we get the following result for convex minimization problem (37). q n = J ∂ϕ λ n exp p n -λ n grad φ(p n ) , p n+1 = exp p n (1γ n ) exp -1 p n q n , for all n ∈ N, where {γ n } is a sequence in (0, 1) and {λ n } is a real positive sequence satisfying the following conditions: (i) 0 < γ 1 ≤ γ n ≤ γ 2 < 1, ∀n ∈ N, (ii) 0 <λ ≤ λ n ≤ 2α < ∞, ∀n ∈ N. Then {p n } is convergent to a solution of the convex minimization problem (37).
Proof By replacing A and B by grad φ and ∂ϕ, respectively, in Theorem 2, we get the required result.

Variational inequalities
A monotone variational inequality problem (VIP) on a Hadamard manifold was initially studied by Németh [44]. Then the problem is to find p * ∈ such that where V : → TM is a single-valued vector field. VIP(V , ) denotes the set of solutions of the problem (39). Let N (p) denote the normal cone of the set at p ∈ : N (p) := ν ∈ T p M : ν, exp -1 p q ≤ 0, ∀q ∈ .