inclusion problems in Hadamard manifolds

R E S E A R C H Open Access

Iterative algorithm for singularities of

inclusion problems in Hadamard manifolds

Parin Chaipunya^1,2 , Konrawut Khammahawong^1,2* and Poom Kumam^1,3*

*Correspondence:

k.konrawut@gmail.com;

poom.kumam@mail.kmutt.ac.th

1Center of Excellence in Theoretical and Computational Science (TaCS-CoE), Faculty of Science, Science Laboratory Building, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thung Khru, Bangkok 10140, Thailand Full list of author information is available at the end of the article

Abstract

The main purpose of this paper is to introduce a new iterative algorithm to solve inclusion problems in Hadamard manifolds. Moreover, applications to convex minimization problems and variational inequality problems are studied. A numerical example also is presented to support our main theorem.

MSC: 47H05; 547J20; 49J53; 51H25; 58A05

Keywords: Hadamard manifolds; Inclusion problems; Inverse-strongly-monotone vector field; Maximal monotone vector field; Proximal point method

1 Introduction

LetHbe a Hilbert space,A:H→Han operator andB:H→2^Ha multivalued operator.

The inclusion problem is to findp^∗∈Hsuch that 0∈(A+B)

p^∗

. (1)

IfA= 0, then the problem (1) becomes the inclusion problem introduced by Rockafel- lar [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 prob- lem (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 mappingJ_λ^B(I–λA), that is,p^∗=J_λ^B(p^∗–λA(p^∗))⇔0∈(A+B)(p^∗), where J_λ^B= (I+λB)^–1. Owing to the fixed point formulation, Lions and Mercier [11] presented the following proximal point method: letp₀∈Hbe an initial point and

pn+1=J_λ^B

pn–λA(pn)

, ∀n∈N, (2)

whereλ> 0. The proximal point method (2) has been extensively studied with many au- thors; see, e.g. [8,12–17]. In particular, Chen and Rockafellar [14] studied convergence rates of the method (2). Afterwards, Tseng [15] proposed the modification for approximat- ing singularities of the inclusion problem (1), also known as Tseng’s splitting algorithm.

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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. Re- cently, 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 optimiza- tion problems can be viewed as convex unconstrained optimization problems by the in- troduction of a suitable Riemannian metric (see, e.g. [18–26] 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 in- clusion 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 singulari- ties 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 prob- lem (1) in the framework of Hadamard manifolds. Very recent, Ansari and Babu [30] pre- sented 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 it- erative 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 con- cepts of geometry and nonlinear analysis in Riemannian manifolds. In Sect.3, we con- struct 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 con- vergence 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 sup- port the Mann-type splitting method.

2 Preliminaries

LetMbe a connected finite-dimensional Riemannian manifold,∇ a Levi-Civita connec- tion, andχ a smooth curve onM.Fis the unique vector field such that∇χF= 0 for all t∈[a,b], where 0 is the zero section of the tangent bundleTM. Then the parallel transport Pχ,χ(b),χ(a):Tχ(a)M→Tχ(b)Mon the tangent bundleTMalongχ: [a,b]→Ris defined by

Pχ,χ(b),χ(a)(ν) =F χ(b)

, ∀a,b∈Randν∈Tχ(a)M.

Ifχis a minimizing geodesic joiningptoq, then we write Pq,pinstead of Pχ,q,p. Note that, for everya,b,b₁,b₂∈R, we have

P_{χ(b2),χ(b1)}◦P_χ(b1),χ(a)= P_χ(b2),χ(a) and P^–1_χ(b),χ(a)= P_χ(a),χ(b).

Also, Pχ(b),χ(a)is an isometry fromTχ(a)MtoTχ(b)M, that is, the parallel transport pre- serves the inner product,

Pχ(b),χ(a)(υ), Pχ(b),χ(a)(ν)

χ(b)=υ,νχ(a), (3)

for allυ,ν∈T_χ(a)M.

A Riemannian manifold is complete if for anyp∈Mall geodesic emanating frompare defined for allt∈R.

Let Mbe a complete Riemannian manifold and p∈M. The exponential mapexp_p : TpM→Mis defined asexp_pν=χν(1,p), then, for any value oft, we haveexp_ptν=χν(t,p).

Note that the mappingexp_pis differentiable onTpMfor everyp∈M. The exponential map has inverseexp^–1_p :M→TpM. Moreover, for anyp,q∈M, we haved(p,q) =exp^–1_p q, whered(·,·) is a Riemannian distance.

A complete simply connected Riemannian manifold of nonpositive sectional curvature is said to be anHadamard manifold.Throughout,Malways denotes a finite-dimensional Hadamard manifold. The following proposition is outstanding and will be helpful.

Proposition 1([31]) Let p∈M.Theexp_p:TpM→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

χ(t) =exp_ptexp^–1_p q, ∀t∈[0, 1].

A geodesic triangle(p1,p2,p3) of a Riemannian manifoldMis a set consisting of three pointsp₁,p₂andp₃, and three minimizing geodesics joining these points.

Proposition 2([31]) Let(p₁,p₂,p₃)be a geodesic triangle.Then d²(p1,p2) +d²(p2,p3) – 2

exp^–1_p

2p1,exp^–1_p

2p3

≤d²(p3,p1) (4)

and

d²(p1,p2)≤ exp^–1_p

1p3,exp^–1_p

1p2

+ exp^–1_p

2p3,exp^–1_p

2p1

. (5)

Moreover,ifθ is the angle at p1,then we have exp^–1_p

1p2,exp^–1_p

1p3

=d(p2,p1)d(p1,p3)cosθ.

The following relation between geodesic triangles in Riemannian manifolds and trian- gles inR²can be found in [32].

Lemma 1([32]) Let(p1,p2,p3)be a geodesic triangle in a Hadamard manifold M.Then there exists a triangle(p1,p2,p3)for(p1,p2,p3)such that d(pi,pi+1) =pi–pi+1,with the indices taken modulo3;it is unique up to an isometry ofR².

The triangle(p1,p2,p3) in Lemma1is said to be acomparison trianglefor(p1,p2,p3).

The pointsp₁,p₂,p₃are called comparison points to the pointsp₁,p₂,p₃, respectively.

Lemma 2 Let(p1,p2,p3)be a geodesic triangle in M and(p1,p2,p3)be its comparison triangle.

(i) Letθ₁,θ₂,θ₃(respectively,θ₁,θ₂,θ₃)be the angles of(p1,p2,p3)(respectively, (p1,p2,p3))at the verticesp1,p2,p3(respectively,p1,p2,p3).Then

θ₁≤θ₁, θ₂≤θ₂ and θ₃≤θ₃.

(ii) Letqbe a point on the geodesic joiningp1top2andqits comparison point in the interval[p1,p2].Ifd(p1,q) =p1–qandd(p2,q) =p2–q,then

d(p3,q)≤ p3–q.

Definition 1 A subsetin a Hadamard manifoldMis calledgeodesic convexif for allp andqin, and for any geodesicχ: [a,b]→M,a,b∈Rsuch thatp=χ(a) andq=χ(b), one hasχ((1 –t)a+tb)∈for allt∈[0, 1].

Definition 2 A functionφ:M→Ris calledgeodesic convexif for any geodesicχinM, the composition functionφ◦χ: [a,b]→Ris convex, that is,

(φ◦χ)

ta+ (1 –t)b

≤t(φ◦χ)(a) + (1 –t)(φ◦χ)(b), a,b∈R, and∀t∈[0, 1].

Proposition 3([31]) Let d:M×M→Rbe the distance function.Then d(·,·)is a geodesic convex function with respect to the product Riemannian metric,that is, for any pair of geodesicsχ₁: [0, 1]→M andχ₂: [0, 1]→M the following inequality holds:

d

χ₁(t),χ₂(t)

≤(1 –t)d

χ₁(0),χ₂(0) +td

χ₁(1),χ₂(1)

, ∀t∈[0, 1].

Particularly, for allq∈M, the functiond(·,q) :M→Ris a geodesic convex function.

We now present the results of parallel transport which will be helpful in the sequel.

Remark1 ([24]) Ifp,q∈Mandν∈TpM, then ν, –exp^–1_p q

=

ν, Pp,qexp^–1_q p

=

Pq,pν,exp^–1_q p

. (6)

Remark2 ([33]) Letp,q,r∈Mandν∈TpM, and using (5) and Remark1, ν,exp^–1_p q

≤

ν,exp^–1_p r +

ν, Pp,rexp^–1_r q

. (7)

Let us end the preliminary section with the following results, which are important in establishing our convergence theorem.

Definition 3([19]) Letbe a nonempty subset ofMand{pn}be a sequence inM. Then {pn}is said to beFejér monotonewith respect toif for allq∈andn∈N,

d(pn+1,q)≤d(pn,q).

Lemma 3([19]) Letbe a nonempty subset of M and{pn} ⊂M be a sequence in M such that{pn}is a Fejér monotone with respect to.Then the following hold:

(i) for everyq∈,{d(p_n,q)}converges;

(ii) {pn}is bounded;

(iii) assume that any cluster point of{pn}belongs to,then{pn}converges to a point in.

3 Problem formulations

Givenis a nonempty subset of a Hadamard manifoldM. Let () denote the set of all single-valued vector fieldsA:→TMsuch thatA(p)∈TpM, for eachp∈.X() denote the set of all multivalued vector fieldsB:→2^TMsuch thatB(p)⊆TpMfor allp∈, and denoteD(B) the domain ofBdefined byD(B) ={p∈:B(p)=∅}.

Let a vector fieldA∈ () and a vector fieldB∈X(). In this paper, we consider the following inclusion problem: findp^∗∈such that

0∈(A+B) p^∗

. (8)

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 fieldA∈ () is called (i) monotoneif

A(p),exp^–1_p q

≤

A(q), –exp^–1_q p

, ∀p,q∈; (ii) β-strongly monotoneif there isβ> 0such that

A(p),exp^–1_p q +

A(q),exp^–1_q p

≤–βd²(p,q), ∀p,q∈; (iii) K-Lipschitz continuousif there isK> 0such that

Pp,qA(q) –A(p)≤Kd(p,q), ∀p,q∈. Definition 5([36]) A vector fieldB∈X() is called

(i) monotoneif for allp,q∈D(B) υ,exp^–1_p q

≤

ν, –exp^–1_q p

, ∀υ∈B(p)and∀ν∈B(q);

(ii) maximal monotoneif it is monotone and for allp∈andυ∈Tp, the condition υ,exp^–1_p q

≤

ν, –exp^–1_q p

, ∀q∈D(B)and∀ν∈B(q), implies thatυ∈B(p).

The concept of the resolvent for multivalued vector fields and firmly nonexpansive map- pings on Hadamard manifolds was introduce by Li et al. [24] and reads as follows.

Definition 6 ([37]) Let a vector fieldB∈X() andλ∈(0,∞). Theλ-resolventofBis multivalued mapJ_λ^B:→2defined by

J_λ^B(p) :=

r∈:p∈exp_rλB(r)

, ∀p∈.

Definition 7([37]) LetT:⊆M→Mbe a mapping. ThenTis said to befirmly nonex- pansiveif for any two pointsp,q∈, the function: [0, 1]→[0, +∞] defined by

(t) :=d

exp_ptexp^–1_p T(p),exp_qtexp^–1_q T(q)

, ∀t∈[0, 1], is nonincreasing.

LetT:→be a nonexpansive mapping, i.e.,d(T(p),T(q))≤d(p,q) for allp,q∈. By Definition7, it is clear that any firmly nonexpansive mappingT 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_λ^Bis single-valued and firmly nonexpansive.

Letbe a nonempty closed geodesic convex subset ofM. Theprojection operator P(·) : M→is defined for anyp∈MbyP(p) :={r:d(p,r)≤d(p,q),∀q∈}. The projection operatorPis 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) Pis single-valued and firmly nonexpansive;

(ii) For allp∈M,r=P(p)if and only ifexp^–1_r p,exp^–1_r q ≤0,for allq∈.

Recently, Ansari et al. [28] obtained the relationship between a fixed point ofT_λ^A,B(see Lemma5) and a singularity of the inclusion problem (8) as follows.

Proposition 5([28]) For each p∈,the following assertions are equivalent:

(i) p∈(A+B)^–1(0);

(ii) p=T_λ^A,B(p),∀λ∈(0,∞).

Moreover, they [28] also provided the following lemma which is useful in establishing the convergence result of the inclusion problem (8).

Lemma 4 ([29]) Let be a nonempty closed subset of a Hadamard manifold M and B∈X()a maximal monotone.Let{λ_n} ⊂(0,∞)withlim_n→∞λ_n=λ> 0and a sequence {pn} ⊂withlim_n→∞pn=p∈such thatlim_n→∞J_λ^B_n(pn) =q.Then q=J_λ^B(p).

Next, let us introduce the concept of an inverse-strongly-monotone vector field in Hadamard manifolds.

Definition 8 A vectorA∈ () is said to beinverse-strongly-monotone if there exists α> 0 such that

A(p),exp^–1_p q +

A(q),exp^–1_q p

≤–αA(p) – Pp,qA(q)², ∀p,q∈;

In this caseα-inverse-strongly-monotone. The reason for us to provide this definition is thatβ-strongly monotone andK-Lipschitz continuous vector field must be_K^β2-inverse- strongly-monotone. (It is seen from the definition.) Moreover, we can see that ifAisα- inverse-strongly-monotone, then it is ¹_α-Lipschitz continuous.

Indeed, letAbeα-inverse-strongly-monotone, then by the definition we have αA(p) – Pp,qA(q)²≤–

A(p),exp^–1_p q –

A(q),exp^–1_q p

=

Pp,qA(q) –A(p),exp^–1_p q

≤Pp,qA(q) –A(p)exp^–1_p q

=Pp,qA(q) –A(p)d(p,q), this implies that

P_p,qA(q) –A(p)≤1 αd(p,q),

for allp,q∈, whereα> 0. Thus,Ais_α¹-Lipschitz continuous.

Conversely, letAbe_α¹-Lipschitz continuous, then by the definition we have –αP_p,qA(q) –A(p)²≥–d(p,q)P_p,qA(q) –A(p)

= –exp^–1_p qPp,qA(q) –A(p)

≥–

Pp,qA(q) –A(p),exp^–1_p q

=

A(p),exp^–1_p q +

A(q),exp^–1_q p ,

for allp,q∈, whereα> 0. Thus,Aisα-inverse-strongly-monotone.

Now, we provide some examples of inverse-strongly-monotone vector fields.

Example1 LetR⁺⁺n is the product space ofR⁺⁺, that is,R⁺⁺n ={p= (p₁,p₂, . . . ,p_n)∈Rⁿ:p_i>

0, i= 1, . . . ,n}. LetM= (R⁺⁺_n ,·,·) with metric defined byυ,ν:=υ^TG(p)ν, forp∈R⁺⁺_n andυ,ν∈TpR⁺⁺n whereG(p) is a diagonal metric defined byG(p) =diag(p^–2₁ ,p^–2₂ , . . . ,p^–2_n ).

Specially,M= (R⁺⁺_n ,·,·) is a Hadamard manifold with sectional curvature zero (see [18]).

LetA:R⁺⁺n →TR⁺⁺n be a single-valued vector field defined by A(p)

i:=a_ib_ip²_ie^–bipi+ 2c_ip_iln(p_i) +d_ip_i, i= 1, 2, . . . ,n,

whereai,bi,di∈R⁺andci∈R⁺⁺satisfyci>ai. Hence,Ais aK-Lipschitz continuous with K< ⁿ_i=1(ai+ 2ci)²; for more details see [22]. Thus, we see thatAis _K¹-inverse-strongly- monotone whereK< ⁿ_i=1(a_i+ 2c_i)².

Example2 Letn= 1 in Example1andA:R⁺⁺→TR⁺⁺ be a single-valued vector field defined by

A(p) := 1

32pln(p) +ln(p).

Hence,Ais a³³₃₂-Lipschitz continuous; for further details see [38]. Thus, we see thatAis

32

33-inverse-strongly-monotone.

We have the following lemma.

Lemma 5 Let A∈ ()be anα-inverse-strongly-monotone vector field,whereα> 0,and B∈X()a maximal monotone vector field.Then the following properties hold:

(i) for eachλ∈[0, 2α],the mappingWλ:→defined byWλ(p) =exp_p–λA(p)is nonexpansive;

(ii) for eachλ> 0,the mappingT_λ^A,B:→defined byT_λ^A,B(p) =J_λ^B(Wλ(p))is well defined and(A+B)^–1(0) =Fix(T_λ^A,B),whereFix(T_λ^A,B)is the set of fixed points of T_λ^A,B;

(iii) for eachλ∈(0, 2α],T_λ^A,Bis nonexpansive.

Proof Conclusion (ii) follows from Proposition5and the maximal monotonicity ofB.

In order to prove (i), let(p,Wλ(p),Wλ(q))⊆Mbe a geodesic triangle with verticesp, Wλ(p) andWλ(q), and(p,Wλ(p),Wλ(q))⊆R² is the corresponding comparison trian- gle. Then we haved(p,Wλ(p)) =p–Wλ(p),d(Wλ(p),Wλ(q)) =Wλ(p) –Wλ(q), and d(Wλ(q),p) =Wλ(q) –p. Again, letting(p,q,Wλ(q))⊆Mbe a geodesic triangle with verticesp,qandWλ(q), and(p,q,Wλ(q))⊆R²be the corresponding comparison trian- gle, one obtains

d(p,q) =p–q, d

q,Wλ(q)

=q–Wλ(q) and d

Wλ(q),p

=Wλ(q) –p. Now,

d²

Wλ(p),Wλ(q)

=Wλ(p) –Wλ(q)²

=Wλ(p) –p

+ (p–q) –

Wλ(q) –q²

=p–q²+Wλ(p) –p –

Wλ(q) –q² + 2

Wλ(p) –p –

Wλ(q) –q ,p–q

=d²(p,q) +Wλ(p) –p –

Wλ(q) –q² + 2

Wλ(p) –p,p–q + 2

Wλ(q) –q,q–p + 2p–q²– 2p–q²+ 2q–p²– 2q–p²

=d²(p,q) +Wλ(p) –p –

Wλ(q) –q² + 2

Wλ(p) –p,p–q

+ 2p–q,p–q– 2d²(p,q) + 2

Wλ(q) –q,q–p

+ 2q–p,q–p– 2d²(q,p)

=d²(p,q) +Wλ(p) –p –

Wλ(q) –q²

+ 2

Wλ(p) –q,p–q

– 2d²(p,q) + 2

Wλ(q) –p,q–p

– 2d²(q,p). (9)

Letθ,θ be the angles at the verticesq,q. By (i) of Lemma2, we getθ≤θ. Besides, by Proposition2, we have

Wλ(p) –q,p–q

=Wλ(p) –qp–qcosθ

=d

Wλ(p),q

d(q,p)cosθ

≤d

Wλ(p),q

d(q,p)cosθ

=

exp^–1_q Wλ(p),exp^–1_q p

. (10)

Repeating the argument above gives Wλ(q) –p,q–p

≤

exp^–1_p Wλ(q),exp^–1_p q

. (11)

Moreover, we have Wλ(p) –p

–

Wλ(q) –q²

=Wλ(p) –p²– 2

Wλ(p) –p,Wλ(q) –q

+Wλ(q) –q²

=d²

Wλ(p),p – 2

Wλ(p) –p,Wλ(q) –q +d²

Wλ(q),q

=exp^–1_p Wλ(p)²– 2

Wλ(p) –p,Wλ(q) –q

+exp^–1_q Wλ(q)²

=–λA(p)²– 2

Wλ(p) –p,Wλ(q) –q

+–λA(q)²

=λ²A(p)²– 2

Wλ(p) –p,Wλ(q) –q

+λ²A(q)². (12)

Consider –2

Wλ(p) –p,Wλ(q) –q

= –2

Wλ(p) –q+q–p,Wλ(q) –q

= 2

q–Wλ(p),Wλ(q) –q + 2

p–q,Wλ(q) –q

= 2

q–Wλ(p),Wλ(q) –q

+ 2q–Wλ(p)² – 2q–Wλ(p)²+ 2

p–q,Wλ(q) –q

= 2

q–Wλ(p),Wλ(q) –Wλ(p) – 2d²

q,Wλ(p) + 2

p–q,Wλ(q) –q

. (13)

Repeating the argument above yields q–Wλ(p),Wλ(q) –Wλ(p)

≤ exp^–1_W

λ(p)q,exp^–1_W

λ(p)Wλ(q)

(14) and

p–q,Wλ(q) –q

≤

exp^–1_q p,exp^–1_q Wλ(q)

. (15)

Substituting (14) and (15) into (13) gives –2

Wλ(p) –p,Wλ(q) –q

≤2 exp^–1_W

λ(p)q,exp^–1_W

λ(p)Wλ(q) – 2d²

q,Wλ(p) + 2

exp^–1_q p,exp^–1_q Wλ(q) . Noting Remarks1and2in the last inequality, we get

–2

Wλ(p) –p,Wλ(q) –q

≤2 exp^–1_W

λ(p)q,exp^–1_W

λ(p)q + 2

exp^–1_W

λ(p)q, P_Wλ(p),qexp^–1_q Wλ(q) – 2d²

q,Wλ(p) + 2

exp^–1_q Wλ(p),exp^–1_q Wλ(q) + 2

Pq,Wλ(p)exp^–1_W

λ(p)p,exp^–1_q Wλ(q)

= –2

exp^–1_q Wλ(p),exp^–1_q Wλ(q) + 2

exp^–1_q Wλ(p),exp^–1_q Wλ(q) – 2

exp^–1_p Wλ(p), Pp,qexp^–1_q Wλ(q)

= –2

–λA(p), Pp,q–λA(q)

= –2λ²

A(p), P_p,qA(q)

. (16)

Substituting (16) into (12) yields Wλ(p) –p

–

Wλ(q) –q²≤λ²A(p)²– 2λ²

A(p), Pp,qA(q)

+λ²A(q)²

=λ²A(p)²– 2

A(p), Pp,qA(q)

+A(q)²

=λ²A(p) – Pp,qA(q)². (17) Combining (9), (10), (11) and (17), we obtain

d²

Wλ(p),Wλ(q)

≤d²(p,q) +λ²A(p) – Pp,qA(q)² + 2

exp^–1_q Wλ(p),exp^–1_q p

– 2d²(p,q) + 2

exp^–1_p Wλ(q),exp^–1_p q

– 2d²(q,p). (18)

From Remarks1and2, the last inequality becomes d²

Wλ(p),Wλ(q)

≤d²(p,q) +λ²A(p) – Pp,qA(q)² + 2

exp^–1_q p,exp^–1_q p + 2

Pq,pexp^–1_p Wλ(p),exp^–1_q p – 2d²(p,q) + 2

exp^–1_p q,exp^–1_p q + 2

Pp,qexp^–1_q Wλ(q),exp^–1_p q

– 2d²(q,p)

=d²(p,q) +λ²A(p) – Pp,qA(q)² + 2

–λA(p), –exp^–1_p q + 2

–λA(q), –exp^–1_q p

=d²(p,q) +λ²A(p) – Pp,qA(q)²+ 2λ

A(p)exp^–1_p q + 2λ

A(q),exp^–1_q p .

So, d²

Wλ(p),Wλ(q)

≤d²(p,q) +λ²A(p) – Pp,qA(q)²+ 2λ

A(p)exp^–1_p q + 2λ

A(q),exp^–1_q p

. (19)

SinceAis anα-inverse-strongly-monotone, we get A(p),exp^–1_p q

+

A(q), +exp^–1_q p

≤–αA(p) – Pp,qA(q)². (20)

Substituting (20) into (19), we deduce that d²

Wλ(p),Wλ(q)

≤d²(p,q) +λ²A(p) – Pp,qA(q)²– 2αλA(p) – Pp,qA(q)²

=d²(p,q) –λ(2α–λ)A(p) – Pp,qA(q)². From the fact theλ∈[0, 2α]

d

Wλ(p),Wλ(q)

≤d(p,q),

henceWλ(p) is nonexpansive. To prove (iii), notice thatBis maximal monotone and so the resolvent J_λ^B is firmly nonexpansive. It follows immediately from (i) thatT_λ^A,B(x) =

J_λ^B(Wλ(x)) is nonexpansive.

4 Mann-type splitting method

In this section, we present the conditions that guarantee the convergence of the Mann- type splitting method in Hadamard manifolds and the proof.

Theorem 2 Letbe a nonempty,closed and geodesic convex subset of a Hadamard man- ifold M.Let A∈ ()be anα-inverse-strongly-monotone vector field,whereα> 0,and B∈X()a maximal monotone vector field with(A+B)^–1(0)=∅.Choose p0∈and define {qn}and{pn}as follows:

qn=J_λ^B_n exp_pn

–λnA(pn) , pn+1=exp_pn(1 –γ_n)exp^–1_p

nqn,

(21)

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 <γ₁≤γ_n≤γ₂< 1,∀n∈N, (ii) 0 <λˆ≤λ_n≤2α<∞,∀n∈N.

Then{pn}is convergent to a solution of the inclusion problem(8).

Proof Letx∈(A+B)^–1(0). From (ii) of Lemma5, we havex=T_λ^A,B_n (x) =J_λ^B(Wλ(x)). By the nonexpansiveness ofJ_λ^B_nandWλn, gives

d(qn,x) =d J_λ^B_n

exp_pn

–λnA(pn) ,J_λ^B_n

exp_x

–λnA(x)

≤d exp_pn

–λnA(pn) ,exp_x

–λnA(x)

≤d(pn,x), ∀n∈N. (22) Letχ: [0, 1]→Mbe geodesic joiningp_ntoq_n. Thus, (21) can be written asp_n+1=χ(1 – γ_n), respectively. By using the geodesic convexity of Riemannian distance, we have

d(p_n+1,x) =d

χ(1 –γ_n),x

≤γ_nd(p_n,x) + (1 –γ_n)d(q_n,x)

≤γ_nd(pn,x) + (1 –γ_n)d(pn,x)

=d(pn,x).

Hence,{pn}is Fejér monotone with respect to (A+B)^–1(0). By (ii) of Lemma3,{pn}is bounded. Hence, there exists a subsequence{pn_k}of {pn}which converges to a cluster pointyof{pn}. Next, we show that

n→∞lim d(p_n,q_n) = 0.

Fixn∈Nand forx∈(A+B)^–1(0). Let(pn,qn,x)⊆Mbe a geodesic triangle with vertices pn,qnandx, and(pn,qn,x)⊆R²be the corresponding comparison triangle, one obtains

d(pn,x) =pn–x, d(qn,x) =qn–x, and d(pn,qn) =pn–qn.

Letp_n+1=γ_np_n+ (1 –γ_n)q_nbe the comparison point ofp_n+1. Using (ii) of Lemma2and (22),

d²(p_n+1,x)≤ pn+1–x²

=γ_np_n+ (1 –γ_n)q_n–x²

=γ_npn–x²+ (1 –γ_n)qn–x²–γ_n(1 –γ_n)pn–qn²

=γ_nd²(pn,x) + (1 –γ_n)d²(qn,x) –γ_n(1 –γ_n)d²(pn,qn)

≤γ_nd²(pn,x) + (1 –γ_n)d²(pn,x) –γ_n(1 –γ_n)d²(pn,qn)

=d²(p_n,x) –γ_n(1 –γ_n)d²(p_n,q_n), ∀n∈N, which implies that

γ_n(1 –γ_n)d²(p_n,q_n)≤d²(p_n,x) –d²(p_n+1,x), ∀n∈N. (23) From the fact that 0 <γ₁≤γ_n≤γ₂< 1, we haveγ₁(1 –γ₂)≤γ_n(1 –γ_n) for alln∈N.

Summing (23) fromi= 0 toi=n, we obtain γ₁(1 –γ₂)

n i=0

d²(pi,qi)≤d²(p0,x) –d²(pn+1,x), ∀n∈N.

Lettingn→ ∞, we have γ₁(1 –γ₂)

∞ i=0

d²(pi,qi)≤d²(p0,x) <∞.

Hence,

n→∞lim d(pn,qn) = 0. (24)

Next, we provey∈(A+B)^–1(0). Sinceλˆ≤λ_n≤2α, we may assume without the loss of generality thatlim_k→∞λ_nk=λfor someλ∈[λ, 2α]. Recall thatˆ qn=J_λ^B_n(exp_pn(–λnA(pn))).

Then, by (24) and Lemma4, we obtainlim_k→∞qnk =yand thaty=J_λ^B(exp_y(–λA(y))). It indicates thaty∈(A+B)^–1(0) by applying (ii) of Lemma5. By (iii) of Lemma3, the se- quence{pn}converges to a singularity of the inclusion problem (8). Therefore, the proof

is completed.

In the order to present an example in support of our main theorem, we need the follow- ing results.

LetMbe a Riemannian manifold andφ:M→Ra differentiable function. The direc- tional derivative ofφatpin directionν∈TpMis

φ(p;ν) := lim

t→0⁺

φ(exp_ptν) –φ(p)

t .

The gradient ofφ atp∈M[39] is given bygradφ(p),ν:=φ(p;ν) for allν∈TpM. If φ:M→Ris a twice differentiable function, then the Hessian ofφatp∈M[40], denoted by Hessφ, is defined by

Hessφ(p) :=∇ν

gradφ(p)

, ∀ν∈T_pM, where∇is the Riemannian connection ofM.

The norm of the Hessian, hessφ, atp∈Mis given by hessφ(p):=suphessφ(p)ν:ν∈T_pM,ν= 1

. (25)

Proposition 6([20]) Let M be an Hadamard manifold andφ:M→Ra twice continu- ously differentiable function.IfHessφ is bounded,then the gradient vector fieldgradφis K -Lipschitz continuous.

Definition 9([31]) Letϕ:→Rbe a geodesic convex function. Takep∈, a vector ν∈T_pMis said to be asubgradientofϕatpif and only if

ϕ(q)≥ϕ(p) +

ν,exp^–1_p q

, ∀q∈. (26)

The set of all subgradients ofϕ, denoted by∂ϕ(p) is said to be thesubdifferentialofϕat p, which is a closed geodesic convex (possibly empty) set.

Lemma 6([24]) Letϕ:→R∪ {+∞}be a proper,lower semicontinuous and geodesic convex function and D(ϕ) =.Then the subdifferential∂ϕ ofϕ is a maximal monotone vector field.

Next, we present an example in the cone of the positive semidefinite matrices with other metrics.

Example3 LetSⁿbe the set of symmetric matrices,Sⁿ₊be the cone of the symmetric posi- tive semidefinite matrices andSⁿ₊₊be the cone of the symmetric positive-definite matrices bothn×n.X,Y∈Sⁿ₊,XX(orXY) means thatY–X∈Sⁿ₊andXY (orX≺Y) means thatY–X∈Sⁿ++.

Following Rothaus [41], letM:= (Sⁿ++,·,·) be the Riemannian manifold endowed with the Riemannian metric defined by

U,V=Tr

VX^–1UX^–1

, X∈MandU,V∈TXM, (27)

whereTr(X) denotes the trace of matrixX∈SⁿandT_XM≈Sⁿ, with the corresponding norm denoted by · . The gradient and the Hessian of a twice differentiable function φ:Sⁿ₊₊→Rare given by

gradφ(X) =Xφ(X)X, (28)

hessφ(X)V=Xφ(X)VX+1 2

Vφ(X)X+Xφ(X)V

, (29)

whereV∈TXM, andφ(X) andφ(X) are the Euclidian gradient and Hessian ofφatX, respectively.

In fact,Mis a Hadamard manifold with curvature is not identically zero; see [42, Theo- rem 1.2. p. 325] for further details. The unique geodesic segment connecting anyX,Y∈M is given by

χ(t) =X^1/2

X^–1/2YX^–1/2t

X^1/2, t∈[0, 1];

see, for example [43]. From the last equation χ(0) =X^1/2ln

X^–1/2YX^–1/2 X^1/2.

Thus, for allX∈M,exp^–1_X :M→TXMandexp_X:TXM→Mare defined, respectively, by exp^–1_X Y=X^1/2ln

X^–1/2YX^–1/2

X^1/2, exp_XV=X^1/2e^{(X–1/2VX–1/2)}X^1/2. (30) Now, since the Riemannian distancedis given byd(X,Y) =exp^–1_X Y, from (27), along with first expression in (30), we have

d²(X,Y) =Tr

ln²X^–1/2YX^–1/2

= n

i=1

ln²η_i

X^–1/2YX^–1/2

, (31)

whereη_i(X^–1/2YX^–1/2) denotes theith eigenvalue of the symmetric matrixX^–1/2YX^–1/2. Following [22], let the functionφ:Sⁿ++→Rdefined by

φ(X) =aln det(X)2

, (32)

wherea> 0. The Euclidian gradient and Hessian ofφare given, respectively, by φ(X) =

2aln det(X)

X^–1, (33)

φ(X)V= 2aTr X^–1V

X^–1– 2aln

det(X)

X^–1VX^–1, (34)

whereX∈Sⁿ₊₊andV∈Sⁿ.

By combining (28), (29), (33) and (34), we obtain, after some calculations, gradφ(X) = 2aln

det(X)

X, (35)

hessφ(X)V= 2aTr X^–1V

X, (36)

for allX∈MandV∈TXM. We further havehessφ(X)V,V= 2aTr(X^–1V)²≥0. Thus, φ is geodesic convex in M. Moreover, (27) together with (36) gives hessφ(X)V= 2aTr(X^–1V) for allX∈MandV∈TXM. If we assume thatV²=Tr(VX^–1VX^–1) = 1, thenTr(X^–1V)≤√

n. Hence, hessφ(X)V≤2a√

n, X∈M,V∈TXM,V= 1.

Therefore, (25) and Proposition6imply thatgradφis Lipschitz with constantK≤2a√ n.

We also havegradφis_K¹-inverse-strongly-monotone vector field with constantK≤2a√ n.

Let={X∈Sⁿ++:ηmin(X)≥1}, whereηmin(X) denotes the minimum eigenvalue of the matrixX, be a nonempty, closed and geodesic convex subset ofMandϕ:→R∪ {+∞}

be a proper, lower semicontinuous and geodesic convex defined by

ϕ(X) =d²(X,I) = n

i=1

ln²η_i X^–1

,

whereη_i(X^–1) is theith eigenvalue of the matrixX^–1. One can see thatI is a minimizer ofϕ, whereIdenotes the identity matrix. By Definition9, the subdifferential ofϕatXis defined by

∂ϕ(X) =

U∈T_XM n

i=1

ln²η_i Y^–1

– n

i=1

ln²η_i X^–1

≥Tr X^1/2ln

X^–1/2YX^–1/2

X^1/2X^–1UX^–1

, ∀Y∈.

The subdifferential∂ϕofϕis a maximal monotone vector field, according to Lemma6.

Moreover, we have J_λ^∂ϕ(X) =arg min

Y∈

ϕ(Y) + 1

2λd²(Y,X)

, ∀λ> 0.

Since the minimizer ofϕisI, it is easy to see that 0∈∂ϕ(I).

LetA:→Sⁿis a _K¹-inverse-strongly-monotone vector field defined by A(X) = 2(ln detX)X,

whereK≤2√

n, andB:→2^Snbe a maximal monotone multivalued field defined by B(X) =∂ϕ(X).

We see that (A+B)^–1(0) ={I}. Choose initial pointX0∈, then Theorem2is applicable leading us to conclude that any sequence generated by Eq. (21) converges to a singularity of the inclusion problem (8).

Remark3 It is worth noting thatφis non-convex with non-Lipschitz continuous gradient onSⁿ₊₊ 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.

5 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.

5.1 Convex minimization problems

Letφ,ϕ:→R∪ {+∞}are proper, lower semicontinuous and geodesic convex functions such thatφis differentiable. We consider the problem of findingp^∗∈such that

φ p^∗

+ϕ p^∗

=min

p∈

φ(p) +ϕ(p)

. (37)

The problem is said to be aconvex minimization problem. We denoteSby the set of min- imizers of the problem (37), that is,S:={p∈:φ(p) +ϕ(p)≤φ(q) +ϕ(q),∀q∈}. It is to see that the problem (37) is equivalent to the following inclusion problem: findp∈such that 0∈gradf(p) +∂g(p), that is,

p∈S ⇐⇒ 0∈gradφ(p) +∂ϕ(p). (38) For further details see [30].

Ifφ:→R∪ {+∞}is a proper, twice continuously differentiable and geodesic convex function such thatHessφis bounded, then, by Proposition6,gradφ isK-Lipschitz con- tinuous vector field. Thengradφis_K¹-inverse-strongly-monotone vector field. Moreover, from Lemma6,∂ϕ is a maximal monotone vector field. By replacingAandBbygradφ and∂ϕ, respectively, in Theorem2, we get the following result for convex minimization problem (37).

Theorem 3 Suppose that S=∅.Letbe a nonempty,closed and geodesic convex subset of a Hadamard manifold M.Letφ:→R∪ {+∞}be a proper,differentiable and geodesic convex function such thatHessφis bounded,ϕ:→R∪ {+∞}a proper,lower semicon- tinuous and geodesic convex function such that D(ϕ) =.Choose p₀∈and define{p_n}as follows:

qn=J_λ^∂ϕ_n exp_pn

–λngradφ(pn) , pn+1=exp_pn(1 –γ_n)exp^–1_pnqn,

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 <γ₁≤γ_n≤γ₂< 1,∀n∈N, (ii) 0 <λˆ≤λ_n≤2α<∞,∀n∈N.

Then{pn}is convergent to a solution of the convex minimization problem(37).