Inertial proximal point algorithm for variational inclusion in hadamard manifolds

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Inertial proximal point algorithm for variational inclusion in Hadamard manifolds

Shih-Sen Chang, Jen-Chih Yao, M Liu & L C Zhao

To cite this article: Shih-Sen Chang, Jen-Chih Yao, M Liu & L C Zhao (2021): Inertial proximal point algorithm for variational inclusion in Hadamard manifolds, Applicable Analysis, DOI:

10.1080/00036811.2021.2016719

To link to this article: https://doi.org/10.1080/00036811.2021.2016719

Published online: 16 Dec 2021.

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Inertial proximal point algorithm for variational inclusion in

Hadamard manifolds

Shih-Sen Changa, Jen-Chih Yaoa, M Liuband L C Zhaob

a Center for General Education, China Medical University, Taichung Taiwan; b Department of Mathematics, Yibin University, Yibin, People’s Republic of China

ABSTRACT

In this paper, we consider the inertial proximal point algorithm for finding

a zero point of variational inclusions on Hadamard manifolds Under

suit-able conditions, it is proved that the sequence generated by the algorithm

converges to an element of the set of zero points of variational inclusion

problem As applications, we utilize our results to study the minimization

problem and saddle point problem in the setting of Hadamard manifolds.

ARTICLE HISTORY

Received 21 September 2021 Accepted 17 November 2021

COMMUNICATED BY

B Mordukhovich

KEYWORDS

Monotone inclusion problem; Hadamard manifold; regularization method; inertial proximal point algorithm; Fejér monotone

SUBJCLASS (2010)

49J53; 58E35; 47J22; 58C30

1 Introduction

During the past two decades, many authors have drawn their attention to the following variational

inclusion problem of finding x∗∈ X such that

where X is a real Banach space, A : X → X is an operator and B : X → 2 X is a set-valued operator This problem includes, as special cases, convex programming, variational inequalities, split feasibil-ity problem and minimization problem To be more precise, some concrete problems in machine learning, image processing and linear inverse problem can be modeled mathematically as the form

of (1)

For solving the problem (1), the forward-backward splitting method (see, for example, [1–3]) is

usually employed and defined by the following manner: x1 ∈ H and

x n+1= (I + rB)−1(x n − rAx n ), n ≥ 1, (2)

where r > 0 This method includes, as special cases, the proximal point algorithm [4] and the gradient method

In recent years, the forward-backward splitting method has been extended by many authors

In Ref [5], Alvarez and Attouch employed the heavy ball method which was studied in Ref [6] for maximal monotone operators by the proximal point algorithm This algorithm is called the inertial

CONTACT Shih-sen Chang changss2013@163.com

proximal point algorithm and it is of the following form:



y n = x n + θ n (x n − x n−1)

x n+1= (I + r n B )−1y n, n≥ 1 (3)

It was proved in Ref [5] that if{r n } is non-decreasing and {θ n } ⊂ [0, 1) with

∞



n=1

then algorithm (3) converges weakly to a solution of the following inclusion problem

Here,θ n is an extrapolation factor and the inertia is represented by the termθ n (x n − x n−1) It is

remarkable that the inertial methodology greatly improves the performance of the algorithm and has

a nice convergence properties

In Ref [7], Moudafi and Oliny proposed the following inertial proximal point algorithm for solving

problem (1) in Hilbert space H:



y n = x n + θ n (x n − x n−1)

x n+1 = (I + r n B )−1(y n − r n Ax n ), n ≥ 1. (6)

where A : H → H and B : H → 2 H They obtained the weak convergence theorem provided r n <

2/L with L the Lipschitz constant of A and the condition (4) holds.

In Ref [1], Lorenz and Pock proposed the following inertial forward-backward algorithm for monotone operators:



y n = x n + θ n (x n − x n−1)

x n+1= (I + r n B)−1(y n − r n Ay n ), n ≥ 1. (7)

where{r n} is a positive real sequence It is observed that algorithm (7) differs from that of Moudafi

and Oliny insofar that they evaluated the operator B as the inertial extrapolate {y n}

Recently, Li et al [8] considered the inclusion problem (5) in the setting of Hadamard manifold M.

They extended the proximal point algorithm from Hilbert spaces to Hadamard manifolds as follows:

0 ∈ λ n B(x n+1) − exp−1

where exp is the exponential mapping Later on, Tang and Huang [9] extended the inexact proximal

point algorithm in the framework of Hadamard manifold M as

where{e n}n≥1can be regarded as an error sequence which satisfies some appropriate conditions with

||e n+1|| ≤ σn d (x n+1, x n ) andn≥0σ2

n < +∞, where d is the Riemannian metric on M They

stud-ied strong convergence of the sequence generated by algorithm (9) to a solution of the inclusion problem (5)

Very recently, Al-Homidan-Ansari-Babu [10], Ansari et al [11–14], Chang et al [15–17], Liu et al [18] and Zhu et al [19] considered the variational inclusion problem (1) in a Hadamard manifold,

where B is a set-valued maximal monotone vector field, and A is a single-valued continuous and

monotone vector field They proposed some Halpern-type and Mann-type iterative methods Under suitable conditions, they proved that the sequence generated by the algorithm converges strongly to

a element of the set of solutions of variational inclusion problem (1)

Motivated by the works of Alvarez et al [5], Tang and Huang [9], Li [8], Ansari et al [11–14], Al-Homidan et al [10], Chang et al [15–17], and the research ongoing in this direction, the purpose

of this article is to propose the following inertial proximal point algorithm for solving variational

inclusion problem (1) in Hadamard manifold M:

− α n P x n+1 ,x nexp−1x n x n−1 ∈ λ n (A + B)(x n+1) − exp−1x n+1 x n, n≥ 1 (10)

where P y,x : T x M → T y M is the parallel transport on the tangent bundle TM along a minimal

geodesic joining x to y (the definition see Section2) Under suitable conditions, it is proved that the sequence{x n} generated by the algorithm (10) converges to an element of the set of solutions of

variational inclusion problem (1) in Hadamard manifold M As applications, we utilize our results to

study the minimization problems and saddle point problems in the setting of Hadamard manifolds

2 Preliminaries

In this section, we recall some notations, terminologies and basic results from Riemannian manifold which can be found in any textbook on Riemannian geometry (see, for example [20])

Let M be a finite-dimensional differentiable manifold, T p M be the tangent space of M at p ∈ M.

We denote by TM=p ∈M T p M the tangent bundle of M An inner product p on T p M is called

a Riemannian metric on T p M A tensor field

every p p is a Riemannian metric on T p M The corresponding norm to the inner

product p on T p M is denoted by|| · ||p We omit the subscript p, if there is no confusion occurs.

A differentiable manifold M endowed with a Riemannian metric

manifold The length of a piecewise smooth curveγ : [0, 1] → M joining p to q (i.e γ (0) = p and

γ (1) = q) is defined as

L(γ ) = 1

The Riemannian distance d (p, q) is the minimal length over the set of all such curves joining p to q,

which induces the original topology on M.

Let∇ be the Levi-Civita connection associated with the Riemannian metric Let γ be a smooth curve in M A vector field X along γ is said to be parallel if ∇ γX= 0, where 0 is the zero tangent

vector Ifγitself is parallel alongγ , 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 M is complete if for any p ∈ M, all geodesics emanating from p are defined for all t∈R A geodesic joining p to q in M is said to be a minimal geodesic if its length is equal to

d (p, q) A Riemannian manifold M equipped with Riemannian distance d is a metric space (M, d) By

Hopf-Rinow Theorem [20], 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.

If M is a complete Riemannian manifold, then the exponential map exp p : T p M → M at p ∈ M

is defined by expp v = γ v (1, p) for all v ∈ T p M, where γ v (·, p) is the geodesic starting from p with

velocity v, that is, γ v (0, p) = p and γ v(0, p) = v.

It is known that expp tv = γ v (t, p) for each real number t It is easy to see that exp p 0 = γ v (0, p) =

p, where 0 is the zero tangent vector Note that the exponential map exp p is differentiable on T p M for

any p ∈ M.

The parallel transport P γ ,γ (b),γ (a) : T γ (a) M → T γ (b) M on the tangent bundle TM along γ :

[a, b]→Rwith respect to∇ is defined by:

P γ ,γ (b),γ (a) (v) = V(γ (b)), ∀a, b ∈Rand v ∈ T γ (a) M,

where V is the unique vector field such that∇γ(t) V = 0 for all t ∈ [a, b] and V(γ (a)) = v If γ is

a minimal geodesic joining x to y, then we write P y,x instead of P γ ,y,x Note that, P y,xis an isometry

from T x M to T y M That is, the parallel transport preserve the inner product

Moreover, from Li et al [8], we have the following result:

Lemma 2.1: If x, y, z ∈ M, then

(i) P x,yexp−1y x= − exp−1

x y, and ||P y,x u || = ||u|| for all u ∈ T x M;

(ii) (see, [8, P 671], see also, [21, Remark 2.1]) if v ∈ T y M, then

v, − exp−1y x y,xexp−1x y x,y v, exp−1x y

Definition 2.2: A complete simply connected Riemannian manifold of non-positive sectional

cur-vature is called a Hadamard manifold

Proposition 2.3 ([ 20]): Let M be a Hadamard manifold and p ∈ M Then exp p : T p M → M is a

dif-feomorphism, and for any two points p, q ∈ M, there exists, a unique normalized geodesic γ : [0, 1] →

M joining p = γ (0) to q = γ (1) which is in fact a minimal geodesic denoted by

This proposition shows that M is diffeomorphic to the Euclidean spaceRm Moreover, Hadamard manifolds and Euclidean spaces have some similar geometrical properties One of the most important properties is described in the following proposition.

Recall that a geodesic triangle

(p1, p2, p3) in a Riemannian manifold M is a set consisting of three

points p1, p2 and p3, and three minimal geodesics γ i joining p i to p i+1, where i = 1, 2, 3(mod3).

Proposition 2.4 ([ 20]): Let

(p1, p2, p3) be a geodesic triangle in a Hadamard manifold M For each

i = 1, 2, 3(mod3), let γ i : [0, l i]→ M be the geodesic joining p i to p i+1and α i be the angle between tangent vectors γ i(0) and −γ i−1(l i−1) Then

(a) α1+ α2 + α3 ≤ π;

(b) l2i + l2

i+1− 2l i l i+1cos α i+1 ≤ l2

i−1

As in Ref [22], Proposition 2.4 (b) can be written in terms of Riemannian distance and exponential mappings as:

d2(p i , p i+1) + d2(p i+1, p i+2) − d2(p i−1, p i ) ≤ 2 exp−1p i+1 p i, exp−1p i+1 p i+2 (13)

In the sequel, unless otherwise specified, we always assume that M is a finite-dimensional Hadamard manifold, and C is a nonempty, bounded, closed and geodesic convex set in M and Fix (S)

is the fixed point set of a mapping S.

Definition 2.5: A function f : C → (−∞, ∞] is said to be geodesic convex if, for any geodesic

γ (λ)(0 ≤ λ ≤ 1) joining x, y ∈ C, the function f ◦ γ is convex, i.e.

f (γ (λ)) ≤ λf (γ (0)) + (1 − λ)f (γ (1)) = λf (x) + (1 − λ)f (y). (14)

In the sequel, we denote by(M) the set of all single-valued vector fields A : M −→ TM such

that A (x) ∈ T x M for each x ∈ M, and D(A) the domain of A defined by

D(A) = {x ∈ M : A(x) ∈ T x M}

Denote byX (M) the set of all set-valued vector fields B : M → TM such that B(x) ⊂ T x M for all

x ∈ M, and D (B) the domain of B defined by D (B) = {x ∈ M : B(x) = ∅}.

Definition 2.6 ([ 23]): A set-valued vector field A ∈ X (M) on a Hadamard manifold M is said to

be

(1) monotone if for any x, y∈D (A)

u, exp−1x y −1y x

(2) maximal monotone if it is monotone and for all x∈D (A) and u ∈ T x M, the condition

u, exp−1x y −1y x D (A) and ∀v ∈ A(y).

implies u ∈ A(x).

Li et al [8] defined the upper Kuratowski semicontinuity for set-valued vector field in the setting

of Hadamard manifolds and gave its relation with the set-valued maximal monotone vector field

Definition 2.7: A set-valued vector field A ∈ X (M) is said to be

(a) upper Kuratowski semicontinuous at x∈D (A) if for any sequences {x n} ⊂D (A) and {u n} ⊂

TM with each u n ∈ A(x n ), the relations lim n→∞x n = x∗ and limn→∞u n = u∗ imply u∗∈

A (x∗);

(b) upper Kuratowski semicontinuous on M if it is upper Kuratowski semicontinuous at each x∈

D (A).

Lemma 2.8 (Theorem 3.7 of [ 10]): Suppose that A ∈ X (M) is monotone and D (A) = M Then the following statements are equivalent:

(i) the vector field A is maximal monotone;

(ii) the vector field A is upper Kuratowski semicontinuous on M, and A (x) is closed and convex for each

x ∈ M.

Definition 2.9: Let X be a complete metric space and Q ⊂ X be a nonempty set A sequence {x n } ⊂ X

is called Fejér monotone with respect to Q if for any y ∈ Q and n ≥ 0,

d(x n+1, y ) ≤ d(x n , y ).

Lemma 2.10 ([ 22 , 24]): Let X be a complete metric space, Q ⊂ X be a nonempty set If {x n } ⊂ X is

Fejér monotone with respect to Q, then {x n } is bounded Moreover, if a cluster point x of {x n } belongs to

Q, then {x n } converges to x.

Lemma 2.11 ([ 25]): Let {ξ n } ⊂ [0, 1) be a sequence If∞n=1ξ n < ∞, then

∞



n=1

ξ n

1− ξ n < +∞, and

∞



n=1



1+ ξ n

1− ξ n

< +∞.

3 Main results

We are now in a position to give the main result of this article

Theorem 3.1: Let M be a finite-dimensional Hadamard manifold Let B ∈ X (M) be a maximal mono-tone vector field with D (B) = M, and A ∈ (M) be a continuous single-valued vector fields with

A(x) ∈ T x M for each x ∈ M such that (A + B) is a maximal monotone set-valued vector field with

D (A + B) = M For any given x0, x1 ∈ M let {x n } be the inertial proximal point sequence generated by

− α n P x n+1 ,x nexp−1x n x n−1 ∈ λ n (A + B)(x n+1) − exp−1x n+1 x n, n≥ 1 (15)

where P x n+1 ,x n : T x n M → T x n+1 M is the parallel transport on the tangent bundle TM along a minimal geodesic γ joining x n to x n+1, and the parameters α n and λ n satisfy the following conditions:

(i) there exists λ > 0 such that λ n ≥ λ, ∀n ≥ 1;

(ii) for each n ≥ 1, α n ≥ 0 and

||α n P x n+1 ,x nexp−1x n x n−1|| ≤ αnd(x n , x n+1) with

∞



n=1

α2

If S : = (A + B)−1(0) = ∅, then there exists x∗∈ S such that {x n } converges strongly to x∗as

Proof: (I) We first examine the case where α n = 0, ∀n ≥ 1 which corresponds to the standard

proximal method Although in this case, the result is well known, the proof gives some guidelines for the general situation

According to the assumptions of operators A and B, the mapping A + B ∈ X (M) is a maximal

monotone vector field withD (A + B) = M Fix z ∈ S = (A + B)−1(0) and define the auxiliary

real sequence

φ n:= 1

2d

2(x n , z ), n = 1, 2,

By (13) we have

φ n+1≤ φ n+ exp−1

x n+1 x n, exp−1n+1z 1

2d

Sinceα n= 0, from (15) we have 1

λ nexp−1x n+1 x n ∈ (A + B)(x n+1) By virtue of the monotonicity

of A + B we deduce that

0= 0, exp−1z x n+1 λ1n exp−1x n+1 x n, − exp−1x n+1 z

i.e exp−1

x n+1 x n, exp−1x n+1 z n+1≤ φ n Therefore we have that

(1) d(x n+1, z ) ≤ d(x n , z ), ∀n ≥ 1, i.e {x n } is Fejér monotone with respect to S By Lemma 2.10, {x n } is bounded and for each z ∈ S, the limit lim n→∞ d(x n , z ) exists; and

(2) from (17), we have∞

n=1 d2(x n , x n+1) ≤ 2φ1 = d2(x1, z) This shows that

lim

n→∞d(x n , x n+1) = lim n→∞|| exp−1

x n+1 x n|| = 0,

Asλ nis bounded away from zero we have

lim

1

λ n

Since{x n} is bounded, by Hopf-Rinow Theorem [20],{x n} is a compact sequence There exists a subsequence{x n } (for simplicity, we denote it by {x k }) of {x n } such that {x k} converges strongly

to some point x∗ Then from (18), we have

lim

1

Since λ1

kexp−1x k+1 x k ∈ (A + B)(x k+1) and A + B is a maximal monotone vector field, by

Lemma 2.8, it is upper Kuratowski semicontinuous Therefore, as x k → x∗and combining (19),

we obtain 0 ∈ (A + B)(x∗), i.e x∗∈ S By Lemma 2.10, it follows that x n → x∗

This completes the proof of Theorem 3.1 whenα n = 0, ∀n ≥ 1.

(II) Now we consider the case α n > 0 for some n ≥ 1.

From (15) we have

1

λ n (exp−1x n+1 x n + V n+1) ∈ (A + B)(x n+1), ∀ n ≥ 1, (20)

where V n+1= −α n P x n+1 ,x nexp−1x n x n−1 For any given z ∈ S, by using the monotonicity of A + B,

we have

0= 0, exp−1z x n+1 λ1n exp−1x n+1 x n + V n+1, − exp−1x n+1 z

This implies that

exp−1

x n+1 x n + V n+1, exp−1x n+1 z (21)

By (13), we have

d2(x n+1, z ) ≤ d2(x n , z ) − d2(x n , x n+1) + 2 exp−1x n+1 x n, exp−1x n+1 z (22) This together with (21) shows

d2(x n+1, z ) ≤ d2(x n , z ) − d2(x n , x n+1) + 2 exp−1x n+1 x n, exp−1x n+1 z

= d2(x n , z ) − d2(x n , x n+1) + 2 exp−1x n+1 x n + V n+1, exp−1x n+1 z

+ 2 V n+1,− exp−1x n+1 z

≤ d2(x n , z ) − d2(x n , x n+1) + 2 V n+1, − exp−1x n+1 z (23)

Next we prove that {x n} is a bounded sequence and

lim

In fact, sinceα n > 0, using the Cauchy-Schwarz inequality, we have

2 V n+1,− exp−1x n+1 z 1

2α2

n ||V n+1|| 2+ 2α2

n|| − exp−1x n+1 z||2 This together with the condition (16) shows that

2 V n+1,− exp−1x n+1 z 1

2d(x n , x n+1) + 2α2

Substituting (25) into (23), we have

d2(x n+1, z ) ≤ d2(x n , z ) − (d2(x n , x n+1) −1

2d

2(x n , x n+1)) + 2α2

nd2(x n+1, z )

= d2(x n , z ) −1

2d

2(x n , x n+1) + 2α2

nd2(x n+1, z )

After simplifying, we have

(1 − 2α2

n ) d2(x n+1, z ) ≤ d2(x n , z ) −1

2d

2(x n , x n+1). (26)

Sinceα n → 0, there exists n0 ≥ 0 such that for all n ≥ n0, 1 − 2α2

n > 0 Therefore, it follows from (26)

that

d2(x n+1, z ) ≤ 1

1− 2α2

n

d2(x n , z ) − 1

2(1 − 2α2

n )d2(x n , x n+1)

≤



1+ 2α n2

1− 2α2

n

d2(x n , z ) −1

2d

Thus we have

d2(x n+1, z ) ≤



1+ 2α2n

1− 2α2

n

Since∞

n=1α2

n < ∞, it follows from Lemma 2.11 that

∞



n =n0

2α2

n

1− 2α2

∞



n =n0



1+ 2α2n

1− 2α2

n

< +∞.

Since ∞n =n0

1+ 2α2

n

1−2α 2

n

< +∞, this implies that {x n} is bounded

On the other hand, from (27) we have that

1

2d

2(x n , x n+1) ≤ d2(x n , z ) − d2(x n+1, z ) + 2α n2

1− 2α2

n

d2(x n , z ), ∀n ≥ n0, and thus

1

2

∞



n =n0

d2(x n , x n+1) ≤

∞



n =n0

(d2(x n , z ) − d2(x n+1, z )) +

∞



n =n0

2α2

n

1− 2α2

n

d2(x n , z )

≤ d2(x n0, z ) +∞

n =n0

2α2

n

1− 2α2

n

( sup

n ≥n0

This implies that limn→∞ d(x n , x n+1) = 0 The conclusion (24) is proved Therefore we have

exp−1x n+1 x n − α n P x n+1 ,x nexp−1x n x n−1→ 0 as n → ∞.

Since{x n} is bounded, by Hopf-Rinow Theorem [20], it is compact There exists a subsequence

{x n j } ⊂ {x n } (for simplicity, we denote it by {x j }) such that x j → x∗ (some point in M) Since

(exp−1

x j+1 x j − α j P x j+1 ,x jexp−1x j x j−1) ∈ (A + B)(x j+1) and A + B is upper Kuratowski semicontinuous.

This shows that 0 ∈ (A + B)(x∗), i.e x∗∈ S.

Now we prove that S contains only a unique cluster point x∗of {x n}.

Suppose on the contrary, let x∗and y∗∈ S be two cluster points of {x n} Set

l1 = limn→∞d2(x n , x∗); l2= limn→∞d2(x n , y∗). (30) Let{x n j } and {x n k } be two subsequences of {x n} such that limj→∞x n j = x∗and lim

k→∞x n k = y∗ By inequality (13), we have

d2(x n j , x∗) − d2(x n j , y∗) ≤ d2(x∗, y∗) + 2 exp−1y∗ x∗,− exp−1y∗ x n j (31) and

d2(x n k , y∗) − d2(x n k , x∗) ≤ −d2(x∗, y∗) + 2 exp−1y∗ x∗, exp−1y∗ x n k (32)

Taking j→ ∞ in (31), it follows from (30) that

l1− l2≤ d2(x∗, y∗) − 2 exp−1y∗ x∗, exp−1y∗ x∗ 2(x∗, y∗). (33)

Also taking k→ ∞ in (32), from (30), we have

These show that d2(x∗, y∗) ≤ l2− l1 ≤ −d2(x∗, y∗) Therefore

l2− l1= d2(x∗, y∗) = 0, i.e x∗= y∗

Hence S contains only a unique cluster point x∗of{x n} Therefore limn→∞x n = x∗

4 Applications

Throughout this section, we assume that M is a finite-dimensional Hadamard manifold and C is a nonempty, bounded, closed and geodesic convex set in M.

4.1 Minimization problem on Hadamard manifolds

Let f : M → (−∞, +∞] be a differentiable function and g : M → (−∞, +∞] be a proper lower

semicontinuous and geodesic convex function Consider the minimization problem: to find a point

x∗∈ M such that

(f + g)(x∗) = min

We denote by the solution set of the minimization problem (35).

Let∇f be the gradient of f and the subdifferential ∂g(x) of g at x ∈ M [23] is defined by

∂g(x) := {v ∈ T x M : v, exp−1x y

It is easy to check that∂g(x) is closed and convex.

Lemma 4.1 ([ 26]): Let g : M → (−∞, +∞] be a proper lower semicontinuous and geodesic convex

function Then, the subdifferential ∂f of f is a maximal monotone vector field And

From Theorem 3.1 and Lemma 4.1, we have the following result:

4.1 Minimization problem on Hadamard manifolds< /b>...

semicontinuous and geodesic convex function Consider the minimization problem: to find a point

x∗∈ M such that

(f + g)(x∗) = min... {x j }) such that x j → x∗ (some point in M) Since

(exp−1

x j+1 x j