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Dong and Yuan Fixed Point Theory and Applications (2015) 2015:125 DOI 10.1186/s13663-015-0374-6

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Accelerated Mann and CQ algorithms for finding a fixed point of a nonexpansive mapping Qiao-Li Dong1,2* and Han-bo Yuan1 *

Correspondence: [email protected] 1 College of Science, Civil Aviation University of China, Tianjin, 300300, China 2 Tianjin Key Lab for Advanced Signal Processing, Civil Aviation University of China, Tianjin, 300300, China

Abstract The purpose of this paper is to present accelerations of the Mann and CQ algorithms. We first apply the Picard algorithm to the smooth convex minimization problem and point out that the Picard algorithm is the steepest descent method for solving the minimization problem. Next, we provide the accelerated Picard algorithm by using the ideas of conjugate gradient methods that accelerate the steepest descent method. Then, based on the accelerated Picard algorithm, we present accelerations of the Mann and CQ algorithms. Under certain assumptions, we show that the new algorithms converge to a fixed point of a nonexpansive mapping. Finally, we show the efficiency of the accelerated Mann algorithm by numerically comparing with the Mann algorithm. A numerical example is provided to illustrate that the acceleration of the CQ algorithm is ineffective. Keywords: nonexpansive mapping; convex minimization problem; Picard algorithm; Mann algorithm; CQ algorithm; conjugate gradient method; steepest descent method

1 Introduction Let H be a real Hilbert space with inner product ·, · and induced norm  · . Suppose that C ⊂ H is nonempty, closed and convex. A mapping T : C → C is said to be nonexpansive if Tx – Ty ≤ x – y for all x, y ∈ C. The set of fixed points of T is defined by Fix(T) := {x ∈ C : Tx = x}. In this paper, we consider the following fixed point problem. Problem . Suppose that T : C → C is nonexpansive with Fix(T) = ∅. Then   find x∗ ∈ C such that T x∗ = x∗ . The fixed point problems for nonexpansive mappings [–] capture various applications in diversified areas, such as convex feasibility problems, convex optimization problems, problems of finding the zeros of monotone operators, and monotone variational inequalities (see [, ] and the references therein). The Picard algorithm [], the Mann algorithm © 2015 Dong and Yuan. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

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[, ], and the CQ method [] are useful fixed point algorithms to solve the fixed point problems. Meanwhile, to guarantee practical systems and networks (see, e.g., [–]) are stable and reliable, the fixed point has to be quickly found. Recently, Sakurai and Liduka [] accelerated the Halpern algorithm and obtained a fast algorithm with strong convergence. Inspired by their work, we focus on the Mann and the CQ algorithms and present new algorithms to accelerate the approximation of a fixed point of a nonexpansive mapping. We first apply the Picard algorithm to the smooth convex minimization problem and illustrate that the Picard algorithm is the steepest descent method [] for solving the minimization problem. Since conjugate gradient methods [] have been widely seen as an efficient accelerated version of most gradient methods, we introduce an accelerated Picard algorithm by combining the conjugate gradient methods and the Picard algorithm. Finally, based on the accelerated Picard algorithm, we present accelerations of the Mann and CQ algorithms. In this paper, we propose two accelerated algorithms for finding a fixed point of a nonexpansive mapping and prove the convergence of the algorithms. Finally, the numerical examples are presented to demonstrate the effectiveness and fast convergence of the accelerated Mann algorithm and the ineffectiveness of the accelerated CQ algorithm.

2 Mathematical preliminaries 2.1 Picard algorithm and our algorithm The Picard algorithm generates the sequence {xn }∞ n= as follows: given x ∈ H, xn+ = Txn ,

n ≥ .

()

The Picard algorithm () converges to a fixed point of the mapping T if T : C → C is contractive (see, e.g., []). When Fix(T) is the set of all minimizers of a convex, continuously Fréchet differentiable functional f over H, that algorithm () is the steepest descent method [] to minimize f over H. Suppose that the gradient of f , denoted by ∇f , is Lipschitz continuous with a constant L >  and define T f : H → H by T f := I – λ∇f ,

()

where λ ∈ (, /L) and I : H → H stands for the identity mapping. Accordingly, T f satisfies the contraction condition (see, e.g., []) and       Fix T f = arg min f (x) := x∗ ∈ H : f x∗ = min f (x) . x∈H

x∈H

Therefore, algorithm () with T := T f can be expressed as follows:  f dn+ := –∇f (xn ), f xn+ := T f (xn ) = xn – λ∇f (xn ) = xn + λdn+ .

()

The conjugate gradient methods [] are popular acceleration methods of the steepest f ,CGD descent method. The conjugate gradient direction of f at xn (n ≥ ) is dn+ := –∇f (xn ) +

Dong and Yuan Fixed Point Theory and Applications (2015) 2015:125

f ,CGD

βn dn that

f ,CGD

, where d

f ,CGD

dn+

=

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:= –∇f (x ) and {βn }∞ n= ⊂ (, ∞), which, together with (), implies

  f T (xn ) – xn + βn dnf ,CGD . λ

() f ,CGD

f

By replacing dn+ := –∇f (xn ) in algorithm () with dn+ erated Picard algorithm as follows:

defined by (), we get the accel-

 f ,CGD   f ,CGD dn+ := λ T f (xn ) – xn + βn dn , f ,CGD xn+ := xn + λdn+ .

()

The convergence condition of Picard algorithm is very restrictive and it does not converge for general nonexpansive mappings (see, e.g., []). So, in  Mann [] introduced the Mann algorithm xn+ = αn xn + ( – αn )Txn ,

n ≥ ,

()

and showed that the sequence generated by it converges to a fixed point of a nonexpansive mapping. In this paper we combine ()-() and the CQ algorithm to present two novel algorithms.

2.2 Some lemmas We will use the following notation: () xn  x means that {xn } converges weakly to x and xn → x means that {xn } converges strongly to x. () ωw (xn ) := {x : ∃xnj  x} denotes the weak ω-limit set of {xn }. Lemma . Let H be a real Hilbert space. There hold the following identities: (i) x – y = x – y – x – y, y, ∀x, y ∈ H, (ii) tx + ( – t)y = tx + ( – t)y – t( – t)x – y , t ∈ [, ], ∀x, y ∈ H. Lemma . Let K be a closed convex subset of a real Hilbert space H, and let PK be the (metric or nearest point) projection from H onto K (i.e., for x ∈ H, PK x is the only point in K such that x – PK x = inf{x – z : z ∈ K}). Given x ∈ H and z ∈ K . Then z = PK x if and only if there holds the relation x – z, y – z ≤  for all y ∈ K. Lemma . (See []) Let K be a closed convex subset of H. Let {xn } be a sequence in H and u ∈ H. Let q = PK u. Suppose that {xn } is such that ωw (xn ) ⊂ K and satisfies the condition xn – u ≤ u – q for all n. Then xn → q. Lemma . (See []) Let C be a closed convex subset of a real Hilbert space H, and let T : C → C be a nonexpansive mapping such that Fix(T) = ∅. If a sequence {xn } in C is such that xn  z and xn – Txn → , then z = Tz.

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Lemma . (See []) Assume that {an } is a sequence of nonnegative real numbers satisfying the property an+ ≤ an + un ,

n ≥ ,

where {un } is a sequence of nonnegative real numbers such that limn→∞ an exists.

∞

n= un

< ∞. Then

3 The accelerated Mann algorithm In this section, we present the accelerated Mann algorithm and give its convergence. Algorithm . Choose μ ∈ (, ], λ > , and x ∈ H arbitrarily, and set {αn }∞ n= ⊂ (, ), ∞ {βn }n= ⊂ [, ∞). Set d := (T(x ) – x )/λ. Compute dn+ and xn+ as follows: ⎧    ⎪ ⎨dn+ := λ T(xn ) – xn + βn dn , yn := xn + λdn+ , ⎪ ⎩ xn+ := μαn xn + ( – μαn )yn .

()

We can check that Algorithm . coincides with the Mann algorithm () when βn :≡  and μ := . In this section we make the following assumptions. ∞ Assumption . The sequences {αn }∞ n= and {βn }n= satisfy ∞ (C) μαn ( – μαn ) = ∞, n= ∞ (C) n= βn < ∞. Moreover, {xn }∞ n= satisfies (C) {T(xn ) – xn }∞ n= is bounded.

Before doing the convergence analysis of Algorithm ., we first show the two lemmas. Lemma . Suppose that T : H → H is nonexpansive with Fix(T) = ∅ and that Assump∞ tion . holds. Then {dn }∞ n= and {xn –p}n= are bounded for any p ∈ Fix(T). Furthermore, limn→∞ xn – p exists. Proof We have from (C) that limn→∞ βn = . Accordingly, there exists n ∈ N such that βn ≤ / for all n ≥ n . Define M := max{max≤k≤n dk , (/λ) supn∈N T(xn ) – xn }. Then (C) implies that M < ∞. Assume that dn  ≤ M for some n ≥ n . The triangle inequality ensures that    T(xn ) – xn + βn dn  ≤ M , T(x dn+  = + β ) – x d n n n n ≤ λ λ

()

which means that dn  ≤ M for all n ≥ , i.e., {dn }∞ n= is bounded. The definition of {yn }∞ implies that n=

yn = xn + λ

  T(xn ) – xn + βn dn λ

= T(xn ) + λβn dn .



()

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The nonexpansivity of T and () imply that, for any p ∈ Fix(T) and for all n ≥ n , yn – p = T(xn ) + λβn dn – p ≤ T(xn ) – T(p) + λβn dn  ≤ xn – p + λM βn .

()

Therefore, we find xn+ – p = μαn (xn – p) + ( – μαn )(yn – p) ≤ μαn xn – p + ( – μαn )yn – p   ≤ μαn xn – p + ( – μαn ) xn – p + λM βn ≤ xn – p + λM βn ,

()

which implies xn – p ≤ x – p + λM

n– 

βk < ∞.

k= ∞ So, we get that {xn }∞ n= is bounded. From () it follows that {yn }n= is bounded. In addition, using Lemma ., (C), and (), we obtain limn→∞ xn – p exists.



Lemma . Suppose that T : H → H is nonexpansive with Fix(T) = ∅ and that Assumption . holds. Then lim xn – T(xn ) = .

n→∞

Proof By ()-() and the nonexpansivity of T, we have xn+ – T(xn+ )       = μαn xn + ( – μαn ) T(xn ) + λβn dn – T μαn xn + ( – μαn ) T(xn ) + λβn dn     = μαn xn – T(xn ) + ( – μαn )λβn dn     + T(xn ) – T μαn xn + ( – μαn ) T(xn ) + λβn dn ≤ μαn xn – T(xn ) + ( – μαn )λβn dn     + xn – μαn xn + ( – μαn ) T(xn ) + λβn dn ≤ μαn xn – T(xn ) + ( – μαn )λβn dn  + ( – μαn ) xn – T(xn ) + ( – μαn )λβn dn  ≤ xn – T(xn ) + λβn dn  ≤ xn – T(xn ) + λβn M , which, with (C) and Lemma ., yields that the limit of xn – T(xn ) exists.

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On the other hand, for any p ∈ Fix(T) and for all n ≥ n , using the triangle inequality, the Cauchy-Schwarz inequality, and Lemma .(ii), we obtain  xn+ – p = μαn xn + ( – μαn )yn – p    = μαn xn + ( – μαn ) T(xn ) + λβn dn – p    = μαn (xn – p) + ( – μαn ) T(xn ) – p + ( – μαn )λβn dn     ≤ μαn (xn – p) + ( – μαn ) T(xn ) – p + ( – μαn )λβn dn   +  μαn (xn – p) + ( – μαn ) T(xn ) – p ( – μαn )λβn dn   ≤ μαn xn – p + ( – μαn ) T(xn ) – p – μαn ( – μαn ) T(xn ) – xn   + ( – μαn ) λ βn M +  μαn xn – p + ( – μαn ) T(xn ) – p × ( – μαn )λβn M  ≤ xn – p – μαn ( – μαn ) T(xn ) – xn   + βn ( – μαn )λ M xn – p + ( – μαn )λβn M  ≤ xn – p – μαn ( – μαn ) T(xn ) – xn + βn M . We have from Lemma . that M := supk≥ ( – μαk )λ{M xk – p + ( – μαk )λβk M } is bounded. Therefore, using (C), we obtain n 

n   μαk ( – μαk ) T(xk ) – xk ≤ x – p – xn+ – p + M βk < ∞,

k=

k=

which, with (C), implies that lim inf T(xn ) – xn = . n→∞

Due to existence of the limit of T(xn ) – xn , we have lim T(xn ) – xn = ,

n→∞

which with Lemma . implies that ωw (xn ) ⊂ Fix(T).



Theorem . Suppose that T : H → H is nonexpansive with Fix(T) = ∅ and that Assumption . holds. Then the sequence {xn } generated by Algorithm . weakly converges to a fixed point of T. Proof To see that {xn } is actually weakly convergent, we need to show that ωω (xn ) consists of exactly one point. Take p, q ∈ ωw (xn ) and let {xni } and {xmj } be two subsequences of {xn } such that xni  p and xmj  q, respectively. Using Lemma . of [] and Lemma ., we have p = q. Hence, the proof is complete. 

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4 The accelerated CQ algorithm In general, the Mann algorithm () has only weak convergence (see [] for an example). However, strong convergence is often much more desirable than weak convergence in many problems that arise in infinite dimensional spaces (see [] and the references therein). In , Nakajo and Takahashi [] introduced the following modification of the Mann algorithm: ⎧ ⎪ x ∈ C chosen arbitrarily, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨yn := αn xn + ( – αn )Txn ,

Cn = {z ∈ C : yn – z ≤ xn – z}, ⎪ ⎪   ⎪ ⎪ Qn = z ∈ C : xn – z, x – xn  ≥  , ⎪ ⎪ ⎪ ⎪ ⎩ xn+ = PCn ∩Qn x ,

()

where C is a nonempty closed convex subset of a Hilbert space H and T : C → C is a nonexpansive mapping, and PK denotes the metric projection from H onto a closed convex subset K of H. Here, we introduce an acceleration of CQ algorithm based on Algorithm . and show its strong convergence. Theorem . Let C be a bounded, closed and convex subset of a Hilbert space H and T : C → C be nonexpansive with Fix(T) = ∅. Assume that {αn }∞ n= is a sequence in (, a] for ⊂ [, ∞) such that lim β = . Define a sequence {xn }∞ some  < a <  and {βn }∞ n→∞ n n= n= in C by the following algorithm: ⎧ ⎪ x ∈ C chosen arbitrarily, ⎪ ⎪ ⎪   ⎪ ⎪  ⎪ ⎪ ⎪dn+ := λ T(xn ) – xn + βn dn , ⎪ ⎪ ⎪ ⎪yn := xn + λdn+ , ⎪ ⎨ zn := αn xn + ( – αn )yn , ⎪ ⎪   ⎪ ⎪ Cn = z ∈ C : zn – z ≤ xn – z + θn , ⎪ ⎪ ⎪ ⎪   ⎪ ⎪ Qn = z ∈ C : xn – z, x – xn  ≥  , ⎪ ⎪ ⎪ ⎪ ⎩x = P n+ Cn ∩Qn x ,

()

where θn = λβn M [λβn M + M ] →  as n → ∞, M = diam C and M := max{max≤k≤n dk , (/λ)M }, n is chosen such that βn ≤ / for all n ≥ n . Then {xn }∞ n= converges in norm to PFix(T) (x ). Proof First observe that Cn is convex. Indeed, the inequality defined in Cn can be rewritten as   (xn – zn ), z ≤ xn  – zn  + θn

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which is affine (and hence convex ) in z. Next we show that Fix(T) ⊂ Cn for all n. Similar to the proof of (), we get dn  ≤ M . Due to xn ∈ C, we have, for all p ∈ Fix(T) ⊂ C, xn – p ≤ M ,

()

and yn – p ≤ xn – p + λβn M , which comes from (). Thus we get zn – p ≤ αn xn – p + ( – αn )yn – p ≤ xn – p + λβn M , and consequently, zn – p ≤ xn – p + λβn M xn – p + (λβn M ) ≤ xn – p + θn , where θn = λβn M [λβn M + M ]. So, p ∈ Cn for all n. Next we show that Fix(T) ⊂ Qn

for all n ≥ .

()

We prove this by induction. For n = , we have Fix(T) ⊂ C = Q . Assume that Fix(T) ⊂ Qn . Since xn+ is the projection of x onto Cn ∩ Qn , we have xn+ – z, x – xn+  ≥ ,

∀z ∈ Cn ∩ Qn .

As Fix(T) ⊂ Cn ∩Qn , the last inequality holds, in particular, for all z ∈ Fix(T). This together with the definition of Qn+ implies that Fix(T) ⊂ Qn+ . Hence () holds for all n ≥ . Notice that the definition of Qn actually implies xn = PQn (x ). This together with the fact that Fix(T) ⊂ Qn further implies xn – x  ≤ p – x ,

p ∈ Fix(T).

Due to q = PFix(T) (x ) ∈ Fix(T), we have xn – x  ≤ q – x .

()

The fact that xn+ ∈ Qn asserts that xn+ – xn , xn – x  ≥ . This together with Lemma .(i) implies  xn+ – xn  = (xn+ – x ) – (xn – x ) = xn+ – x  – xn – x  – xn+ – xn , xn – x  ≤ xn+ – x  – xn – x  .

()

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This implies that the sequence {xn – x }∞ n= is increasing. Recall (), we get that limn→∞ xn – x  exists. It turns out from () that lim xn+ – xn  = .

n→∞

By the fact xn+ ∈ Cn we get zn – xn+  ≤ xn – xn+  + θn , and thus zn – xn+  ≤ xn – xn+  +



θn .

()

On the other hand, since zn = αn xn + ( – αn )T(xn ) + ( – αn )λβn dn and αn ≤ a, we have T(xn ) – xn =

 zn – xn – ( – αn )λβn dn  – αn

 zn – xn  + λβn dn  –a    zn – xn+  + xn+ – xn  + λβn M ≤ –a     ≤ xn+ – xn  + θn + λβn M → , –a

≤

()

where the last inequality comes from (). Lemma . and () then guarantee that every weak limit point of {xn }∞ n= is a fixed point of T. That is, ωw (xn ) ⊂ Fix(T). This fact, with inequality () and Lemma ., ensures the strong convergence of {xn }∞  n= to q = PFix(T) x .

5 Numerical examples and conclusion In this section, we compare the original algorithms and the accelerated algorithms. The codes were written in Matlab . and run on personal computer. Firstly, we apply the Mann algorithm () and Algorithm . to the following convex feasibility problem (see [, ]). Problem . (From []) Given a nonempty, closed convex set Ci ⊂ RN (i = , , . . . , m), find x∗ ∈ C :=

m 

Ci ,

i=

where one assumes that C = ∅. Define a mapping T : RN → RN by  T := P

 m   Pi , m i=

()

where Pi = PCi (i = , , . . . , m) stands for the metric projection onto Ci . Since Pi (i = , , . . . , m) is nonexpansive, T defined by () is also nonexpansive. Moreover, we find

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Table 1 Computational results for Problem 5.1 with different dimensions Initial point

rand(N, 1)

Algorithm 3.1

200 × rand(N, 1)

5e

5,000e

N=5 m=5

Iter. 2 Sec. 0 Mann algorithm Iter. 64 Sec. 0

62 0 68 0

5 0 73 0

9 0 74 0

N = 1,000 m = 50

Algorithm 3.1

Iter. Sec. Mann algorithm Iter. Sec.

4 0.0156 505 1.2948

448 1.0608 515 1.2480

7 0.0312 478 1.2012

425 1.0608 429 1.1076

N = 50 m = 1,000

Algorithm 3.1

6,814 38.9846 8,438 53.6799

7 0.1092 7,771 43.4463

4 0.0312 8,692 49.7799

9 0.1248 6,913 38.7818

Iter. Sec. Mann algorithm Iter. Sec.

that Fix(T) = Fix(P )

m 

Fix(Pi ) = C

i=

m 

Ci = C.

i=

Set λ := , μ := ., αn := /(n + ) (n ≥ ), and βn := /(n + ) in Algorithm . and αn := μ/(n + ) in the Mann algorithm (). In the experiment, we set Ci (i = , , . . . , m) as a closed ball with center ci ∈ RN and radius ri > . Thus, Pi (i = , , . . . , m) can be computed with Pi (x) :=

 ci + x

ri (x – ci ) ci –x

if ci – x > ri , if ci – x ≤ ri .

√ √ We set ri :=  (i = , , . . . , m), c :=  and ci ∈ (–/ N, / N)N (i = , . . . , m) were randomly chosen. Set e := (, , . . . , ). In Table , ‘Iter.’ and ‘Sec.’ denote the number of iterations and the cpu time in seconds, respectively. We took T(xn ) – xn  < ε = – as the stopping criterion. Table  illustrates that, with a few exceptions, Algorithm . significantly reduces the running time and iterations needed to find a fixed point compared with the Mann algorithm. The advantage is more obvious, as the parameters N and m become larger. It is worth further research on the reason of emergence of a few exceptions. Next, we apply the CQ algorithm () and the accelerated CQ algorithm () to the following problem. Problem . (From []) Let C be the unit closed ball S(, ) = {x ∈ R |x ≤ } and T : S(, ) → S(, ) be defined by T : (x , x , x )T → ( √ sin(x + z), √ sin(x + z), √ (x + y))T . Then   find x∗ ∈ S(, ) such that T x∗ = x∗ . He and Yang [] showed that T is nonexpansive and has at least a fixed point in S(, ).  Take the sequence αn = n in () and (), and βn = ×n  , λ = .. in (). We tested four different initial points and the numerical results are listed in Table .

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Table 2 Computational results for Problem 5.2 with different initial points Initial point CQ algorithm Accelerated CQ algorithm

Iter. Sec. Iter. Sec.

(1, 0, 0)

(0, 1, 0)

(0, 0.5, 0.5)

(0.5, 0.5, 0.5)

652 0.1092 577 0.0936

167 0.0312 273 0.0468

1,441 0.2652 1,430 0.2496

38 0 84 0.0156

Table  shows that the acceleration of the CQ algorithm is ineffective, that is, the accelerated CQ algorithm does not in fact accelerate the CQ algorithm from running time or the number of iterations. The acceleration may be eliminated by the projection onto the sets Cn and Qn .

6 Concluding remarks In this paper, we accelerate the Mann and CQ algorithms to obtain the accelerated Mann and CQ algorithms, respectively. Then we present the weak convergence of the accelerated Mann algorithm and the strong convergence of the accelerated CQ algorithm under some conditions. The numerical examples illustrate that the acceleration of the Mann algorithm is effective, however, the acceleration of the CQ algorithm is ineffective. Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript. Acknowledgements The authors express their thanks to the reviewers, whose constructive suggestions led to improvements in the presentation of the results. Supported by the National Natural Science Foundation of China (No. 61379102) and Fundamental Research Funds for the Central Universities (No. 3122013D017), in part by the Foundation of Tianjin Key Lab for Advanced Signal Processing. Received: 26 February 2015 Accepted: 30 June 2015 References 1. Bauschke, HH, Combettes, PL: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, Berlin (2011) 2. Goebel, K, Kirk, WA: Topics in Metric Fixed Point Theory. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1990) 3. Goebel, K, Reich, S: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Dekker, New York (1984) 4. Takahashi, W: Nonlinear Functional Analysis. Yokohama Publishers, Yokohama (2000) 5. Bauschke, HH, Borwein, JM: On projection algorithms for solving convex feasibility problems. SIAM Rev. 38(3), 367-426 (1996) 6. Picard, E: Mémoire sur la théorie des équations aux dérivées partielles et la méthode des approximations successives. J. Math. Pures Appl. 6, 145-210 (1890) 7. Krasnosel’skii, MA: Two remarks on the method of successive approximations. Usp. Mat. Nauk 10, 123-127 (1955) 8. Mann, WR: Mean value methods in iteration. Proc. Am. Math. Soc. 4, 506-510 (1953) 9. Nakajo, K, Takahashi, W: Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. J. Math. Anal. Appl. 279, 372-379 (2003) 10. Iiduka, H: Iterative algorithm for solving triple-hierarchical constrained optimization problem. J. Optim. Theory Appl. 148, 580-592 (2011) 11. Iiduka, H: Fixed point optimization algorithm and its application to power control in CDMA data networks. Math. Program. 133, 227-242 (2012) 12. Iiduka, H: Iterative algorithm for triple-hierarchical constrained nonconvex optimization problem and its application to network bandwidth allocation. SIAM J. Optim. 22(3), 862-878 (2012) 13. Iiduka, H: Fixed point optimization algorithms for distributed optimization in networked systems. SIAM J. Optim. 23, 1-26 (2013) 14. Sakurai, K, Liduka, H: Acceleration of the Halpern algorithm to search for a fixed point of a nonexpansive mapping. Fixed Point Theory Appl. 2014, 202 (2014) 15. Nocedal, J, Wright, SJ: Numerical Optimization, 2nd edn. Springer Series in Operations Research and Financial Engineering. Springer, Berlin (2006) 16. Yao, Y, Marino, G, Muglia, L: A modified Korpelevich’s method convergent to the minimum-norm solution of a variational inequality. Optimization 63(4), 1-11 (2012)

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