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The Method of Alternating Resolvents RevisitedOganeditse A. Boikanyo a & Gheorghe Moroşanu b

a Department of Mathematics, University of Botswana, Gaborone, Botswanab Department of Mathematics and Its Applications, Central European University, Budapest,HungaryVersion of record first published: 10 Oct 2012.

To cite this article: Oganeditse A. Boikanyo & Gheorghe Moroşanu (2012): The Method of Alternating Resolvents Revisited,Numerical Functional Analysis and Optimization, 33:11, 1280-1287

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Numerical Functional Analysis and Optimization, 33(11):1280–1287, 2012Copyright © Taylor & Francis Group, LLCISSN: 0163-0563 print/1532-2467 onlineDOI: 10.1080/01630563.2012.693564

THE METHOD OF ALTERNATING RESOLVENTS REVISITED

Oganeditse A. Boikanyo1 and Gheorghe Morosanu2

1Department of Mathematics, University of Botswana, Gaborone, Botswana2Department of Mathematics and Its Applications, Central European University,Budapest, Hungary

� The purpose of this article is to prove a strong convergence result associated with ageneralization of the method of alternating resolvents introduced by the authors in convergence ofthe method of alternating resolvents [4] under minimal assumptions on the control parametersinvolved. Thus, this article represents a significant improvement of the article mentioned above.

Keywords Alternating resolvents; Maximal monotone operator; Nonexpansive map;Proximal point algorithm; Resolvent operator.

Mathematics Subject Classification 47J25; 47H05; 47H09.

1. INTRODUCTION

In the sequel, H will be a real Hilbert space with inner product〈·, ·〉 and induced norm ‖ · ‖. We recall that a map T : H → H is callednonexpansive if for every x , y ∈ H we have ‖Tx − Ty‖ ≤ ‖x − y‖. Anoperator A : D(A) ⊂ H → 2H is said to be monotone if

〈x − x ′, y − y′〉 ≥ 0, ∀(x , y), (x ′, y′) ∈ G(A)�

In other words, its graph G(A) = �(x , y) ∈ H × H : x ∈ D(A), y ∈ Ax� is amonotone subset of the product space H × H . An operator A is calledmaximal monotone if in addition to being monotone, its graph is notproperly contained in the graph of any other monotone operator. Notethat if A is maximal monotone, then so is A−1. For a maximal monotoneoperator A, the resolvent of A, defined by J A� := (I + �A)−1, is well defined

Received 19 August 2011; Accepted 8 May 2012.Address correspondence to Oganeditse A. Boikanyo, Department of Mathematics, University

of Botswana, Private Bag 00704, Gaborone, Botswana; E-mail: boikanyoa@gmail.com

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The Method of Alternating Resolvents Revisited 1281

on the whole space H , is single-valued and nonexpansive for every � > 0.It is known that the Yosida approximation of A, an operator defined byA� := �−1(I − J A� ) (where I is the identity operator) is maximal monotonefor every � > 0.

The method of alternating resolvents is an iterative procedure forfinding a point in the intersection A−1(0) ∩ B−1(0) =: F (where A and Bare maximal monotone operators), and it is defined as follows:

x0 �→ x1 = J A� x0 �→ x2 = J B� x1 �→ x3 = J A� x2 �→ x4 = J B� x3 �→ � � � ,

for some � > 0 and any starting point x0 ∈ H . Bauscheke et al. [1] showedthat such a method converges weakly to some point of F . In fact, theyshowed that the method of alternating resolvents given above convergesweakly to some point of FixJ A� J

B� – the fixed point set of the composition

mapping J A� JB� – provided that this set is not empty. For a generalized

version of this method, see [2]. In the particular case when A and B arenormal cones, the method of alternating resolvents reduces to the wellknown method of alternating projections which was introduced by vonNeumann in 1933. The latter method converges weakly [5] to some pointin F , but not strongly in general [6]. This fact prompted the currentauthors to construct sequences generated from the method of alternatingresolvents which converge strongly, see [2, 4]. One such sequence wasdefined [4] as

x2n+1 = J A�n (�nu + (1 − �n)x2n + en) for n=0,1, � � � , (1)

x2n = J B�n (�nu + (1 − �n)x2n−1 + e ′n) for n=1,2, � � � , (2)

for any given u, x0 ∈ H , where (en) and (e ′n) are sequences of

computational errors, �n , �n ∈ (0, 1) and �n , �n ∈ (0,∞). In this article,we investigate if the method initiated in [9] can be extended to thecase of two operators, namely, to the scheme (1), (2) in order to obtainstrong convergence results of sequences generated by it under minimalassumptions on the control parameters �n , �n , �n , and �n , thereby refiningthe previously obtained results associated with the strong convergence ofthe method of alternating resolvents [2, 4].

2. PRELIMINARY RESULTS

The following two lemmas will be crucial in proving our main results.

Lemma 1 ([4]). Let (sn) be a sequence of non-negative real numbers satisfying

sn+1 ≤ (1 − �n)(1 − �n)sn + �nbn + �ncn + dn , n ≥ 0, (3)

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1282 O. A. Boikanyo and G. Morosanu

where (�n), (�n), (bn), (cn), and (dn) satisfy the conditions: (i) �n , �n ∈ [0, 1],with

∏∞n=0(1 − �n) = 0, (ii) lim supn→∞ bn ≤ 0, (iii) lim supn→∞ cn ≤ 0, and

(iv) dn ≥ 0 for all n ≥ 0 with∑∞

n=0 dn < ∞. Then limn→∞ sn = 0.

Remark 2. If limn→∞ an = 0, then∏∞

n=0(1 − an) = 0 if and only if∑∞n=0 an = ∞.

Lemma 3 ([7]). Let (sn) be a sequence of real numbers that does not decrease atinfinity, in the sense that there exists a subsequence (snj ) of (sn) such that snj ≤ snj+1

for all j ≥ 0. Define an integer sequence (�(n))n≥n0 as

�(n) = max{n0 ≤ k ≤ n : sk < skj+1

}�

Then �(n) → ∞ as n → ∞ and for all n ≥ n0

max �s�(n), sn� ≤ s�(n)+1� (4)

The next lemma is well known, it can be found for example in [8,p. 20].

Lemma 4. Any maximal monotone operator A : D(A) ⊂ H → 2H satisfies thedemicloseness principle. In other words, given any two sequences (xn) and (yn)satisfying xn → x and yn ⇀ y with (xn , yn) ∈ G(A), then (x , y) ∈ G(A).

Lemma 5 ([10]). For any x ∈ H and � ≥ � > 0,

‖x − J�x‖ ≤ 2‖x − J�x‖,where A : D(A) ⊂ H → 2H is a maximal monotone operator.

3. MAIN RESULTS

We begin by proving a strong convergence result associated with theexact iterative process

v2n+1 = J A�n (�nu + (1 − �n)v2n) for n = 0, 1, � � � , (5)

v2n = J B�n (�nu + (1 − �n)v2n−1) for n = 1, 2, � � � , (6)

where �n , �n ∈ (0, 1) and v0,u ∈ H are given. The proof of the followingtheorem makes use of the ideas of the articles [3, 4, 7, 9]

Theorem 6. Let A : D(A) ⊂ H → 2H and B : D(B) ⊂ H → 2H be maximalmonotone operators with A−1(0) ∩ B−1(0) =: F �= ∅. For arbitrary but fixed vectorsv0,u ∈ H , let (vn) be the sequence generated by (5), (6), where �n , �n ∈ (0, 1) and

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The Method of Alternating Resolvents Revisited 1283

�n , �n ∈ (0,∞). Assume that (i) limn→∞ �n = 0 and limn→∞ �n = 0, (ii) either∑∞n=0 �n = ∞ or

∑∞n=0 �n = ∞, and (iii) both (�n) and (�n) are bounded from

below away from zero. Then (vn) converges strongly to the point of F nearest to u.

Proof. We already know that (vn) is bounded [4]. Next, we show that forany p ∈ F

(1 + �n)∥∥v2n+1 − p

∥∥2 ≤ (1 − �n)∥∥v2n − p

∥∥2 + 2�n〈u − p, v2n+1 − p〉− (1 − �n) ‖v2n+1 − v2n‖2 (7)

holds. Indeed, multiplying

v2n+1 − p + �nAv2n+1 � �n(u − p) + (1 − �n)(v2n − p)

scalarly by v2n+1 − p and using the monotonicity of A, we obtain

2∥∥v2n+1 − p

∥∥2 ≤ 2�n〈u − p, v2n+1 − p〉 + 2(1 − �n)〈v2n − p, v2n+1 − p〉= (1 − �n)(

∥∥v2n − p∥∥2 + ∥∥v2n+1 − p

∥∥2 − ‖v2n+1 − v2n‖2)

+ 2�n〈u − p, v2n+1 − p〉�Rearranging terms, we readily get (7). Using similar arguments as above,one can prove that for any p ∈ F

(1 + �n)∥∥v2n − p

∥∥2 ≤ (1 − �n)∥∥v2n−1 − p

∥∥2 + 2�n〈u − p, v2n − p〉− (1 − �n) ‖v2n − v2n−1‖2 �

Using this inequality in (7), we get

(1 + �n)∥∥v2n+1 − p

∥∥2 ≤ (1 − �n)(1 − �n)∥∥v2n−1 − p

∥∥2 + 2�n〈u − p, v2n+1 − p〉− (1− �n)‖v2n+1 − v2n‖2 + 2�n(1− �n)〈u − p, v2n − p〉− (1 − �n)(1 − �n) ‖v2n − v2n−1‖2 � (8)

Denote sn := ‖v2n−1 − PF u‖2. Then it follows from (8) and theboundedness of (vn) that

sn+1 − sn + ‖v2n − v2n−1‖2 + ‖v2n+1 − v2n‖2 ≤ (�n + �n)M , (9)

for some positive constant M . On the other hand, we have from (5)∥∥v2n+1 − J A� v2n+1

∥∥ ≤ 2∥∥v2n+1 − J A�n v2n+1

∥∥≤ 2 ‖�n(u − v2n) + (v2n − v2n+1)‖≤ (�n + ‖v2n − v2n+1‖)M ′, (10)

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1284 O. A. Boikanyo and G. Morosanu

where � > 0 is the greatest lower bound of �n , M ′ is a positive constantand the first inequality follows from Lemma 5. Similarly, starting from (6),we arrive at

∥∥v2n − J B� v2n∥∥ ≤ (�n + ‖v2n − v2n−1‖)M ′, (11)

where � > 0 is the greatest lower bound of �n . We next consider twopossible cases on the sequence (sn).

Case 1: (sn) is eventually decreasing (i.e., there exists N ≥ 0 suchthat (sn) is decreasing for all n ≥ N ). In this case, (sn) is convergent.Passing to the limit in (9), we get

‖vn+1 − vn‖ → 0 as n → ∞� (12)

Therefore, it follows from (10) that∥∥v2n+1 − J A� v2n+1

∥∥ → 0 as n → ∞�

Using the fact that A−1� is demiclosed, where A� is the Yosida approximation

of A, we get w((v2n+1)) ⊂ A−1(0). Recall that w((xn)) denotes the setof all weak cluster points of the sequence (xn). Similarly, from (11) and(12), we derive w((v2n)) ⊂ B−1(0). These two inclusions and (12) implythat w((vn)) ⊂ A−1(0) ∩ B−1(0). Extract a subsequence (v2nk+1) of (v2n+1)converging weakly to some y ∈ F such that

lim supn→∞

〈u − PF u, v2n+1 − PF u〉 = limk→∞

〈u − PF u, v2nk+1 − PF u〉= 〈u − PF u, y − PF u〉 ≤ 0,

where PF u denotes the projection of u on the set F (which is closed andconvex). On the other hand, from (8), we derive

‖v2n+1 − PF u‖2 ≤ (1− �n)(1− �n) ‖v2n−1 −PF u‖2 + 2�n〈u −PF u, v2n+1 −PF u〉+ 2�n(1 − �n)〈u − PF u, (v2n − v2n+1) + (v2n+1 − PF u)〉,

and hence from Lemma 1, we get v2n+1 → PF u as n → ∞. By virtue of(12), we also have v2n → PF u as n → ∞. Hence vn → PF u as desired.

Case 2: (sn) is not eventually decreasing, that is, there is asubsequence (snj ) of (sn) such that snj < snj+1 for all j ≥ 0. We thereforedefine an integer sequence (�(n)) as in Lemma 3 so that for all n ≥ n0,s�(n) ≤ s�(n)+1 holds. Note that from (9), we have

limn→∞

∥∥v2�(n)+1 − v2�(n)∥∥ = 0�

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The Method of Alternating Resolvents Revisited 1285

It then follows from (10) that w((v2�(n)+1)) ⊂ A−1(0). Similarly, from (11),we have w((v2�(n))) ⊂ B−1(0). On the other hand, the above limit impliesthat w((v2�(n)+1)) ⊂ B−1(0). Therefore, w((v2�(n)+1)) ⊂ F . Consequently,

lim supn→∞

〈u − PF u, v2�(n)+1 − PF u〉 ≤ 0�

On the other hand, since s�(n) ≤ s�(n)+1 holds, it follows from (8) that

(2��(n) + ��(n) − ��(n)��(n))s�(n)+1 ≤ 2��(n)〈u − PF u, v2�(n)+1 − PF u〉+ 2��(n)(1− ��(n))〈u −PF u, v2�(n) − v2�(n)+1〉+ 2��(n)(1− ��(n))〈u −PF u, v2�(n)+1 −PF u〉,

which implies, according to the above pieces of information, that s�(n)+1 →0 as n → ∞. Hence, from (4) it follows that sn → 0 as n → ∞; that is,v2n+1 → PF u as n → ∞. Now, using again (9), we get v2n → PF u as n → ∞.This shows that vn → PF u as n → ∞, and the proof is complete. �

Remark 7. Theorem 6 is a refinement of Theorem 7 [4].

Theorem 8. Let A : D(A) ⊂ H → 2H and B : D(B) ⊂ H → 2H be maximalmonotone operators with A−1(0) ∩ B−1(0) =: F �= ∅. For arbitrary but fixed vectorsx0,u ∈ H , let (xn) be the sequence generated by (1), (2), where �n , �n ∈ (0, 1) and�n , �n ∈ (0,∞). Assume that (i) limn→∞ �n = 0 and limn→∞ �n = 0, (ii) either∑∞

n=0 �n = ∞ or∑∞

n=0 �n = ∞, and (iii) both (�n) and (�n) are bounded frombelow away from zero. In addition, if any of the following conditions is satisfied,

(a)∑∞

n=0 ‖en‖ < ∞ and∑∞

n=1 ‖e ′n‖ < ∞;

(b)∑∞

n=0 ‖en‖ < ∞ and ‖e ′n‖/�n → 0;

(c)∑∞

n=0 ‖en‖ < ∞ and ‖e ′n‖/�n → 0;

(d) ‖en‖/�n → 0 and∑∞

n=1 ‖e ′n‖ < ∞;

(e) ‖en‖/�n → 0 and∑∞

n=1 ‖e ′n‖ < ∞;

(f) ‖en‖/�n → 0 and ‖e ′n‖/�n → 0;

(g) ‖en‖/�n → 0 and ‖e ′n‖/�n → 0;

(h) ‖en‖/�n → 0 and ‖e ′n‖/�n → 0;

(i) ‖en‖/�n → 0 and ‖e ′n‖/�n → 0;

(j) ‖en‖/�n → 0 and ‖e ′n‖/�n−1 → 0;

(k) ‖en−1‖/�n → 0 and ‖e ′n‖/�n−1 → 0;

(l) ‖en−1‖/�n → 0 and ‖e ′n‖/�n → 0;

(m)∑∞

n=0 ‖en‖ < ∞ and ‖e ′n‖/�n−1 → 0;

(n) ‖en−1‖/�n → 0 and∑∞

n=1 ‖e ′n‖ < ∞,

then (xn) converges strongly to the point of F nearest to u.

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1286 O. A. Boikanyo and G. Morosanu

Proof. In view of Theorem 6, it is enough to show that ‖xn − vn‖ → 0 asn → ∞. Since the resolvent of A is nonexpansive, we derive from (1) and(5) that

‖x2n+1 − v2n+1‖ ≤ (1 − �n) ‖x2n − v2n‖ + ‖en‖ � (13)

Similarly, from (2) and (6), we have

‖x2n − v2n‖ ≤ (1 − �n) ‖x2n−1 − v2n−1‖ + ∥∥e ′n

∥∥ � (14)

These two inequalities imply that

‖x2n+1 − v2n+1‖ ≤ (1 − �n)(1 − �n) ‖x2n−1 − v2n−1‖ + ‖en‖ + ∥∥e ′n

∥∥ �

Therefore, if the error sequence satisfy any of the conditions (a)–(i),then it readily follows from Lemma 1 that ‖x2n+1 − v2n+1‖ → 0 as n → ∞.Passing to the limit in (14), we derive ‖x2n − v2n‖ → 0 as well. If the errorsequence satisfy any of the conditions (j)–(n), then from (13) and (14),we have

‖x2n − v2n‖ ≤ (1 − �n−1)(1 − �n) ‖x2n−2 − v2n−2‖ + ‖en−1‖ + ∥∥e ′n

∥∥ �

It then follows from Lemma 1 that ‖x2n − v2n‖ → 0 as n → ∞. Passing tothe limit in (13), we derive ‖x2n+1 − v2n+1‖ → 0 as well. This completes theproof of the theorem. �

Remark 9. Theorem 8 improves Theorem 8 [4]. It also containsTheorems 2–3 [4] as special cases since for �n = 0 for all n ∈ �, algorithm(1), (2) reduces to algorithm (14), (15) introduced in [2], and for �n =�n for all n ∈ �, algorithm (1), (2) reduces to algorithm (24), (25)introduced in [2].

REFERENCES

1. H. H. Bauschke, P. L. Combettes and S. Reich (2005). The asymptotic behavior of thecomposition of two resolvents. Nonlinear Anal. 60:283–301.

2. O. A. Boikanyo and G. Morosanu (2011). On the method of alternating resolvents. NonlinearAnal. 74:5147–5160.

3. O. A. Boikanyo and G. Morosanu (submitted). A generalization of the regularization proximalpoint method. J. Nonlinear Anal. Appl. DOI: 10.5899/2012/jnaa-00129.

4. O. A. Boikanyo and G. Morosanu (to appear). Strong convergence of the method of alternatingresolvents. J. Nonlinear Convex Anal.

5. L. M. Bregman. (1965). The method of successive projection for finding a common point ofconvex sets. Sov. Math. Dokl. 6:688–692.

6. H. Hundal (2004). An alternating projection that does not converge in norm. Nonlinear Anal.57:35–61.

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The Method of Alternating Resolvents Revisited 1287

7. P. E. Maingé (2008). Strong convergence of projected subgradient methods for nonsmoothand nonstrictly convex minimization. Set-Valued Anal. 16:899–912.

8. G. Morosanu (1988). Nonlinear Evolution Equations and Applications. Reidel, Dordrecht.9. F. Wang and H. Cui (to appear). On the contraction-proximal point algorithms with multi-

parameters. J. Glob. Optim. DOI: 10.1007/s10898-011-9772-4.10. H. K. Xu (2006). A regularization method for the proximal point algorithm. J. Glob. Optim.

36:115–125.

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