A variable metric extension of the forward–backward–forward algorithm for monotone operators ∗

arXiv:1210.2986v2 [math.OC] 31 Oct 2012

A variable metric extension of the forward–backward–forward algorithm for monotone operators

B`˘ang Cˆong V˜u

UPMC Universit´e Paris 06

Laboratoire Jacques-Louis Lions – UMR CNRS 7598 75005 Paris, France

vu@ljll.math.upmc.fr

Abstract

We propose a variable metric extension of the forward–backward-forward algorithm for finding a zero of the sum of a maximally monotone operator and a monotone Lipschitzian op- erator in Hilbert spaces. In turn, this framework provides a variable metric splitting algorithm for solving monotone inclusions involving sums of composite operators. Monotone operator splitting methods recently proposed in the literature are recovered as special cases.

Keywords: variable metric, composite operator, duality, monotone inclusion, monotone op- erator, operator splitting, primal-dual algorithm

Mathematics Subject Classifications (2010) 47H05, 49M29, 49M27, 90C25

1 Introduction

A basic problem in applied monotone operator theory is to find a zero of a maximally monotone operator A on a real Hilbert space H. This problem can be solved by the proximal point algo- rithm proposed in [17] which requires only the resolvent of A, provided it is easy to implement numerically. In order to get more efficient proximal algorithms, some authors have proposed the use of variable metric or preconditioning in such algorithms [3, 5, 6, 10, 13, 15, 16].

This problem was then extended to the problem of finding a zero of the sum of a maximally monotone operator A and a cocoercive operator B (i.e., B⁻¹ is strongly monotone). In such

∗This work was partially supported by Grant 102.01-2012.15 of the Vietnam National Foundation for Science and Technology Development (NAFOSTED).

instances, the forward-backward splitting algorithm [1, 8, 12, 18] can be used. Recently, this algorithm has been investigated in the context of variable metric [11]. In the case when B is only Lipschitzian and not cocoercive, the problem can be solved by the forward-backward-forward splitting algorithm [4, 19]. New applications of this basic algorithm to more complex monotone inclusions are presented in [4, 9].

In the present paper, we propose a variable metric version of the forward-backward-forward splitting algorithm. In Section 2, we recall notation and background on convex analysis and monotone operator theory. In Section 3, we present our variable metric forward-backward-forward splitting algorithm. In Section 4, the results of Section 3 are used to develop a variable metric primal–dual algorithm for solving the type of composite inclusions considered in [9].

2 Notation and background

Throughout, H, G, and (Gⁱ)1≤i≤m are real Hilbert spaces. Their scalar products and associated norms are respectively denoted by h· | ·iand k · k. We denote by B(H,G) the space of bounded linear operators from H to G. The adjoint of L ∈ B(H,G) is denoted by L^∗. We set B(H) = B(H,H). The symbols⇀and→denote respectively weak and strong convergence, and Id denotes the identity operator, andB(x;ρ) denotes the closed ball of centerx∈ H and radiusρ ∈ ]0,+∞[.

The interior ofC ⊂ H is denoted by intC. We denote by ℓ¹₊(N) the set of summable sequences in [0,+∞[.

Let M1 and M2 be self-adjoint operators in B(H), we write M1 < M2 if and only if (∀x ∈ H) hM₁x|xi ≥ hM₂x|xi.Letα∈]0,+∞[. We set

P_α(H) =

M ∈B(H)|M^∗ =M and M <αId . (2.1)

Moreover, for everyM ∈P_α(H), we define respectively a scalar product and a norm by (∀x∈ H)(∀y ∈ H) hx|yiM =hM x|yi and kxk^M =p

hM x|xi. (2.2)

Let A:H → 2^H be a set-valued operator. The domain is domA =

x∈ H | Ax6=∅ , and the graph of A is graA =

(x, u)∈ H × H | u∈Ax . The set of zeros of A is zerA =

x∈ H | 0∈Ax , and the range of A is ranA =

u∈ H | (∃x∈ H)u∈Ax . The inverse of A is A⁻¹:H 7→2^H:u7→

x∈ H |u∈Ax , and the resolvent ofA is

J_A= (Id +A)⁻¹. (2.3)

Moreover,A is monotone if

(∀(x, y)∈ H × H)(∀(u, v)∈Ax×Ay) hx−y |u−vi ≥0, (2.4) and maximally monotone if it is monotone and there exists no monotone operator B: H → 2^H such that graA⊂graB and A6=B. We say thatAis uniformly monotone at x∈domA if there exists an increasing functionφA: [0,+∞[→[0,+∞] vanishing only at 0 such that

∀u∈Ax

∀(y, v)∈graA

hx−y |u−vi ≥φ_A(kx−yk). (2.5)

3 Variable metric forward-backward-forward splitting algorithm

The forward-backward-forward splitting algorithm was first proposed in [19] to solve inclusion involving the sum of a maximally monotone operator and a monotone Lipschitzian operator. In [4], it was revisited to include computational errors. Below, we extend it to a variable metric setting.

Theorem 3.1 Let K be a real Hilbert space with the scalar product hhh· | ·iii and the associated norm|||| · ||||. Let α and β be in]0,+∞[, let (ηn)n∈N be a sequence in ℓ¹₊(N), and let(Un)n∈N be a sequence inB(K) such that

µ= sup

n∈NkU_nk<+∞ and (1 +η_n)U_n+1 <U_n∈P_α(K). (3.1) Let A: K → 2^K be maximally monotone, let B:K → K be a monotone and β-Lipschitzian operator on K such that zer(A+B) 6= ∅. Let (an)n∈N, (bn)n∈N, and (cn)n∈N be absolutely summable sequences in K. Let x₀ ∈ K, let ε ∈ ]0,1/(βµ+ 1)[, let (γn)n∈N be a sequence in [ε,(1−ε)/(βµ)], and set

(∀n∈N)







y_n=xn−γnUn(Bxn+an) p_n =JγnU_nAy_n+b_n

q_n=p_n−γnUn(Bp_n+cn) x_n+1 =xn−y_n+q_n.

(3.2)

Then the following hold for some x∈zer(A+B).

(i) P

n∈N||||x_n−p_n||||² <+∞ and P

n∈N||||y_n−q_n||||² <+∞. (ii) xn⇀x and p_n⇀x.

(iii) Suppose that one of the following is satisfied:

(a) limd_zer(A+B)(xn) = 0.

(b) A+B is demiregular (see [1, Definition 2.3]) at x.

(c) A or B is uniformly monotone atx.

(d) int zer(A+B)6=∅and there exists(ν_n)_n∈N∈ℓ¹₊(N) such that(∀n∈N) (1 +ν_n)U_n U_n+1.

Thenxn→x andp_n→x.

Proof. It follows from [11, Lemma 3.7] that the sequences (xn)n∈N, (y_n)n∈N, (p_n)n∈N and (q_n)n∈N

are well defined. Moreover, using [10, Lemma 2.1(i)(ii)] and (3.1), we obtain

∀(zn)n∈N∈K^N X

n∈N

||||zn||||<+∞ ⇔ X

n∈N

||||zn||||^U−_n¹ <+∞ (3.3) and

∀(zn)n∈N∈K^N X

n∈N

||||zn||||<+∞ ⇔ X

n∈N

||||zn||||^Un <+∞. (3.4)

Let us set

(∀n∈N)











 e

y_n=xn−γnUnBxn

e

p_n=J_γnU_nAye_n e

q_n=pe_n−γnU_nBep_n e

x_n+1 =xn−ye_n+eq_n,

and







un=γ_n⁻¹U⁻¹_n (xn−ep_n) +Bpe_n−Bxn

e_n=ex_n+1−x_n+1 dn=q_n−eq_n+ye_n−y_n.

(3.5)

Then (3.5) yields

(∀n∈N) u_n=γ_n⁻¹U⁻¹_n ( ye_n−ep_n) +Bpe_n∈Aep_n+Bpe_n, (3.6) and (3.5), (3.2), Lemma [11, Lemma 3.7(ii)], and the Lipschitzianity ofB onK yield

(∀n∈N)









||||y_n−ye_n||||U⁻¹_n ≤(βµ)⁻¹||||an||||^Un

||||p_n−ep_n||||^U−_n¹ ≤ ||||b_n||||^U−_n¹+ (βµ)⁻¹||||a_n||||^Un

||||q_n−eq_n||||U⁻_n¹ ≤2

||||b_n||||U⁻_n¹ + (βµ)⁻¹||||a_n||||^Un

+ (βµ)⁻¹||||c_n||||^Un. (3.7) Since (an)n∈N, (bn)n∈N, and (cn)n∈N are absolutely summable sequences in K, we derive from (3.3), (3.4), (3.5), and (3.7) that





 P

n∈N||||p_n−pe_n||||<+∞ and P

n∈N||||p_n−pe_n||||U⁻_n¹ <+∞ P

n∈N||||q_n−eq_n||||<+∞ and P

n∈N||||q_n−qe_n||||^U−1_n <+∞ P

n∈N||||d_n||||<+∞ and P

n∈N||||d_n||||^U−_n¹ <+∞.

(3.8)

Now, let x ∈ zer(A + B). Then, for every n ∈ N, (x,−γnU_nBx) ∈ gra(γnU_nA) and (3.5) yields (ep_n,ye_n− ep_n) ∈ gra(γnUnA). Hence, by monotonicity of UnA with respect to the scalar product hhh· | ·iiiU⁻_n¹, we have hhhep_n−x | pe_n−ey_n−γ_nU_nBxiiiU⁻_n¹ ≤ 0. More- over, by monotonicity of U_nB with respect to the scalar product hhh· | ·iiiU⁻¹

n , we also have hhhep_n−x|γnU_nBx−γnU_nBep_niii^U−_n¹ ≤0.By adding the last two inequalities, we obtain

(∀n∈N) hhhep_n−x|pe_n−ey_n−γnU_nBep_niii^U−_n¹ ≤0. (3.9) In turn, we derive from (3.5) that

(∀n∈N) 2γnhhhep_n−x|UnBxn−UnBpe_niiiU⁻¹_n

= 2hhhep_n−x|pe_n−ey_n−γnUnBep_niiiU⁻¹_n

+ 2hhhep_n−x|γnU_nBx_n+ye_n−ep_niii^U−_n¹

≤2hhhep_n−x|γnU_nBx_n+ey_n−ep_niii^U−_n¹

= 2hhhep_n−x|x_n−pe_niii^U−_n¹

=||||x_n−x||||²U⁻¹_n − ||||ep_n−x||||²U⁻¹_n − ||||x_n−ep_n|||²U⁻¹_n . (3.10)

Hence, using (3.5), (3.10), the β-Lipschitz continuity of B, (3.1), and [10, Lemma 2.1(ii)], for everyn∈N, we obtain

||||ex_n+1−x||||²U⁻¹

n =||||eq_n+xn−ye_n−x||||²U⁻¹

n

=||||(pe_n−x) +γnU_n(Bxn−Bep_n)||||²U⁻_n¹

=||||ep_n−x||||²U⁻¹

n + 2γ_nhhhep_n−x|Bx_n−Bep_niii

+γ²_n||||Un(Bxn−Bpe_n)||||²U⁻_n¹

≤ ||||x_n−x||||²U⁻¹

n − ||||x_n−ep_n||||²U⁻¹

n +γ_n²µβ²||||x_n−ep_n||||²

≤ ||||xn−x||||²U⁻¹

n −µ⁻¹||||xn−ep_n||||²+γ_n²µβ²||||xn−ep_n||||². (3.11) Hence, it follows from (3.1) and [10, Lemma 2.1(i)] that

(∀n∈N) ||||ex_n+1−x||||²U⁻¹

n+1 ≤(1+ηn)||||xn−x||||²U⁻_n¹−µ⁻¹(1−γ²_nβ²µ²)||||xn−ep_n||||². (3.12) Consequently,

(∀n∈N) ||||ex_n+1−x||||U⁻¹

n+1 ≤(1 +ηn)||||xn−x||||^U−1_n . (3.13) For everyn∈N, set

εn=p µα⁻¹

2 ||||b_n||||^U−_n¹+(βµ)⁻¹||||a_n||||^Un

+(βµ)⁻¹||||c_n||||^Un+(βµ)⁻¹||||a_n||||^Un

. (3.14) Then (εn)n∈N is summable by (3.3) and (3.4). We derive from [10, Lemma 2.1(ii)(iii)], and (3.8) that

(∀n∈N) ||||e_n||||U⁻¹

n+1 =||||ex_n+1−x_n+1||||U⁻¹

n+1

≤√

α⁻¹||||ex_n+1−x_n+1||||

≤p

µα⁻¹||||ex_n+1−x_n+1||||U⁻_n¹

≤p

µα⁻¹(||||ey_n−y_n||||^U−_n¹+||||eq_n−q_n||||^U−_n¹)

≤εn. (3.15)

In turn, we derive from (3.13) that (∀n∈N) ||||x_n+1−x||||U⁻¹

n+1 ≤ ||||xe_n+1−x||||U⁻¹

n+1 +||||ex_n+1−x_n+1||||U⁻¹

n+1

≤ ||||xe_n+1−x||||U⁻¹

n+1 +εn

≤(1 +η_n)||||x_n−x||||U⁻_n¹ +ε_n. (3.16) This shows that (xn)n∈N is | · |–quasi-Fej´er monotone with respect to the target set zer(A+B) relative to (U⁻_n¹)n∈N. Moreover, by [10, Proposition 3.2], (||||xn−x||||^U−1_n )n∈N is bounded. In turn, sinceB and (J_γnU_nA)_n∈N are Lipschitzian, and (∀n∈ N) x= J_γnU_nA(x−γ_nU_nBx), we deduce from (3.5) that (ey_n)n∈N,(ep_n)n∈N, and (qe_n)n∈N are bounded. Therefore,

τ = sup

n∈N{||||xn−ey_n+eq_n−x||||^U−_n¹,||||xn−x||||^U−_n¹,1 +ηn}<+∞. (3.17)

Hence, using (3.5), Cauchy-Schwarz for the norms (|||| · ||||^U−_n¹)n∈N, and (3.11), we get (∀n∈N) ||||x_n+1−x||||²U⁻¹

n =||||x_n−y_n+q_n−x||||²U⁻¹

n

=||||eq_n+xn−ye_n−x+dn||||²U⁻¹

n

≤ ||||eq_n+x_n−ye_n−x||||²U⁻¹_n + 2τ||||d_n||||^U−_n¹+||||d_n||||²U⁻¹_n

≤ ||||xn−x||||²U⁻¹

n −µ⁻¹(1−γ_n²β²µ²)||||xn−pe_n||||²+ε_1,n, (3.18) where (∀n ∈N) ε1,n = 2τ||||d_n||||^U−_n¹ +||||d_n||||²U⁻¹

n . In turn, for every n∈ N, by (3.1) and [10, Lemma 2.1(i)],

||||x_n+1−x||||²U⁻¹

n+1 ≤(1 +ηn)||||x_n+1−x||||²U⁻¹_n

≤ ||||x_n−x||||²U⁻¹

n −µ⁻¹(1−γ_n²β²µ²)||||x_n−ep_n||||²+τ ε_1,n+τ²η_n. (3.19) Since (τ ε1,n+τ²ηn)n∈N∈ℓ¹₊(N) by (3.8), it follows from [7, Lemma 3.1] that

X

n∈N

||||x_n−ep_n||||² <+∞. (3.20)

(i): It follows from (3.20) and (3.8) that X

n∈N

||||xn−p_n||||² ≤2X

n∈N

||||xn−ep_n||||²+ 2X

n∈N

|||||p_n−ep_n||||² <+∞. (3.21) Furthermore, we derive from (3.8) and (3.5) that

X

n∈N

||||y_n−q_n||||² =X

n∈N

||||eq_n−ey_n+d_n||||²

=X

n∈N

||||ep_n−x_n+γnU_n(Bxn−Bpe_n) +d_n||||²

≤3 X

n∈N

||||xn−ep_n||||²+||||γnUn(Bxn−Bpe_n)||||²+||||dn||||²

<+∞. (3.22)

(ii): Let x be a weak cluster point of (xn)n∈N. Then there exists a subsequence (xkn)n∈N

that converges weakly to x. Therefore pe_kn ⇀ x by (3.20). Furthermore, it follows from (3.5) that u_kn → 0. Hence, since (∀n ∈ N) (ep_kn,u_kn) ∈ gra(A+B), we obtain, x ∈ zer(A+B) [2, Proposition 20.33(ii)]. Altogether, it follows [10, Lemma 2.3(ii)] and [10, Theorem 3.3] that xn ⇀xand hence thatp_n⇀xby (i).

(iii)(a): Since A and B are maximally monotone and domB = K, A+B is maximally monotone [2, Corollary 24.4(i)], zer(A+B) is therefore closed [2, Proposition 23.39]. Hence, the claims follow from (i), (3.16), and [10, Proposition 3.4].

(iii)(b): By (i), x_n⇀x, and hence (3.20) implies that ep_n⇀x. Furthermore, it follows from (3.5) thatun→0. Hence, since (∀n∈N) (ep_n,un)∈gra(A+B) and sinceA+B is demiregular at x, by [1, Definition 2.3], pe_n→x, and therefore (3.20) implies that x_n→x.

(iii)(c): If A or B is uniformly monotone at x, then A +B is uniformly monotone at x.

Therefore, the result follows from [1, Proposition 2.4(i)].

(iii)(d): Suppose thatz∈int zer(A+B) and fixρ∈ ]0,+∞[ such thatB(z;ρ)⊂zer(A+B).

It follows from (3.16) and [10, Proposition 3.2] that ε= sup

x∈B(z;ρ)

sup

n∈N||||x_n−x||||^U−_n¹ ≤(1/√

α) sup

n∈N||||x_n−z||||+ sup

x∈B(z;ρ)||||x−z||||

<+∞ (3.23) and from (3.16) that

(∀n∈N)(∀x∈B(z;ρ))||||x_n+1−x||||²U⁻¹

n+1 ≤ ||||x_n−x||||²U⁻¹

n + 2ε(εη_n+ε_n) + (εη_n+ε_n)². (3.24) Hence, the claim follows from (i), [10, Lemma 2.1], and [10, Proposition 4.3].

Remark 3.2 Here are some remarks.

(i) In the case when (∀n ∈ N) Un = Id, the standard forward-backward-forward splitting algorithm (3.2) reduces to algorithm proposed in [4, Eq. (2.3)], which was proposed initially in the error-free setting in [19].

(ii) An alternative variable metric splitting algorithm proposed in [14] can be used to find a zero of the sum of a maximally monotone operator A and a Lipschitzian monotone operator B in instance when K is finite-dimensional. This algorithm uses a different error model and involves more iteration-dependent variables than (3.2).

Example 3.3 Let f: K → [−∞,+∞] be a proper lower semicontinuous convex function, let α ∈ ]0,+∞[, let β ∈ ]0,+∞[, let B:K → K be a monotone and β-Lipschitzian operator, let (ηn)n∈N ∈ℓ¹₊(N), and let (Un)n∈N be a sequence in P_α(K) that satisfies (3.1). Furthermore, let x₀ ∈K, let ε∈]0,min{1,1/(µβ+ 1)}[, where µis defined as in (3.1), let (γ_n)_n∈N be a sequence in [ε,(1−ε)/(βµ)]. Suppose that the variational inequality

find x¯ ∈K such that (∀y∈K) hx¯−y|Bx¯i+f(¯x)≤f(y) (3.25) admits at least one solution and set

(∀n∈N)









y_n=xn−γnUnBxn

p_n= arg min

x∈K f(x) +_2γ¹

n||||x−y_n||||²U⁻¹

n

q_n=p_n−γnUnBp_n

x_n+1=x_n−y_n+q_n.

(3.26)

Then (x_n)_n∈N converges weakly to a solution ¯x to (3.25).

Proof. Set A=∂f and (∀n∈N)an= 0,bn= 0,cn= 0 in Theorem 3.1(ii).

4 Monotone inclusions involving Lipschitzian operators

The applications of the forward-backward-forward splitting algorithm considered in [4, 9, 19] can be extended to a variable metric setting using Theorem 3.1. As an illustration, we present a variable metric version of the algorithm proposed in [9, Eq. (3.1)]. Recall that the parallel sum ofA: H →2^H and B:H →2^H is [2]

AB= (A⁻¹+B⁻¹)⁻¹. (4.1)

Problem 4.1 Let H be a real Hilbert space, let m be a strictly positive integer, let z ∈ H, let A:H → 2^H be maximally monotone operator, let C:H → H be monotone and ν₀-Lipschitzian for some ν₀ ∈ ]0,+∞[. For every i ∈ {1, . . . , m}, let Gⁱ be a real Hilbert space, let ri ∈ Gⁱ, let Bi:Gⁱ→2^Gi be maximally monotone operator, letDi:Gⁱ →2^Gi be monotone and such thatD_i⁻¹ isν_i-Lipschitzian for someν_i ∈]0,+∞[, and letL_i: H → Gi is a nonzero bounded linear operator.

Suppose that z∈ran

A+

Xm i=1

L^∗_i (B_i D_i)(L_i· −r_i) +C

. (4.2)

The problem is to solve the primal inclusion findx∈ H such thatz∈Ax+

Xm i=1

L^∗_i (Bi Di)(Lix−ri)

+Cx, (4.3)

and the dual inclusion

find v₁ ∈ G1, . . . , v_m ∈ Gm such that (∃x∈ H)

(z−Pm

i=1L^∗_ivi ∈Ax+Cx,

(∀i∈ {1, . . . , m})v_i ∈(B_iD_i)(L_ix−r_i).

(4.4) As shown in [9], Problem 4.1 covers a wide class of problems in nonlinear analysis and convex optimization problems. However, the algorithm in [9, Theorem 3.1] is studied in the context of a fixed metric. The following result extends this result to a variable metric setting.

Corollary 4.2 Letαbe in]0,+∞[, let(η_0,n)n∈Nbe a sequence inℓ¹₊(N), let(Un)n∈Nbe a sequence in P_α(H), and for every i ∈ {1, . . . , m}, let (ηi,n)n∈N be a sequence in ℓ¹₊(N), let (Ui,n)n∈N be a sequence inP_α(Gi) such that µ= sup_n∈N{kU_nk,kU_1,nk, . . . ,kU_m,nk}<+∞ and

(∀n∈N) (1 +η0,n)Un+1 <Un, and (∀i∈ {1, . . . , m}) (1 +ηi,n)Ui,n+1 <Ui,n. (4.5) Let (a1,n)n∈N,(b1,n)n∈N, and (c1,n)n∈N be absolutely summable sequences in H, and for every i ∈ {1, . . . , m}, let (a_2,i,n)n∈N,(b_2,i,n)n∈N, and (c_2,i,n)n∈N be absolutely summable sequences in Gi. Furthermore, set

β= max{ν₀, ν₁, . . . , ν_m}+ vu ut

Xm i=1

kL_ik², (4.6)

let x0 ∈ H, let (v1,0, . . . , vm,0)∈ G¹⊕. . .⊕ G^m, let ε ∈]0,1/(1 +βµ)[, let (γn)n∈N be a sequence in [ε,(1−ε)/(βµ)]. Set

(∀n∈N)



















y_1,n =xn−γnUn Cxn+Pm

i=1L^∗_ivi,n+a_1,n p_1,n =J_γnUnA(y_1,n+γ_nU_nz) +b_1,n

for i= 1, . . . , m







y_2,i,n =v_i,n+γ_nU_i,n L_ix_n−D⁻_i ¹v_i,n+a_2,i,n p_2,i,n=J_γ

nUi,nB⁻_i¹(y_2,i,n−γnUi,nri) +b_2,i,n q_2,i,n=p_2,i,n+γnUi,n Lip_1,n−D_i⁻¹p_2,i,n+c_2,i,n v_i,n+1=vi,n−y_2,i,n+q_2,i,n

q1,n =p1,n−γnUn Cp1,n+Pm

i=1L^∗_ip2,i,n+c1,n x_n+1=xn−y_1,n+q_1,n.

(4.7)

Then the following hold.

(i) P

n∈Nkxn−p_1,nk² <+∞ and (∀i∈ {1, . . . , m}) P

n∈Nkvi,n−p_2,i,nk² <+∞.

(ii) There exist a solution xto (4.3)and a solution (v1, . . . , vm) to (4.4)such that the following hold.

(a) x_n⇀ x and p_1,n ⇀ x.

(b) (∀i∈ {1, . . . , m})vi,n⇀ vi and p_2,i,n ⇀ vi.

(c) Suppose thatA or C is uniformly monotone at x, then xn→x and p_1,n →x.

(d) Suppose thatB_j⁻¹ or D_j⁻¹ is uniformly monotone atvj, for some j ∈ {1, . . . , m}, then v_j,n→v_j and p_2,j,n →v_j.

Proof. All sequences generated by algorithm (4.7) are well defined by [11, Lemma 3.7]. We define K=H ⊕ G1⊕ · · · ⊕ G^m the Hilbert direct sum of the Hilbert spacesH and (Gⁱ)₁≤i≤m, the scalar product and the associated norm ofK respectively defined by

hhh· | ·iii: ((x,v),(y,w))7→ hx|yi+ Xm i=1

hv_i |w_ii and |||| · ||||: (x,v)7→

vu utkxk²+

Xm i=1

kv_ik², (4.8) wherev= (v₁, . . . , vm) and w= (w₁, . . . , wm) are generic elements in G1⊕ · · · ⊕ G^m. Set











A:K→2^K: (x, v₁, . . . , vm)7→(−z+Ax)×(r₁+B₁⁻¹v₁)×. . .×(rm+B_m⁻¹vm) B:K→K: (x, v₁, . . . , vm)7→

Cx+Pm

i=1L^∗_ivi, D⁻₁¹v₁−L₁x, . . . , D⁻_m¹vm−Lmx

(∀n∈N) U_n:K→K: (x, v₁, . . . , vm)7→ Unx, U_1,nv₁, . . . Um,nvm

.

(4.9)

Since A is maximally monotone [2, Propositions 20.22 and 20.23], B is monotone and β- Lipschitzian [9, Eq. (3.10)] with domB =K,A+Bis maximally monotone [2, Corollary 24.24(i)].

Now set (∀n ∈ N) ηn = max{η_0,n, η_1,n, . . . , ηm,n}. Then (ηn)n∈N ∈ ℓ¹₊(N). Moreover, we derive from our assumptions on the sequences (U_n)_n∈N and (U_1,n)_n∈N, . . . ,(U_m,n)_n∈N that

µ= sup

n∈NkUnk<+∞ and (1 +ηn)U_n+1<Un∈P_α(K). (4.10)

In addition, [2, Propositions 23.15(ii) and 23.16] yield (∀γ ∈]0,+∞[)(∀n∈N)(∀(x, v1, . . . , vm)∈ K)

JγU_nA(x, v₁, . . . , vm) =

JγUnA(x+γUnz), J_γU

i,nB⁻_i¹(vi−γUi,nri)

1≤i≤m

. (4.11)

It is shown in [9, Eq. (3.12)] and [9, Eq. (3.13)] that under the condition (4.2), zer(A+B)6=∅. Moreover, [9, Eq. (3.21)] and [9, Eq. (3.22)] yield

(x, v₁, . . . , vm)∈zer(A+B)⇒xsolves (4.3) and (v₁, . . . , vm) solves (4.4). (4.12) Let us next set

(∀n∈N)













xn= (xn, v_1,n, . . . , vm,n) y_n= (y_1,n, y_2,1,n, . . . , y_2,m,n) p_n= (p_1,n, p_2,1,n, . . . , p_2,m,n) q_n = (q_1,n, q_2,1,n, . . . , q_2,m,n)

and







an= (a_1,n, a_2,1,n, . . . , a_2,m,n) b_n= (b_1,n, b_2,1,n, . . . , b_2,m,n) c_n = (c_1,n, c_2,1,n, . . . , c_2,m,n).

(4.13)

Then our assumptions imply that X

n∈N

||||an||||<∞, X

n∈N

||||bn||||<∞, and X

n∈N

||||cn||||<∞. (4.14) Furthermore, it follows from the definition ofB, (4.11), and (4.13) that (4.7) can be rewritten in Kas

(∀n∈N)







y_n=x_n−γ_nU_n(Bx_n+a_n) p_n=JγnU_nAy_n+b_n

q_n=p_n−γnUn(Bp_n+cn) x_n+1=x_n−y_n+q_n,

(4.15)

which is (3.2). Moreover, every specific conditions in Theorem 3.1 are satisfied.

(i): By Theorem 3.1(i),P

n∈N||||xn−p_n||||² <∞.

(ii)(a)&(ii)(b): These assertions follow from Theorem 3.1(ii).

(ii)(c): Theorem 3.1(ii) shows that (x, v1, . . . , vm) ∈ zer(A+B). Hence, it follows from [9, Eq (3.19)] that (x, v₁, . . . , vm) satisfies the inclusions

(−Pm

i=1L^∗_ivi−Cx∈ −z+Ax

(∀i∈ {1, . . . , m})Lix−D_i⁻¹vi ∈ri+B_i⁻¹vi. (4.16) For everyn∈Nand every i∈ {1, . . . , m}, set

(ye_1,n =xn−γnUn Cxn+Pm

i=1L^∗_ivi,n

pe_1,n =J_γnUnA(ye_1,n+γ_nU_nz) and

(ey_2,i,n=vi,n+γnUi,n Lixn−D_i⁻¹vi,n

pe_2,i,n =J_γ

nU_i,nB_i⁻¹(ye_2,i,n−γ_nU_i,nr_i).

(4.17)

Then, using [11, Lemma 3.7], we get e

p1,n−p1,n→0 and (∀i∈ {1, . . . , m}) pe2,i,n−p2,i,n→0, (4.18) in turn, by (i),(ii)(a), and (ii)(b), we obtain

ep_1,n−xn→0, pe_1,n⇀ x, and (∀i∈ {1, . . . , m}) pe_2,i,n−vi,n→0, pe_2,i,n ⇀ vi. (4.19) Furthermore, we derive from (4.17) that

(∀n∈N)

(γ_n⁻¹U_n⁻¹(xn−pe1,n)−Pm

i=1L^∗_ivi,n−Cxn∈ −z+Ape1,n

(∀i∈ {1, . . . , m})γ⁻¹_n U_i,n⁻¹(vi,n−pe_2,i,n) +Lixn−D_i⁻¹vi,n∈ri+B_i⁻¹pe_2,i,n. (4.20) Since A is uniformly monotone at x, using (4.16) and (4.20), there exists an increasing function φA: [0,+∞[→[0,+∞] vanishing only at 0 such that, for every n∈N,

φA(kep_1,n−xk)6

* e

p_1,n−x|γ_n⁻¹U_n⁻¹(xn−pe_1,n)− Xm i=1

(L^∗_ivi,n−L^∗_ivi)−(Cxn−Cx)¯ +

=

pe_1,n−x|γ_n⁻¹U_n⁻¹(xn−pe_1,n)

− Xm i=1

hpe_1,n−x|L^∗_ivi,n−L^∗_ivii −χn, (4.21) where we denote ∀n ∈ N

χn = hpe_1,n−x¯|Cxn−Cx¯i. Since (B_i⁻¹)₁≤i≤m are monotone, for everyi∈ {1, . . . , m}, we obtain

(∀n∈N) 06 D

pe_2,i,n−vi |Lixn+γ_n⁻¹U_i,n⁻¹(vi,n−pe_2,i,n)−Lix−(D_i⁻¹vi,n−D_i⁻¹v¯i)E

=D e

p2,i,n−vi |Li(xn−x) +γ_n⁻¹U_i,n⁻¹(vi,n−pe2,i,n)E

−βi,n, (4.22)

where ∀n ∈N

βi,n = e

p_2,i,n−vi |D_i⁻¹vi,n−D⁻¹_i v¯i

. Now, adding (4.22) from i= 1 to i=m and (4.21), we obtain, for everyn∈N,

φA(kep_1,n−xk)≤ e

p_1,n−x|γ_n⁻¹U_n⁻¹(xn−pe_1,n) +

* e

p_1,n−x| Xm i=1

L^∗_i(ep_2,i,n−vi,n) +

+ Xm

i=1

D e

p2,i,n−vi |Li(xn−pe1,n) +γ_n⁻¹U_i,n⁻¹(vi,n−pe2,i,n)E

−χn− Xm i=1

βi,n. (4.23) For everyn∈Nand every i∈ {1, . . . , m}, we expandχnand βi,n as

(χn=hxn−x |Cxn−Cxi+hep_1,n−xn|Cxn−Cxi, β_i,n=

v_i,n−v_i |D_i⁻¹v_i,n−D⁻_i ¹v_i +

pe_2,i,n−v_i,n|D_i⁻¹v_i,n−D_i⁻¹v_i

. (4.24)

By monotonicity ofC and (D_i⁻¹)_1≤i≤m, (∀n∈N)

(hxn−x|Cxn−Cxi ≥0, (∀i∈ {1, . . . , m})

vi,n−vi|D_i⁻¹vi,n−D_i⁻¹vi

≥0. (4.25)

Therefore, for every n∈N, we derive from (4.24) and (4.23) that φA(kep_1,n−xk)≤φA(kep_1,n−xk) +hxn−x|Cxn−Cxi+

Xm i=1

vi,n−vi|D_i⁻¹vi,n−D_i⁻¹vi

≤ e

p_1,n−x|γ_n⁻¹U_n⁻¹(xn−pe_1,n) +

* e

p_1,n−x| Xm i=1

L^∗_i(ep_2,i,n−vi,n) +

+ Xm

i=1

D

pe_2,i,n−v_i |L_i(x_n−pe_1,n) +γ_n⁻¹U_i,n⁻¹(v_i,n−pe_2,i,n)E

− hep_1,n−xn|Cxn−Cxi − Xm i=1

pe_2,i,n−vi,n|D_i⁻¹vi,n−D⁻¹_i vi

. (4.26)

We set

ζ = max

1≤i≤msup

n∈N{kxn−x||,kep_1,n−xk,kvi,n−vik,kep_2,i,n−vik}. (4.27) Then it follows from (ii)(a), (ii)(b), and (4.19) that ζ < ∞, and from [10, Lemma 2.1(ii)] that (∀n ∈ N) kγ_n⁻¹U_n⁻¹k ≤ (εα)⁻¹ and (∀i ∈ {1, . . . , m}) kγ_n⁻¹U_i,n⁻¹k ≤ (εα)⁻¹. Therefore, using the Cauchy-Schwarz inequality, and the Lipschitzianity of C and (D_i⁻¹)_1≤i≤m, we derive from (4.26) that

φA(kep_1,n−xk)≤(εα)⁻¹ζkxn−pe_1,nk+ζ Xm i=1

kLik kxn−ep_1,nk+ (εα)⁻¹kvi,n−ep_2,i,nk +ζ

Xm i=1

kL^∗_ikkep_2,i,n−v_i,nk+ν₀kep_1,n−x_nk+ Xm

i=1

ν_ikep_2,i,n−v_i,nk

→0. (4.28)

We deduce from (4.28) and (4.19) that φ_A(kep_1,n −xk) → 0, which implies that ep_1,n → x. In turn,xn → x and pn →x. Likewise, if C is uniformly monotone at x, there exists an increasing functionφC: [0,+∞[→[0,+∞] that vanishes only at 0 such that

φ_C(kx_n−xk)≤(εα)⁻¹ζkx_n−pe_1,nk+ζ Xm

i=1

kL_ik kx_n−pe_1,nk+ (εα)⁻¹kv_i,n−pe_2,i,nk +ζ

Xm

i=1

kL^∗_ikkep_2,i,n−vi,nk+ν₀kep_1,n−xnk+ Xm i=1

νikep_2,i,n−vi,nk

→0, (4.29)

in turn,xn→x and pn→x.

(ii)(d): Proceeding as in the proof of (ii)(c), we obtain the conclusions.

Acknowledgement. I thank Professor Patrick L. Combettes for bringing this problem to my attention and for helpful discussions.

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