A reflected forward-backward splitting method for monotone inclusions involving Lipschitzian operators

TL;DR

This paper introduces a new splitting method for solving complex monotone inclusions involving Lipschitzian and cocoercive operators, extending existing algorithms and improving step size ranges for better convergence.

Contribution

It proposes a novel reflected forward-backward splitting method for monotone inclusions with Lipschitzian operators, including cases with cocoercivity, and demonstrates its application to composite problems.

Findings

Developed a simple method for zero points of sums of two monotone operators with Lipschitzian components.

Extended the method to improve step size ranges when the Lipschitzian operator is cocoercive.

Applied the new splitting method to composite monotone inclusions with promising results.

Abstract

The proximal extrapolated gradient method \cite{Malitsky18a} is an extension of the projected reflected gradient method \cite{Malitsky15}. Both methods were proposed for solving the classic variational inequalities. In this paper, we investigate the projected reflected gradient method, in the general setting, for solving monotone inclusions involving Lipschitzian operators. As a result, we obtain a simple method for finding a zero point of the sum of two monotone operators where one of them is Lipschizian. We also show that one can improve the range of the stepsize of this method for the case when the Lipschitzian operator is restricted to be cocoercive. A nice combination of this method and the forward-backward splitting was proposed. As a result, we obtain a new splitting method for finding a zero point of the sum of three operators ( maximally monotone + monotone Lipschitzian + cocoercive). Application to composite monotone inclusions are demonstrated.

Equations

$$\text{find x∈H such that}0\in A\overline{x}+B\overline{x}.$$

$$(\forall x\in \mathcal{H})(\forall y\in \mathcal{H})\Vert Bx-By\Vert \le \mu \Vert x-y\Vert ,$$

$$\{\begin{array}{l}{y}_n=x_n-\gamma B{x}_n\\ z_n=(\mathrm{Id}+\gamma A{)}^{-1}{y}_n\\ r_n=z_n-\gamma B{z}_n\\ x_{n+1}=x_n+r_n-y_n.\end{array}$$

$$\gamma \in \left]0,+\infty \right[,x_{n+1}=(\mathrm{Id}+\gamma A{)}^{-1}(x_n-2\gamma B{x}_n+\gamma B{x}_{n-1}).$$

$$\gamma \in \left]0,+\infty \right[,x_{n+1}=(\mathrm{Id}+\gamma N_K{)}^{-1}(x_n-\gamma B(2x_n-x_{n-1})).$$

$$\{\begin{array}{l}{y}_n=2x_n-x_{n-1}\\ \gamma \in \left]0,+\infty \right[\\ x_{n+1}=(\mathrm{Id}+\gamma A{)}^{-1}(x_n-\gamma B{y}_n).\end{array}$$

$$\big{(}\forall u\in Ax\big{)}\big{(}\forall(y,v)\in\operatorname{gra}A\big{)}\quad\left\langle{x-y}\mid{u-v}\right\rangle\geq 0,$$

$$J_A=(\mathrm{Id}+A{)}^{-1},$$

$$(\forall (x,y)\in {\mathcal{H}}^2)⟨x-y\mid Bx-By⟩\ge \beta \Vert Bx-By{\Vert }^2.$$

$$\partial f\colon\mathcal{H}\to 2^{\mathcal{H}}\colon x\mapsto\big{\{}{u\in\mathcal{H}}\;|\;{(\forall y\in\mathcal{H})\quad\left\langle{y-x}\mid{u}\right\rangle+f(x)\leq f(y)}\big{\}}$$

$${\mathrm{prox}}_f=J_{\partial f}:\mathcal{H}\to \mathcal{H}:x\mapsto \underset{y\in \mathcal{H}}{\mathrm{argmin}}f(y)+\frac{1}{2}\Vert x-y{\Vert }^2.$$

$$⟨x_n+\gamma B{y}_{n-1}-x_{n-1}\mid x_{n+1}-y_n⟩=-⟨p_n\mid x_{n+1}-y_n⟩.$$

$$⟨y_n-x_n\mid x_{n+1}-y_n⟩=-⟨p_n+\gamma B{y}_{n-1}\mid x_{n+1}-y_n⟩,$$

$$⟨x_{n+1}-x_n\mid x-x_{n+1}⟩=-⟨p_{n+1}+\gamma B{y}_n\mid x-x_{n+1}⟩.$$

$$\{\begin{array}{l}2⟨y_n-x_n\mid x_{n+1}-y_n⟩=\Vert x_n-x_{n+1}{\Vert }^2-\Vert x_n-y_n{\Vert }^2-\Vert x_{n+1}-y_n{\Vert }^2\\ 2⟨x_{n+1}-x_n\mid x-x_{n+1}⟩=\Vert x_n-x{\Vert }^2-\Vert x_n-x_{n+1}{\Vert }^2-\Vert x_{n+1}-x{\Vert }^2.\end{array}$$

$$\Vert x_{n+1}-x{\Vert }^2+\Vert x_n-y_n{\Vert }^2+\Vert x_{n+1}-y_n{\Vert }^2=\Vert x_n-x{\Vert }^2+2{\Gamma }_n+2\gamma ⟨B{y}_{n-1}-B{y}_n\mid x_{n+1}-y_n⟩,$$

$${\Gamma }_n$$

$$=⟨p_n\mid x_{n+1}-y_n⟩+⟨p_{n+1}\mid x-x_{n+1}⟩+\gamma ⟨B{y}_n\mid x-y_n⟩.$$

$$⟨p_{n+1}\mid x-x_{n+1}⟩$$

$$=\gamma ⟨Bx\mid x_{n+1}-x⟩.$$

$$\gamma ⟨B{y}_n\mid x-y_n⟩\le \gamma ⟨Bx\mid x-y_n⟩-\gamma \beta \Vert B{y}_n-Bx{\Vert }^2.$$

$${\Gamma }_n$$

$$=⟨p_n+\gamma Bx\mid x_{n+1}-x_n⟩-⟨p_n+\gamma Bx\mid x_n-x_{n-1}⟩-\beta \gamma \Vert B{y}_n-Bx{\Vert }^2$$

$$\le ⟨p_{n+1}+\gamma Bx\mid x_{n+1}-x_n⟩-⟨p_n+\gamma Bx\mid x_n-x_{n-1}⟩-\beta \gamma \Vert B{y}_n-Bx{\Vert }^2.$$

$$T_{n+1}+\Vert x_n-y_n{\Vert }^2+\Vert x_{n+1}-y_n{\Vert }^2+2\beta \gamma \Vert B{y}_n-Bx{\Vert }^2\le T_n+2\gamma ⟨B{y}_{n-1}-B{y}_n\mid x_{n+1}-y_n⟩.$$

$$-2⟨p_{n+1}+\gamma Bx\mid x_{n+1}-x_n⟩$$

$$-\Vert p_{n+1}+\gamma Bx+x_{n+1}-x_n{\Vert }^2$$

$$=\Vert x_{n+1}-x_n{\Vert }^2+\Vert p_{n+1}+\gamma Bx{\Vert }^2-{\gamma }^2\Vert B{y}_n-Bx{\Vert }^2.$$

$$\Vert x_{n+1}-x{\Vert }^2+\Vert x_{n+1}-y_n{\Vert }^2+\Vert x_{n+1}-x_n{\Vert }^2+\Vert p_{n+1}+\gamma Bx{\Vert }^2+\gamma (2\beta -\gamma )\Vert B{y}_n-Bx{\Vert }^2$$

$$\le \Vert x_n-x{\Vert }^2+\Vert p_n+\gamma Bx{\Vert }^2-{\gamma }^2\Vert B{y}_{n-1}-Bx{\Vert }^2+2\gamma ⟨B{y}_{n-1}-B{y}_n\mid x_{n+1}-y_n⟩.$$

$$q_n$$

$$\le \frac{2{\gamma }^2}{1-ϵ}\Vert B{y}_{n-1}-Bx{\Vert }^2+\frac{2{\gamma }^2}{1-ϵ}\Vert B{y}_n-Bx{\Vert }^2+(1-ϵ)\Vert x_{n+1}-y_n{\Vert }^2,$$

$$\Vert x_{n+1}-x{\Vert }^2+\Vert x_{n+1}-x_n{\Vert }^2+\Vert p_{n+1}+\gamma Bx{\Vert }^2+{\gamma }^2\frac{1+ϵ}{1-ϵ}\Vert B{y}_n-Bx{\Vert }^2+ϵ\Vert x_{n+1}-y_n{\Vert }^2$$

$$\le \Vert x_n-x{\Vert }^2+\Vert p_n+\gamma Bx{\Vert }^2+{\gamma }^2\frac{1+ϵ}{1-ϵ}\Vert B{y}_{n-1}-Bx{\Vert }^2-\gamma (2\beta -\frac{4\gamma }{1-ϵ})\Vert B{y}_n-Bx{\Vert }^2.$$

$$\{\begin{array}{l}\Vert x_n-x{\Vert }^2+\Vert p_n+\gamma Bx{\Vert }^2+{\gamma }^2\frac{1+ϵ}{1-ϵ}\Vert B{y}_{n-1}-Bx{\Vert }^2\to \overline{\xi }\in \mathbb{R},\\ {\sum }_{n\in \mathbb{N}}\Vert B{y}_n-Bx{\Vert }^2<\infty ,\\ {\sum }_{n\in \mathbb{N}}\Vert x_n-x_{n+1}{\Vert }^2<+\infty ,\\ {\sum }_{n\in \mathbb{N}}\Vert y_n-x_{n+1}{\Vert }^2<+\infty .\end{array}$$

Full text

Keywords: monotone inclusion, monotone operator, operator splitting, cocoercive, forward-backward-forward method, forward-backward algorithm, composite operator, duality, primal-dual algorithm

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

1 Introduction

The forward-backward-forward splitting method (FBFS) or Tseng’s splitting method was firstly appeared in [24]. This method was proposed to find a zero point of the sum of two monotone operators acting on a real Hilbert space $(\mathcal{H},\left\langle{\cdot}\mid{\cdot}\right\rangle)$, namely,

[TABLE]

under the assumption that $A\colon\mathcal{H}\to 2^{\mathcal{H}}$ is a maximally monotone, $B:\mathcal{H}\to\mathcal{H}$ be a monotone and $\mu$-Lipschitzian, i.e.,

[TABLE]

and that such a solution exists. The FBFS method operates according to the routine

[TABLE]

The weak convergence of $(x_{n})_{n\in\mathbb{N}}$ to a solution of (1.1) was proved under the condition $0<\gamma<1/\mu$. Inexact version of the FBF method was investigated in [5]. Then, variable metric version and the stochastic version the FBF method are in [27] and [25], respectively. One of the most important example of $B$ is the case when $B$ is a linear skew operator [5] where monotone plus skew model plays a central role in solving primal-dual monotone inclusions and primal-dual convex optimization problems. The main idea of [5] was then developed rapidly in [8]. Several developments and extensions of [8] are in [7, 3, 10, 27].

The advantage of this framework is its generality and the main disadvantage of (1.3) is that it requires two calls of $B$ per one iteration. This issue was recently resolved in [16]. Specifically, they propose a forward-reflected-backward splitting method (FRBS) for solving (1.1), namely,

[TABLE]

The weak convergence of the iterates generated by (1.4) is proved under the condition $\gamma\in\left]0,1/(2\mu)\right[$. If $B$ is linear and $A$ is the normal cone of some non-empty closed convex set $K$, the FRBS method admits the same structure as the reflected projected gradient methods for variational inequalities [14], namely,

[TABLE]

For any $\mu$-Lipschitzian monotone operator $B$, the weak convergence of the iterates generated by (1.5) is proved under the condition $\gamma\in\left]0,(\sqrt{2}-1)/\mu\right[$. When $N_{K}$ is replaced by a subdifferential of some proper lower semicontinuous convex function $f$, line-search versions (1.5) are proposed in [15].

The objective of this paper is two-folds. We firstly investigate the convergence of (1.5) for (1.1) for any maximally monotone operator $A$, i.e., we propose to investigate the convergence of the following reflected forward-backward splitting method (RFBS) for (1.1):

[TABLE]

Secondly, we investigate the problem of improving the range of the stepsize $\gamma$ for the cases where $B$ is cocoercive operator.

In Section 2, we prove the weak convergence of (1.6) and exploit the cocoercivity of $B$ to improve the range of the stepsize. We propose a combination of (1.6) and the standard forward-backward splitting and prove its convergence in Section 3. The last Section is an application to composite monotone inclusions involving the parallel sums and Lipschitzian monotone operators.

Notations. (See [1]) The scalar products and the associated norms of all Hilbert spaces used in this paper are denoted respectively by $\left\langle{\cdot}\mid{\cdot}\right\rangle$ and $\|\cdot\|$. We denote by $\mathcal{B}(\mathcal{H},\mathcal{G})$ the space of all bounded linear operators from $\mathcal{H}$ to $\mathcal{G}$. The symbols $\rightharpoonup$ and $\to$ denote respectively weak and strong convergence. Let $A\colon\mathcal{H}\to 2^{\mathcal{H}}$ be a set-valued operator. The domain of $A$ is denoted by $\operatorname{dom}(A)$ that is a set of all $x\in\mathcal{H}$ such that $Ax\not={\varnothing}$. The range of $A$ is \operatorname{ran}(A)=\big{\{}{u\in\mathcal{H}}\;|\;{(\exists x\in\mathcal{H})u\in Ax}\big{\}}. The graph of $A$ is \operatorname{gra}(A)=\big{\{}{(x,u)\in\mathcal{H}\times\mathcal{H}}\;|\;{u\in Ax}\big{\}}. The inverse of $A$ is A^{-1}\colon u\mapsto\big{\{}{x}\;|\;{u\in Ax}\big{\}}. The zero set of $A$ is $\operatorname{zer}(A)=A^{-1}0$. We say that $A$ is monotone if

[TABLE]

and it is maximally monotone if there exists no monotone operator $B$ such that $\operatorname{gra}(B)$ properly contains $\operatorname{gra}(A)$. The resolvent of $A$ is

[TABLE]

where $\operatorname{Id}$ denotes the identity operator on $\mathcal{H}$. A single-valued operator $B\colon\mathcal{H}\to\mathcal{H}$ is $\beta$-cocoercive, for some $\beta\in\left]0,+\infty\right[$, if

[TABLE]

The parallel sum of two operator $A\colon\mathcal{H}\to 2^{\mathcal{H}}$ and $B\colon\mathcal{H}\to 2^{\mathcal{H}}$ is A\mbox{\footnotesize\square}B=(A^{-1}+B^{-1})^{-1}. The class of all lower semicontinuous convex functions $f\colon\mathcal{H}\to\left]-\infty,+\infty\right]$ such that \operatorname{dom}f=\big{\{}{x\in\mathcal{H}}\;|\;{f(x)<+\infty}\big{\}}\neq{\varnothing} is denoted by $\Gamma_{0}(\mathcal{H})$. Now, let $f\in\Gamma_{0}(\mathcal{H})$. The subdifferential of $f\in\Gamma_{0}(\mathcal{H})$ is the maximally monotone operator

[TABLE]

Moreover, the proximity operator of $f$ is

[TABLE]

Various closed-form expressions of the proximity operators are in [1, Chapter 24].

2 Weak convergence

In this section, we prove the weak convergence of RFBS in (1.6) under the condition $\gamma\in\left]0,(\sqrt{2}-1)/\mu\right[$. Moreover, we also exploit the cocoercivity of $B$ to improve the range of the stepsize $\gamma$. Throughout this section, $(x_{n})_{n\in\mathbb{N}}$ and $(y_{n})_{n\in\mathbb{N}}$ are generated by (1.6).

Theorem 2.1

The following hold.

1. (i)

Suppose that $B$ is $\beta$-cocoercive and $\gamma\in\left]0,\beta(1-\epsilon)/2\right]$, for some $\epsilon\in\left]0,1\right[$, then $x_{n}\rightharpoonup\overline{x}\in\operatorname{zer}(A+B)$. 2. (ii)

Suppose that $\gamma\in\left]0,(\sqrt{2}-1)/\mu\right[$, then $x_{n}\rightharpoonup\overline{x}\in\operatorname{zer}(A+B)$.

Proof. Let $x\in\operatorname{zer}(A+B)$.

(i): Suppose that $B$ is $\beta$-cocoercive. For any $n\in\mathbb{N}$, set $p_{n+1}=x_{n}-\gamma By_{n}-x_{n+1}$. Then $p_{n+1}\in\gamma Ax_{n+1}$ We have

[TABLE]

Note that $x_{n}-x_{n-1}=y_{n}-x_{n}$, then it follows that

[TABLE]

and

[TABLE]

We have

[TABLE]

In turn,

[TABLE]

where

[TABLE]

Since $\gamma A$ is monotone and $-\gamma Bx\in\gamma Ax$, we obtain

[TABLE]

Since $B$ is $\beta$-cocoercive, we also have

[TABLE]

Adding (2.7) and (2.8), and using the monotonicity of $\gamma A$, we get

[TABLE]

Let us set $T_{n}=\|x_{n+1}-x\|^{2}-2\left\langle{p_{n+1}+\gamma Bx}\mid{x_{n+1}-x_{n}}\right\rangle$, it follows from (2.9) and (2.5) that

[TABLE]

We have $p_{n+1}+x_{n+1}-x_{n}=-\gamma By_{n}$ and hence

[TABLE]

Therefore, (2.10) becomes

[TABLE]

Let us estimate the term $q_{n}=2\gamma\left\langle{By_{n-1}-By_{n}}\mid{x_{n+1}-y_{n}}\right\rangle$. We have

[TABLE]

and thus, we derive from (2.12) that

[TABLE]

Since $\gamma<(1-\epsilon)\beta/2$, we obtain

[TABLE]

Since $p_{n+1}+\gamma Bx=\gamma(Bx-By_{n})+x_{n}-x_{n+1}\to 0$, it follows that $\|x_{n}-x\|^{2}\to\overline{\xi}\in\mathbb{R}$ and hence $(x_{n})_{n\in\mathbb{N}}$ is bounded. Let $\overline{x}$ be a weak cluster point of $(x_{n})_{n\in\mathbb{N}}$, then there exists a subsequence $(x_{k_{n}})_{n\in\mathbb{N}}$ of $(x_{n})_{n\in\mathbb{N}}$ such that $x_{k_{n}}\rightharpoonup\overline{x}$. Note that $By_{k_{n}}\to Bx$ and $x_{n}-y_{n}=x_{n-1}-x_{n}\to 0$. Since $B$ is maximally monotone, its graph is closed in $\mathcal{H}^{strong}\times\mathcal{H}^{weak}$, we obtain $Bx=B\overline{x}$ and thus $By_{k_{n}}\to B\overline{x}$. Since $A$ is maximally monotone, its graph is closed in $\mathcal{H}^{strong}\times\mathcal{H}^{weak}$, passing limit from

[TABLE]

we obtain $\overline{x}\in\operatorname{zer}(A+B)$. Therefore, using Opial’s result [17], we obtain $x_{n}\rightharpoonup\overline{x}\in\operatorname{zer}(A+B)$.

(ii) Let us consider the general case where $B$ is $\mu$-Lipschitzian. Set $\beta=0$ in (2.12), we obtain

[TABLE]

Since $B$ is $\mu$-Lipschitz continuous, we obtain

[TABLE]

Therefore, (2.17) becomes,

[TABLE]

Set

[TABLE]

Then, we can rewrite (2.19) as

[TABLE]

We have

[TABLE]

Then, since $\gamma<(\sqrt{2}-1)/\mu$, we have $\gamma<2/\mu$ and hence there exists $\epsilon>0$ such that

[TABLE]

In turn, we derive from (2.21) that

[TABLE]

Since $(E_{n})_{n\in\mathbb{N}}$ converges, it is bounded and therefore, it follows from (2.23) that $(\|x_{n}-x\|)_{n\in\mathbb{N}}$ and $(x_{n})_{n\in\mathbb{N}}$ are bounded. Hence, $(y_{n})_{n\in\mathbb{N}}$ and $(By_{n}-Bx)_{n\in\mathbb{N}}$ is also bounded. Since $x_{n}-y_{n}\to 0$, we obtain $\left\langle{y_{n}-x_{n}}\mid{By_{n-1}-Bx}\right\rangle\to 0$ We have

[TABLE]

Therefore, $\|x_{n}-x\|\to\overline{\zeta}$. Let $\overline{x}$ be a weak cluster point of $(x_{n})_{n\in\mathbb{N}}$, then there exists $x_{k_{n}}\rightharpoonup\overline{x}$. Note that $x_{k_{n}}-x_{k_{n}+1}-\gamma By_{k_{n}}+\gamma Bx_{k_{n}+1}\to 0$ and

[TABLE]

Since $A+B$ is maximally monotone, its graph is closed in $\mathcal{H}^{strong}\times\mathcal{H}^{weak}$. Therefore, it follows from (2.26) that $\overline{x}\in\operatorname{zer}(A+B)$. Using Opial’s result [17, 18], we obtain $x_{n}\rightharpoonup\overline{x}\in\operatorname{zer}(A+B)$.

Remark 2.2

Here are some remarks.

1. (i)

Special cases of Theorem (ii) are in [14] when $A=N_{S}$, the normal cone operator to a closed convex set $S\subset\mathcal{H}$. The line-search versions for the case where $A=\partial f$ for some $f\in\Gamma_{0}(\mathbb{R}^{d})$ are in [15]. The connection to the existing work concerning with solving variational inequalities can be found in [14, 15]. For the compactness, we do not cited all of them here. 2. (ii)

In the case, $B$ is $\beta$-cocoercive, $\mu=1/\beta$ and the range of the step size is relaxed from $\left]0,(\sqrt{2}-1)/\mu\right[$ to $\left]0,0.5\beta\right[$ which is relatively small in comparison to the standard forward-backward splitting [1]. 3. (iii)

In the case, $B$ is linear, (1.6) is exactly the same as the one in [16] where the convergence is proved under the condition $\gamma\in\left]0,0.5/\mu\right[$. The computational cost of (1.6) and [16] is much cheaper than that of FBFS in [24].

Example 2.3

Let $f$ be in $\Gamma_{0}(\mathcal{H})$, and let $h\colon\mathcal{H}\to\mathbb{R}$ be a convex differentiable function with $\mu$-Lipschitz continuous gradient. The problem is to

[TABLE]

under the assumption that $(\exists x\in\mathcal{H})\;0\in\partial f(x)+h(x)$. Let $(x_{0},x_{-1})\in\mathcal{H}^{2}$ and $\gamma\in\left]0,0.5/\mu\right[$. Iterate

[TABLE]

Then $x_{n}\rightharpoonup x\in\operatorname{zer}(\partial f+\nabla h)$ solves (2.27).

Proof. Set $B=\nabla h$ and $A=\partial f$. Then $B$ is $1/\mu$-cocoercive by Baillon–Haddad’s theorem [1, Corollary 18.17] and $A$ is maximally monotone [1]. Therefore, the result follows from Theorem 2.1 (i).

3 Semi-reflected forward-backward splitting (SRFB)

In this section, we propose a new splitting method that combines the forward-backward splitting and RFB in (1.6) for solving the following inclusion.

Problem 3.1

Let $\mathcal{H}$ be a real Hilbert space, $A\colon\mathcal{H}\to 2^{\mathcal{H}}$ be a maximally monotone, let $B:\mathcal{H}\to\mathcal{H}$ be a monotone and $\mu$-Lipschitzian and $C\colon\mathcal{H}\to\mathcal{H}$ be a $\beta$-cocoercive. Here, $\beta$ and $\mu$ are strictly positive real numbers. The problem is to find $\overline{x}\in\mathcal{H}$ such that

[TABLE]

Throughtout, we assume that such a solution exists.

Recently, splitting methods for the sum of three operators are of great interest in the literature [6, 4, 9, 20, 21, 23, 12, 16]. Problem 3.1 was investigated in [6, 16, 23]. We propose the following method called ”Semi-reflected forward-backward splitting (SRFB)”: Let $(x_{0},x_{-1})\in\mathcal{H}^{2}$, iterate

[TABLE]

Remark 3.2

The iteration (3.2) are different from the ones in [6, 23] and:

1. (i)

If $B$ is linear, (3.2) is the same as the one in [16]. 2. (ii)

If $B=0$, (3.2) reduces to the forward-backward splitting [13, 1]. 3. (iii)

If $C=0$, (3.2) reduces to (1.6).

Theorem 3.3

Let $\zeta\in\left]0,1/2\right[$ and $\xi\in\left]0,+\infty\right[$, let $\gamma>0$ be such that

[TABLE]

Then $x_{n}\rightharpoonup\overline{x}\in\operatorname{zer}(A+B+C)$.

Proof. Let $x\in\operatorname{zer}(A+B+C)$. Let us set $e_{n+1}=x_{n}-\gamma By_{n}-x_{n+1}-\gamma Cx_{n}$. By definition of $J_{\gamma A}$, we have

[TABLE]

Therefore, using $y_{n}-x_{n}=x_{n}-x_{n-1}$, we have

[TABLE]

which implies that

[TABLE]

We have

[TABLE]

Combining (3.6) and (3.7), we obtain

[TABLE]

where we set

[TABLE]

Let us estimate $\Gamma_{1,n}$. We have

[TABLE]

Since $A$ is monotone and $-\gamma Bx-\gamma Cx\in\gamma Ax$, and $e_{n+1}\in\gamma Ax_{n+1}$, we obtain

[TABLE]

Since $B$ is monotone, we obtain

[TABLE]

Adding (3.11) and (3.12), we get

[TABLE]

where we set

[TABLE]

Let us estimate $\Gamma_{2,n}$, we have

[TABLE]

We also have

[TABLE]

which implies that

[TABLE]

Since $A$ is monotone and $e_{n+1}=p_{n+1}-\gamma Cx_{n}\in\gamma Ax_{n+1}$, we have

[TABLE]

Set $s_{n}=\left\langle{p_{n}+\gamma Bx}\mid{x_{n}-x_{n-1}}\right\rangle$. Adding (3.17) and (3.18), we obtain

[TABLE]

Using Cauchy-Schwatz’s inequality, we obtain, for any $\xi>0$,

[TABLE]

which implies that

[TABLE]

By the definition of $s_{n+1}$, for any $\zeta\in\left]0,1\right[$,

[TABLE]

where $t_{n+1}=2(1-\zeta)\|x_{n}-x_{n+1}\|^{2}+2\gamma\left\langle{By_{n}-Bx}\mid{x_{n+1}-x_{n}}\right\rangle$. Therefore,

[TABLE]

We also have

[TABLE]

Let us set

[TABLE]

Then we derive from (3.23), (3.24) and (3.8) that

[TABLE]

We next have by the definition of $t_{n+1}$,

[TABLE]

Therefore, by the definition of $\alpha_{n+1}$, under the condition $\gamma\mu<1-\zeta$, we get

[TABLE]

Thus, under the condition of $\gamma$, We obtain, for any $x\in\operatorname{zer}(A+B+C)$,

[TABLE]

Since $(\alpha_{n})_{n\in\mathbb{N}}$ converges, it is bounded. By (3.28), $(x_{n})_{n\in\mathbb{N}}$ is also bounded, so is $(y_{n}-x)_{n\in\mathbb{N}}$. Since $B$ is Lipschitz, we get $(By_{n}-Bx)_{n\in\mathbb{N}}$ is bounded and thus $t_{n}\to 0.$ Since $y_{n}-x_{n}=x_{n}-x_{n-1}$, we obtain

[TABLE]

Let $x_{n_{k}}\rightharpoonup x^{*}.$ We have

[TABLE]

Since $A+B+C$ is maximally monotone, it graph is closed in $\mathcal{H}^{weak}\times\mathcal{H}^{strong}$, it follows from (3.32) that hence $0\in(A+B+C)x^{*}$. By the Optial’s result, we obtain $x_{n}\rightharpoonup\overline{x}$.

4 Composite monotone inclusions

In this section, we focus on the following structured primal-dual monotone inclusions [8] which cover a wide class of convex optimization problem [19, 2, 23, 27, 10, 25].

Problem 4.1

Let $B\colon\mathcal{H}\to\mathcal{H}$ be a monotone and $\mu_{0}$-Lipschitzian, $\mu_{0}\in\left]-\infty,+\infty\right[$, and $A\colon\mathcal{H}\to 2^{\mathcal{H}}$ be maximally monotone. Let $m$ be a strictly positive integer and let $(\mathcal{G}_{i})_{1\leq i\leq m}$ be real Hilbert spaces. For every $i\in\{1,\ldots,m\}$, let $A_{i}\colon\mathcal{G}_{i}\to 2^{\mathcal{G}_{i}}$ be a maximally monotone, and let $B_{i}\colon\mathcal{G}_{i}\to 2^{\mathcal{G}_{i}}$ be a maximally monotone such that $B_{i}^{-1}$ is $\mu_{i}$-Lipschitzian operator for some $\mu_{i}\in\left]-\infty,+\infty\right[$, let $L_{i}\colon\mathcal{H}\to\mathcal{G}_{i}$ be a bounded linear operator such that $0\not=\sum_{i=1}^{m}\|L_{i}\|^{2}$. Suppose that

[TABLE]

The primal inclusion is to find $\overline{x}$ such that

[TABLE]

and the dual inclusion is to find $(\overline{v}_{i})_{1\leq i\leq m}\in(\mathcal{G}_{i})_{1\leq i\leq m}$ such that

[TABLE]

Corollary 4.2

Set

[TABLE]

Let $\gamma\in\left]0,(\sqrt{2}-1)/\mu\right[$, $(x_{0},x_{-1})\in\mathcal{H}^{2}$ and, for every $i\in\{1,\ldots,m\}$, let $(v_{i,0},v_{i,-1})\in\mathcal{G}_{i}^{2}$. Iterate, for every $n\in\mathbb{N}$,

[TABLE]

Then $x_{n}\rightharpoonup\overline{x}$ solves (4.2) and $(v_{1,n},\ldots,v_{m,n})\rightharpoonup(\overline{v}_{1},\ldots,\overline{v}_{m})$ solves (4.3).

Proof. We use the technique in [8]. Let $\mathcal{K}=\mathcal{H}\oplus\mathcal{G}_{1}\oplus\ldots\oplus\mathcal{G}_{m}$ be the Hilbert direct sum of the Hilbert spaces $\mathcal{H}$ and $(\mathcal{G}_{i})_{1\leq i\leq m}$, where the scalar product and the the associated norm of $\mathcal{G}$ are respectively defined as

[TABLE]

and

[TABLE]

Let us define

[TABLE]

It is shown in [8, Eq. (3.12)] and [8, Eq. (3.13)] that under the condition (4.1), $\operatorname{zer}(\boldsymbol{A}+\boldsymbol{B})\not={\varnothing}$. Furthermore, [8, Eq. (3.21)] and [8, Eq. (3.22)] yield

[TABLE]

It is show in [8] that $\boldsymbol{B}$ is monotone and $\mu$-Lipschitzian and sing [1, Proposition 20.23] and [1, Proposition 20.22], $\boldsymbol{A}$ is also a maximally monotone operator. Furthermore, it follows from [1, Proposition 23.18] that

[TABLE]

For every $n\in\mathbb{N}$, set $\boldsymbol{x}_{n}=(x_{n},v_{1,n},\ldots,v_{m,n}).$ Then the propose algorithm can be rewritten in the space $\mathcal{K}$ as follows

[TABLE]

In view of Theorem 2.1(ii), $(\boldsymbol{x}_{n})_{n\in\mathbb{N}}$ converges weakly to $\overline{\boldsymbol{x}}=(\overline{x},\overline{v}_{1},\ldots,\overline{v}_{m})$ in $\operatorname{zer}(\boldsymbol{A}+\boldsymbol{B})$. By (4.9), it follows that $x_{n}\rightharpoonup\overline{x}$ solves (4.2) and $(v_{1,n},\ldots,v_{m,n})\rightharpoonup(\overline{v}_{1},\ldots,\overline{v}_{m})$ solves (4.3).

Remark 4.3

Here are some remarks:

1. (i)

The iteration (4.5) is different from the one in [8] and (4.5) requires only one call of $B,(B_{i})_{1\leq i\leq m},(L_{i})_{1\leq i\leq m}$ per itearation. 2. (ii)

When $B,(B_{i})_{1\leq i\leq m}$ are restricted to be cocoercive, (4.5) is different from the one in [26]. 3. (iii)

Using the same idea as in [8], concretes applications to minimization problem involving the parallel sums are straightforward and we omit them here.

Acknowledgments. The work of B. Cong Vu and Volkan Cevher were supported by European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement no 725594 - time-data).