Distributed Mirror Descent Algorithm with Bregman Damping for Nonsmooth Constrained Optimization

Distributed optimization has served as a hot topic in recent years for its broad applications in various fields such as sensor networks and smart grids [1, 2, 3, 4, 5, 6, 7]. Under multi-agent frameworks, the global cost function consists of agents’ local ones, and each agent shares limited information with its neighbors through a network to achieve an optimal solution. Meanwhile, distributed continuous-time algorithms have been well developed thanks to system dynamics and control theory [8, 9, 10, 11, 12].

Up to now, various approaches have been employed for distributed design of constrained optimization. Intuitively, implementing projection operations on local constraints is a most popular method, such as projected proportional-integral protocol [8] and projected dynamics with constraints based on KKT conditions [9]. In addition, other approaches such as distributed primal-dual dynamics and penalty-based algorithms [10, 11, 12] also perform well provided that constraints are endowed with specific expressions. However, time complexity in finding optimal solutions with complex or high-dimensional constraints forces researchers to exploit efficient approaches for special constraint structures, such as the unit simplex and the Euclidean sphere.

In fact, the mirror descent (MD) method serves as a powerful tool for solving constrained optimization. As we know, first introduced in [13], MD is regarded as a generalization of (sub)gradient methods. With mapping the variables into a conjugate space, MD employs the Bregman divergence and performs well in handling local constraints with specific structures [14, 15, 16]. This process results in a faster convergence rate than that of projected (sub)gradient descent algorithms, especially for large-scale optimization problems. Undoubtedly, as such an important tool, MD has played a crucial role in various distributed algorithm designs, as given in [17, 18, 19, 20].

In recent years, continuous-time MD-based algorithms have also attracted much attention. For example, [14] proposed the acceleration of a continuous-time MD algorithm, and afterward, [21] showed continuous-time stochastic MD for strongly convex functions, while [22] proposed a discounted continuous-time MD dynamics to approximate the exact solution. In the distributed design, although [19] presented a distributed MD dynamics with integral feedback, the result merely achieved optimal consensus and part variables turn to be unbounded. Due to the booming development and extensive demand of distributed design, distributed continuous-time MD-based methods need more exploration and development actually.

Therefore, we study continuous-time MD-based algorithms to solve distributed nonsmooth optimization with local and coupled constraints. The main contributions of this note can be summarized as follows. We propose a distributed continuous-time mirror descent algorithm by introducing the Bregman damping, which can be regarded as a generalization of classic distributed projection-based dynamics [8, 23] by taking the Bregman damping in a quadratic form. Moreover, our algorithm well inherits the good capabilities of MD-based approaches to rapidly compute explicit solutions to the problems with some concrete constraint structures like the unit simplex or the Euclidean sphere. With the designed Bregman damping, our MD-based algorithm makes all the variables’ trajectories bounded, which could not be ensured in [14, 19], and avoids the inaccuracy of the convergent point occurred in [22].

The remaining part is organized as follows. Section II gives related preliminary knowledge. Next, Section III formulates the distributed optimization and provides our algorithm, while Section IV presents the main results. Then, Section V provides illustrative numerical examples. Finally, Section VI gives the conclusion.

In this section, we give necessary notations and related preliminary knowledge.

In this paper, we consider a nonsmooth optimization problem with both local and coupled constraints. There are $N$ agents indexed by $\mathcal{V}=\{1,\dots,N\}$ in a network $\mathcal{G}(\mathcal{V},\mathcal{E})$. For agent $i$, the decision variable is $x_{i}$, the local feasible set is $\Omega_{i}\subseteq\mathbb{R}^{n}$, and the local cost function is $f_{i}:\mathbb{R}^{n}\rightarrow\mathbb{R}$. Define $\bm{\Omega}=\prod_{i=1}^{N}\Omega_{i}$ and $\bm{x}=\text{col}\{x_{i}\}_{i=1}^{N}$. All agents cooperate to solve the following distributed optimization:

	$\displaystyle\min_{\bm{x}\in\bm{\Omega}}$	$\displaystyle\;\sum_{i=1}^{N}f_{i}(x_{i})$
	s.t.	$\displaystyle\;\sum_{i=1}^{N}g_{i}(x_{i})\leq 0_{p},\quad\sum_{i=1}^{N}A_{i}x_{i}-b_{i}=0_{q},$		(3)

where $g_{i}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{p}$, $A_{i}\in\mathbb{R}^{q\times n}$ and $b_{i}\in\mathbb{R}^{q}$, for $i\in\mathcal{V}$. Except for the local constraint $\bm{\Omega}$, other constraints in (III) are said to be coupled ones since the solutions rely on global information. In the multi-agent network, agent $i$ only has the local decision variable $x_{i}\in\Omega_{i}$, and moreover, the local information $f_{i}$, $g_{i}$, $A_{i}$ and $b_{i}$. Thus, agents need communication with neighbors through the network $\mathcal{G}$.

Actually, MD replaces the Euclidean regularization in (sub)gradient descent algorithms with Bregman divergence. In return, different generating functions of Bregman divergence may efficiently bring explicit solutions on different special feasible sets. For example, if $\phi(x)=\frac{1}{2}\|x\|^{2}$ and $\Omega$ is convex and closed, then

$\displaystyle\Pi_{\Omega}^{\phi}(z)=\mathop{\text{argmin}}_{x\in\Omega}\frac{1}{2}\|x-z\|^{2}=P_{\Omega}(z),$

(4)

which actually turns into the classical Euclidean regularization with projection operations. Furthermore, if $\Omega=\{x\in\mathbb{R}^{n}_{+}:\sum_{k=1}^{n}x^{k}=1\}$ and $\phi(x)=\sum_{k=1}^{n}x^{k}\log(x^{k})$ with the convention $0\log 0=0$, then

$\displaystyle\Pi_{\Omega}^{\phi}(z)=\text{col}\Big{\{}\frac{\exp(z^{k})}{\sum_{j=1}^{n}\exp(z^{j})}\Big{\}}_{k=1}^{n},$

(5)

which is the well-known KL-divergence on the unit simplex.

Assign a generating function $\phi_{i}:\mathbb{R}^{n}\rightarrow\mathbb{R}$ of the Bregman divergence to each agent $i\in\mathcal{V}$. Then we consider the following assumptions for (III).

For designing a distributed algorithm, we introduce auxiliary variables $\omega_{i}\in\mathbb{R}^{p}$, $\nu_{i}\in\mathbb{R}^{q}$, $\lambda_{i}\in\mathbb{R}^{p}$, $\mu_{i}\in\mathbb{R}^{q}$, $y_{i}\in\mathbb{R}^{n}$ and $\gamma_{i}\in\mathbb{R}^{p}$ for each agent $i\in\mathcal{V}$. Moreover, we employ the gradient $\nabla\phi_{i}$ of generating functions as the Bregman damping in the algorithm, which ensures the trajectories’ boundedness [14, 19] and avoids the convergence inaccuracy [22]. Recall that $a_{ij}$ is the $(i,j)$-th entry of the adjacency matrix $\mathcal{A}$ and $\Pi_{\Omega_{i}}^{\phi_{i}}(\cdot)$ is defined in (1). Then we propose a distributed mirror descent algorithm with Bregman damping (MDBD) for (III).

In Algorithm 1, for each agent $i\in\mathcal{V}$, information like ${\partial}f_{i}$, ${\partial}g_{i}$, $A_{i}$ and $b_{i}$ serves as private knowledge, and values like $\omega_{i}$, $\nu_{i}$, $\lambda_{i}$ and $\mu_{i}$ should be exchanged with neighbors through the network $\mathcal{G}$. Moreover, generating function $\phi_{i}$ and Bregman damping $\nabla\phi_{i}$ can be determined privately and individually, not necessarily identical. It follows from Lemma 2 that the existence of a Caratheodory solution to Algorithm 1 can be guaranteed.

For simplicity, define

$$\bm{\lambda}=\text{col}\big{\{}\lambda_{i}\big{\}}_{i=1}^{N}\in\mathbb{R}^{Np},~{}~{}\bm{\mu}=\text{col}\big{\{}\mu_{i}\big{\}}_{i=1}^{N}\in\mathbb{R}^{Nq},$$

$$\bm{\omega}=\text{col}\big{\{}\omega_{i}\big{\}}_{i=1}^{N}\in\mathbb{R}^{Np},~{}~{}\bm{\nu}=\text{col}\big{\{}\nu_{i}\big{\}}_{i=1}^{N}\in\mathbb{R}^{Nq},$$

$$\bm{y}=\text{col}\big{\{}y_{i}\big{\}}_{i=1}^{N}\in\mathbb{R}^{Nn},~{}~{}\bm{\gamma}=\text{col}\big{\{}\gamma_{i}\big{\}}_{i=1}^{N}\in\mathbb{R}^{Np}.$$

Let $\bm{\Theta}=\bm{\Omega}\times\mathbb{R}^{Np}_{+}\times\mathbb{R}^{N(p+2q)}$, and moreover,

$$\bm{z}=\text{col}\{\bm{x},\bm{\lambda},\bm{\mu},\bm{\omega},\bm{\nu}\},\quad\bm{s}=\text{col}\{\bm{y},\bm{\gamma},\bm{\mu},\bm{\omega},\bm{\nu}\}.$$

Take the Lagrangian function $\mathcal{L}:\bm{\Theta}\rightarrow\mathbb{R}$ as

	$\displaystyle\mathcal{L}(\bm{z})=$	$\displaystyle\sum_{i=1}^{N}f_{i}(x_{i})+\sum_{i=1}^{N}\lambda_{i}^{T}\big{(}g_{i}(x_{i})-\sum_{j=1}^{N}a_{ij}(\omega_{i}-\omega_{j})\big{)}$
		$\displaystyle+\sum_{i=1}^{N}\mu_{i}^{T}\big{(}A_{i}x_{i}-b_{i}-\sum_{j=1}^{N}a_{ij}(\nu_{i}-\nu_{j})\big{)}.$		(6)

For such a distributed convex optimization (III) with a zero dual gap, $\bm{x}^{\star}\in\bm{\Omega}$ is an optimal solution to problem (III) if and only if there exist auxiliary variables $(\bm{\lambda}^{\star},\bm{\mu}^{\star},\bm{\omega}^{\star},\bm{\nu}^{\star})\in\mathbb{R}^{Np}_{+}\times\mathbb{R}^{N(p+2q)}$, such that $\bm{z}^{\star}=\text{col}\{\bm{x}^{\star},\bm{\lambda}^{\star},\bm{\mu}^{\star},\bm{\omega}^{\star},\bm{\nu}^{\star}\}$ is a saddle point of $\mathcal{L}$ [23], that is, for arbitrary $\bm{z}\in\bm{\Theta}$,

	$\displaystyle\mathcal{L}(\bm{x}^{\star},\bm{\lambda},\bm{\mu},\bm{\omega}^{\star},\bm{\nu}^{\star})\leq\mathcal{L}(\bm{x}^{\star},\bm{\lambda}^{\star},$	$\displaystyle\bm{\mu}^{\star},\bm{\omega}^{\star},\bm{\nu}^{\star})$
	$\displaystyle\leq$	$\displaystyle\mathcal{L}(\bm{x},\bm{\lambda}^{\star},\bm{\mu}^{\star},\bm{\omega},\bm{\nu}).$

Define

$\displaystyle F(\bm{z})=\begin{bmatrix}\text{col}\big{\{}{\partial}f_{i}(x_{i})+{\partial}g_{i}(x_{i})^{\rm T}\lambda_{i}+A_{i}^{\rm T}\mu_{i}\big{\}}_{i=1}^{N}\\ \text{col}\big{\{}-g_{i}(x_{i})\big{\}}_{i=1}^{N}+\bm{L}_{p}\bm{w}\\ \text{col}\big{\{}-A_{i}x_{i}+b_{i}\big{\}}_{i=1}^{N}+\bm{L}_{q}\bm{\nu}\\ -\bm{L}_{p}\bm{\lambda}\\ -\bm{L}_{q}\bm{\mu}\end{bmatrix},$

(7)

where $\bm{L}_{p}=L_{N}\otimes I_{p}$ and $\bm{L}_{q}=L_{N}\otimes I_{q}$. In fact, $\bm{z}^{\star}$ is a saddle point of $\mathcal{L}$ if and only if $-F(\bm{z}^{\star})\in\mathcal{N}_{\bm{\Theta}}({\bm{z}}^{\star})$, which was obtained in [27, 23, 10].

Hence, Algorithm 1 can be presented in the following compact form:

$$\left\{\begin{aligned} \dot{\bm{s}}\;{\in}&-F(\bm{z})+\nabla\Phi(\bm{z})-\bm{s},\\ \bm{z}=&\;\Pi_{\bm{\Theta}}^{\Phi}(\bm{s}),\end{aligned}\right.$$

(8)

where

	$\displaystyle\nabla\Phi(\bm{z})\triangleq$	$\displaystyle\text{col}\Big{\{}\text{col}\{\nabla\phi_{i}(x_{i})\}_{i=1}^{N},\text{col}\{\lambda_{i}\}_{i=1}^{N},$
		$\displaystyle\quad\text{col}\{\mu_{i}\}_{i=1}^{N},\text{col}\{\omega_{i}\}_{i=1}^{N},\text{col}\{\nu_{i}\}_{i=1}^{N}\Big{\}},$		(9a)
	$\displaystyle\Pi_{\bm{\Theta}}^{\Phi}(\bm{s})\triangleq$	$\displaystyle\text{col}\Big{\{}\text{col}\{\Pi_{\Omega_{i}}^{\phi_{i}}(y_{i})\}_{i=1}^{N},\text{col}\{[\gamma_{i}]^{+}\}_{i=1}^{N},$
		$\displaystyle\quad\text{col}\{\mu_{i}\}_{i=1}^{N},\text{col}\{\omega_{i}\}_{i=1}^{N},\text{col}\{\nu_{i}\}_{i=1}^{N}\}\Big{\}}.$		(9b)

In this section, we investigate the convergence of MDBD. Though Bregman damping improves the convergence of MDBD, the process also brings challenges for the convergence analysis. The following lemma shows the relationship between MDBD and the saddle points of the Lagrangian function $\mathcal{L}$.

Proof. For $\tilde{\bm{z}}=\Pi_{\bm{\Theta}}^{\Phi}(\bm{s}^{\star})$, the first-order condition is

$\displaystyle-F(\bm{z}^{\star})+\nabla\Phi(\bm{z}^{\star})-\nabla\Phi(\tilde{\bm{z}})\in\mathcal{N}_{\bm{\Theta}}(\tilde{\bm{z}}).$

(11)

We firstly show the sufficiency. Given $\bm{z}^{\star}$, suppose that there exists $\bm{s}^{\star}\in-F(\bm{z}^{\star})+\nabla\Phi(\bm{z}^{\star})$ such that $\bm{z}^{\star}=\Pi_{\bm{\Theta}}^{\Phi}(\bm{s}^{\star})$. Thus, (11) holds with $\tilde{\bm{z}}=\bm{z}^{\star}$, and $-F(\bm{z}^{\star})\in\mathcal{N}_{\bm{\Theta}}({\bm{z}}^{\star})$, which means that $\bm{z}^{\star}$ is a saddle point of $\mathcal{L}$.

Secondly, we show the necessity. Suppose $-F(\bm{z}^{\star})\in\mathcal{N}_{\bm{\Theta}}({\bm{z}}^{\star})$ and take $\bm{s}^{\star}\in-F(\bm{z}^{\star})+\nabla\Phi(\bm{z}^{\star})$. Recall that (11) holds with $\tilde{\bm{z}}=\bm{z}^{\star}$, which implies that $\bm{z}^{\star}$ is a solution to $\Pi_{\bm{\Theta}}^{\Phi}(\bm{s}^{\star})$. Furthermore, since $\phi_{i}(\cdot)$ and $\frac{1}{2}\|\cdot\|^{2}$ are strongly convex, the solution to $\Pi_{\bm{\Theta}}^{\Phi}(\bm{s}^{\star})$ is unique. Therefore, $\bm{z}^{\star}=\Pi_{\bm{\Theta}}^{\Phi}(\bm{s}^{\star})$. $\square$

The following theorem shows the correctness and the convergence of Algorithm 1.

Proof. (i) Firstly, we show that the output $\bm{z}(t)$ is bounded. By Lemma 4, take $\bm{z}^{\star}$ as a saddle point of $\mathcal{L}$ and thus, there exists $\bm{s}^{\star}\in-F(\bm{z}^{\star})+\nabla\Phi(\bm{z}^{\star})$ such that $\bm{z}^{\star}=\Pi_{\bm{\Theta}}^{\Phi}(\bm{s}^{\star})$. Take $\phi^{*}_{i}$ as the convex conjugate of $\phi_{i}$, and construct a Lyapunov candidate function as

	$\displaystyle V_{1}=$	$\displaystyle\sum_{i=1}^{N}D_{\phi_{i}^{*}}(y_{i}-y_{i}^{\star})+\frac{1}{2}\|\bm{\gamma}-\bm{\gamma}^{\star}\|^{2}+\frac{1}{2}\|\bm{\mu}-\bm{\mu}^{\star}\|^{2}$
		$\displaystyle+\frac{1}{2}\|\bm{\omega}-\bm{\omega}^{\star}\|^{2}+\frac{1}{2}\|\bm{\nu}-\bm{\nu}^{\star}\|^{2}.$		(12)

Since $x_{i}=\Pi_{\Omega_{i}}^{\phi_{i}}(y_{i})$, it follows from Lemma 1 that

	$\displaystyle\phi^{*}_{i}(y_{i})=$	$\displaystyle x_{i}^{T}y_{i}-\phi_{i}(x_{i}),$		(13a)
	$\displaystyle\phi^{*}_{i}(y_{i}^{\star})=$	$\displaystyle x_{i}^{\star T}y_{i}^{\star}-\phi_{i}(x_{i}^{\star}).$		(13b)

Thus, by substituting (13), the Bregman divergence becomes

	$\displaystyle D_{\phi_{i}^{*}}(y_{i}-y_{i}^{\star})=$	$\displaystyle\phi^{*}_{i}(y_{i})-\phi^{*}_{i}(y_{i}^{\star})-\nabla\phi^{*}_{i}(y_{i}^{\star})^{T}(y_{i}-y_{i}^{\star})$
	$\displaystyle=$	$\displaystyle\phi_{i}(x_{i}^{\star})-\phi_{i}(x_{i})-(x_{i}^{\star}-x_{i})^{T}y_{i}.$

Since $\phi_{i}(\cdot)$ is strongly convex for $i\in\mathcal{V}$, there exists a positive constant $\sigma$ such that

$\displaystyle\sum_{i=1}^{N}\!D_{\phi_{i}^{*}}\!(y_{i}\!-\!y_{i}^{\star})\!\geq\!\frac{\sigma}{2}\!\|\bm{x}\!-\!\bm{x}^{\star}\!\|^{2}\!+\!\sum_{i=1}^{N}(x_{i}^{\star}\!-\!x_{i})^{T}\!(\nabla\phi_{i}(x_{i})\!-\!y_{i}).$

In fact, $\nabla\phi^{*}_{i}(y_{i})=\mathop{\text{argmin}}_{x\in{\Omega_{i}}}\{-x^{T}y_{i}+\phi_{i}(x)\}$, which leads to

	$\displaystyle 0\leq$	$\displaystyle\big{(}\nabla\phi_{i}(\nabla\phi^{*}_{i}(y_{i}))-y_{i}\big{)}^{T}\big{(}\nabla\phi^{*}_{i}(y_{i}^{\star})-\nabla\phi_{i}^{*}(y_{i})\big{)}$
	$\displaystyle=$	$\displaystyle(\nabla\phi_{i}(x_{i})-y_{i})^{T}(x_{i}^{\star}-x_{i}).$

Thus, $\sum_{i=1}^{N}D_{\phi_{i}^{*}}(y_{i}-y_{i}^{\star})\geq\frac{\sigma}{2}\|\bm{x}-\bm{x}^{\star}\|^{2}$. In addition,

$\displaystyle\|\bm{\lambda}-\bm{\lambda}^{\star}\|^{2}=\|[\bm{\gamma}]^{+}-[\bm{\gamma}^{\star}]^{+}\|^{2}\leq\|\bm{\gamma}-\bm{\gamma}^{\star}\|^{2}.$

Therefore,

	$\displaystyle V_{1}(\bm{s}(t))\geq$	$\displaystyle\frac{\kappa}{2}\Big{(}\|\bm{x}-\bm{x}^{\star}\|^{2}+\|\bm{\lambda}-\bm{\lambda}^{\star}\|^{2}+\|\bm{\mu}-\bm{\mu}^{\star}\|^{2}$
		$\displaystyle+\|\bm{\omega}-\bm{\omega}^{\star}\|^{2}+\|\bm{\nu}-\bm{\nu}^{\star}\|^{2}\Big{)},$		(14)

where $\kappa=\min\{\sigma,1\}$. This means $V_{1}(\bm{s}(t))\geq\frac{\kappa}{2}\|\bm{z}-\bm{z}^{\star}\|^{2}$, that is, $V_{1}$ is radially unbounded in $\bm{z}$. Clearly, the function $V_{1}$ along (20) satisfies

	$\displaystyle\mathcal{L}_{\mathcal{F}}$	$\displaystyle V_{1}=\big{\{}\beta\in\mathbb{R},\beta=\sum_{i=1}^{N}\big{(}\nabla\phi_{i}^{*}(y_{i})-\nabla\phi_{i}^{*}(y_{i}^{\star})\big{)}^{T}\dot{y}_{i}$
	$\displaystyle+$	$\displaystyle(\bm{\gamma}\!-\!\bm{\gamma}^{\star})^{T}\dot{\bm{\gamma}}+(\bm{\mu}\!-\!\bm{\mu}^{\star})^{T}\dot{\bm{\mu}}+(\bm{\omega}\!-\!\bm{\omega}^{\star})^{T}\dot{\bm{\omega}}+(\bm{\nu}\!-\!\bm{\nu}^{\star})^{T}\dot{\bm{\nu}}$
	$\displaystyle=$	$\displaystyle\big{\{}\beta\in\mathbb{R},\beta=\big{(}\bm{z}\!-\!\bm{z}^{\star}\big{)}^{T}\!\big{(}-\bm{\eta}\!+\!\nabla\Phi(\bm{z})\!-\!\bm{s}\big{)},\;\bm{\eta}\in F(\bm{z})\big{\}}.$

Combining the convexity of $f_{i}$ and $g_{i}$ with the property for saddle point $\bm{z}^{\star}$,

	$\displaystyle-(\bm{z}-$	$\displaystyle\bm{z}^{\star}\big{)}^{T}\bm{\eta}$		(15)
	$\displaystyle\leq$	$\displaystyle\mathcal{L}(\bm{x}^{\star},\bm{\lambda},\bm{\mu},\bm{\omega}^{\star},\bm{\nu}^{\star})-\mathcal{L}(\bm{x},\bm{\lambda}^{\star},\bm{\mu}^{\star},\bm{\omega},\bm{\nu})\leq 0.$

Thus,

	$\displaystyle{\beta}$	$\displaystyle\leq(\bm{z}-\bm{z}^{\star})^{T}(\nabla\Phi(\bm{z})-\bm{s})$		(16)
		$\displaystyle=\sum_{i=1}^{N}(x_{i}-x_{i}^{\star})^{T}(\nabla\phi_{i}(x_{i})-y_{i})+(\bm{\lambda}-\bm{\lambda}^{\star})^{T}(\bm{\lambda}-\bm{\gamma}).$

On the one hand, for $i\in\mathcal{P}$, we consider a differentiable function

$\displaystyle J(\alpha)=\phi_{i}\big{(}\alpha x_{i}^{\star}+(1-\alpha)x_{i}\big{)}-\big{(}\alpha x_{i}^{\star}+(1-\alpha)x_{i}\big{)}^{T}y_{i},$

with a constant $\alpha\in[0,1]$. Correspondingly, we have

$\displaystyle J^{\prime}(\alpha)=(x_{i}^{\star}-x_{i})^{T}\Big{(}\nabla\phi_{i}(\alpha x_{i}^{\star}+(1-\alpha)x_{i}\big{)}-y_{i}\Big{)}.$

Recalling $x_{i}=\Pi_{\Omega_{i}}^{\phi_{i}}(y_{i})=\mathop{\text{argmin}}_{x\in\Omega_{i}}\big{\{}-x^{T}y_{i}+\phi_{i}(x)\big{\}}$, $J(0)\leq J(\alpha)$, $\forall\alpha\in[0,1]$, because of the convexity of $\Omega_{i}$. This yields $J^{\prime}(\alpha)\big{|}_{0^{+}}\geq 0$, that is,

$\displaystyle J^{\prime}(\alpha)|_{0^{+}}=(x_{i}^{\star}-x_{i})^{T}(\nabla\phi_{i}(x_{i})-y_{i})\geq 0.$

On the other hand,

$\displaystyle(\bm{\lambda}-\bm{\lambda}^{\star})^{T}(\bm{\lambda}-\bm{\gamma})=([\bm{\gamma}]^{+}-\bm{\lambda}^{\star})^{T}([\bm{\gamma}]^{+}-\bm{\gamma})\leq 0.$

Therefore, ${\beta}\leq 0$, which implies that the output $\bm{z}(t)$ is bounded.

Secondly, we show that $\bm{s}(t)$ is bounded. Actually, it follows from (IV) and the statement above that $\bm{\gamma}(t)$ is bounded. Thereby, we merely need to consider $\bm{y}$. Take another Lyapunov candidate function as

$\displaystyle V_{2}=\frac{1}{2}\|\bm{y}\|^{2},$

(17)

which is radially unbounded in $\bm{y}$. Along the trajectories of Algorithm 1, the derivative of $V_{2}$ satisfies

	$\displaystyle\mathcal{L}_{\mathcal{F}}V_{2}=\big{\{}\zeta\in$	$\displaystyle\mathbb{R}:\zeta\in\sum_{i=1}^{N}y_{i}^{T}\big{(}-\partial f_{i}(x_{i})-\partial g_{i}(x_{i})^{\rm T}\lambda_{i}$
		$\displaystyle-A_{i}^{\rm T}\mu_{i}+\nabla\phi_{i}(x_{i})\big{)}-\|y_{i}\|^{2}\big{\}}.$

It is clear that $\zeta\leq-\|\bm{y}\|^{2}+m\|\bm{y}\|=-2V_{2}+m\sqrt{2V_{2}}$ for a positive constant $m$, since $\bm{x}$, $\bm{\lambda}$ and $\bm{\mu}$ have been proved to be bounded. On this basis, it can be easily verified that $V_{2}$ is bounded, so is $\bm{y}$. Together, $\bm{s}(t)$ is bounded.

For convenience, we define

	$\displaystyle\hat{\bm{x}}\triangleq$	$\displaystyle\frac{1}{t}\int_{0}^{t}\!\bm{x}(\tau)d\tau,\;\hat{\bm{\lambda}}\triangleq\frac{1}{t}\!\int_{0}^{t}\bm{\lambda}(\tau)d\tau,\;\hat{\bm{\mu}}\triangleq\frac{1}{t}\!\int_{0}^{t}\bm{\mu}(\tau)d\tau,$
	$\displaystyle\hat{\bm{\omega}}\triangleq$	$\displaystyle\frac{1}{t}\int_{0}^{t}\bm{\omega}(\tau)d\tau,\;\hat{\bm{\nu}}\triangleq\frac{1}{t}\int_{0}^{t}\bm{\nu}(\tau)d\tau.$

Then we describe the convergence rate of Algorithm 1.