Primal-Dual Algorithm for Distributed Optimization with Coupled Constraints

Abstract

This paper focuses on distributed consensus optimization problems with coupled constraints over time-varying multi-agent networks, where the global objective is the finite sum of all agents’ private local objective functions, and decision variables of agents are subject to coupled equality and inequality constraints and a compact convex subset. Each agent exchanges information with its neighbors and processes local data. They cooperate to agree on a consensual decision vector that is an optimal solution to the considered optimization problems. We integrate ideas behind dynamic average consensus and primal-dual methods to develop a distributed algorithm and establish its sublinear convergence rate. In numerical simulations, to illustrate the effectiveness of the proposed algorithm, we compare it with some related methods by the Neyman–Pearson classification problem.

Appendices

Proof of Lemma 4.1

For simplicity, denote $\delta _k=V_{k+1}^{-1}\left( \nabla \textrm{L}(\omega ^{k+1})-\nabla \textrm{L}(\omega ^k)\right)$, (4.5) can be equivalently written as $s^{k+1}=\mathcal {R}_ks^k+\delta _k$. Furthermore, $s^{k+1}=\varPhi _\mathcal {R}(k,1)s^1+\sum _{l=1}^{k-1}\varPhi _\mathcal {R}(k,l+1)\delta _l+\delta _k$. Since $\{\frac{1}{2m}v_k\}$ is an absolute probability sequence for the matrix sequence $\{\mathcal {R}_k\}$, it follows that

$$\begin{aligned} 1_{2m}\bar{s}^{k+1}=&\frac{1}{2m}1_{2m}v_{k+1}^\top s^{k+1}\\ =&\frac{1}{2m}1_{2m}v_{k+1}^\top \left( \varPhi _\mathcal {R}(k,1)s^1+\delta _k\right) +\frac{1}{2m}1_{2m}v_{k+1}^\top \sum _{l=1}^{k-1}\varPhi _\mathcal {R}(k,l+1)\delta _l\\ =&\frac{1}{2m}1_{2m}v_1^\top s^1+\frac{1}{2m}\sum _{l=1}^{k-1}1_{2m}v_l^\top \delta _l +\frac{1}{2m}1_{2m}v_{k+1}^\top \delta _k. \end{aligned}$$

Applying the subadditivity of $\Vert \cdot \Vert$ yields

$$\begin{aligned} \begin{aligned} \Vert s^{k+1}-1_{2m}\bar{s}^{k+1}\Vert \le&\left\| \left( \varPhi _\mathcal {R}(k,1)-\frac{1}{2m}1_{2m}v_1^\top \right) s^1\right\| +\left\| \left( I_{2m}-\frac{1}{2m}1_{2m}v_{k+1}^\top \right) \delta _k\right\| \\&\quad +\sum _{l=1}^{k-1}\left\| \left( \varPhi _\mathcal {R}(k,l+1)-\frac{1}{2m}1_{2m}v_l^\top \right) \delta _l\right\| . \end{aligned} \end{aligned}$$

(A.1)

For the matrix sequence $\{\varPhi _\mathcal {R}(k,s)\}$, it holds the similar results in Lemma C.3, i.e.,

$$\begin{aligned} \left\| \varPhi _\mathcal {R}(k,s)-\frac{1}{2m}1_{2m}v_s^\top \right\| \le C_1\rho ^{k-s}, \quad \forall \, k\ge s, \end{aligned}$$

where $\rho \in (0,1)$ and $C_1$ is some positive constant. Due to the compactness of the subset $\mathcal {X}$, it is clear that $\delta _k$ is bounded. Denote the upper bound of $\Vert \delta _k\Vert$ by $\widetilde{C}$. Therefore,

$$\begin{aligned} \Vert s^{k+1}-1_{2m}\bar{s}^{k+1}\Vert&\le \rho ^{k-1}\Vert s^1\Vert +\widetilde{C}\sum _{l=1}^{k-1}\rho ^{k-1-l}+\widetilde{C}\\&\le \Vert s^1\Vert +\frac{\widetilde{C}}{1-\rho }+\widetilde{C}=M,\quad \forall \, k\ge 1. \end{aligned}$$

Proof of Lemma 4.2

Firstly, we need to verify the following inequality,

$$\begin{aligned} \left\| \omega ^{k+1}-1_{2m}\bar{\omega }^{k+1}\right\| \le C_2\beta ^{k-1}+C_3\sum _{l=1}^{k-1}\beta ^{k-1-l}\alpha _l+C_4\alpha _k. \end{aligned}$$

Then, combining the conditions on the step-size sequence $\{\alpha _k\}$ in Assumption 3.1 and the results of Lemma C.4, it is easy to demonstrate the claimed results.

(a): From (4.4), there exist $e^k$ such that $\omega ^{k+1}=\mathcal {A}_k\omega ^k-\alpha _kV_ks^k+e^k$. Similar to the inequality (A.1), it holds that

$$\begin{aligned} \begin{aligned}&\left\| \omega ^{k+1}-1_{2m}\bar{\omega }^{k+1}\right\| \\&\quad \le \left\| \left( \varPhi _\mathcal {A}(k,1)-1_{2m}\phi _1^\top \right) \omega ^1\right\| +\left\| \left( I_{2m}-1_{2m}\phi _k^\top \right) \left( e^k-\alpha _kV_ks^k\right) \right\| \\&\qquad +\sum _{l=1}^{k-1}\left\| \left( \varPhi _\mathcal {A}(k,l+1)-1_{2m}\phi _l^\top \right) \left( e^l-\alpha _lV_ls^l\right) \right\| . \end{aligned} \end{aligned}$$

(B.1)

Since $\omega ^{k+1}$ is the projection of $\mathcal {A}_k\omega ^k-\alpha _kV_ks^k$ onto $\varOmega$, for any $\omega \in \varOmega$, one has

$$\begin{aligned} \left\langle \mathcal {A}_k\omega ^k-\alpha _kV_ks^k-\omega ^{k+1},\omega -\omega ^{k+1}\right\rangle \le 0, \end{aligned}$$

which implies that

$$\begin{aligned}&\Vert \omega ^{k+1}-\mathcal {A}_k\omega ^k\Vert ^2 =\Vert e^k-\alpha _kV_ks^k\Vert ^2\\&\quad =\Vert e^k\Vert ^2-2\left\langle \alpha _kV_ks^k,e^k\right\rangle +\Vert \alpha _kV_ks^k\Vert ^2\\&\quad =\Vert e^k\Vert ^2+2\left\langle \omega ^{k+1}-\mathcal {A}_k\omega ^k-e^k,e^k\right\rangle +\Vert \alpha _kV_ks^k\Vert ^2\\&\quad \le \alpha _k^2\Vert V_ks^k\Vert ^2-\Vert e^k\Vert ^2. \end{aligned}$$

From Lemma 4.1, it follows that

$$\begin{aligned} \Vert e^k\Vert \le \alpha _k\Vert V_ks^k\Vert \le \alpha _k\Vert s^k\Vert \le M\alpha _k, \quad \forall \, k\ge 0. \end{aligned}$$

In view of the results in Lemmas C.3 and C.4, one can get a modified version of the inequality (B.1) as follows,

$$\begin{aligned} \Vert \omega ^{k+1}-1_{2m}\bar{\omega }^{k+1}\Vert \le C\Vert \omega ^1\Vert \beta ^{k-1}+2M\alpha _k+ 2M\sum _{l=1}^{k-1}\beta ^{k-1-l}\alpha _l. \end{aligned}$$

(B.2)

Under Assumption 3.1 and Lemma C.4 (a), it’s clear that

$$\begin{aligned} \lim _{k\rightarrow \infty }\Vert \omega ^k-1_{2m}\bar{\omega }^k\Vert =0. \end{aligned}$$

(b): Multiplying the inequality (B.2) by $\alpha _{k+1}$ at the both sides yields

$$\begin{aligned}&\alpha _{k+1}\Vert \omega ^{k+1}-1_{2m}\bar{\omega }^{k+1}\Vert \\&\quad \le C_2\beta ^{k-1}\alpha _{k+1}+2C_3\sum _{l=1}^{k-1} \beta ^{k-1-l}\alpha _l\alpha _{k+1}+C_4\alpha _k\alpha _{k+1}\\&\quad \le C_2\left( \beta ^{2(k-1)}+\alpha _{k+1}^2\right) + C_3\sum _{l=1}^{k-1}\beta ^{k-1-l}\left( \alpha _l^2+\alpha _{k+1}^2\right) +C_4\left( \alpha _k^2+\alpha _{k+1}^2\right) \\&\quad \le C_1\beta ^{2(k-1)}+C_2\sum _{l=1}^{k-1}\beta ^{k-1-l}\alpha _l^2+C_4\alpha _k^2+C_5\alpha _{k+1}^2, \end{aligned}$$

where $C_2, C_3, C_4, C_5$ are some positive constants. Once again, under Assumption 3.1 and Lemma C.4 (b), it is concluded that

$$\begin{aligned} \sum _{k=0}^\infty \alpha _k\Vert \omega ^k-1_{2m}\bar{\omega }^k\Vert <\infty . \end{aligned}$$

In Lemma 4.2, the claimed results about the sequence $\{s^k\}$ is parallel with $\{\omega ^k\}$ and its proof is omitted here.

Supporting Lemmas

Lemma C.1

([29], Lemma 4) Under Assumptions 2.1 and 2.2, it holds that

• (a) $\lim _{k\rightarrow \infty }\varPhi _A(k,s)=1_{2m}\mu _s^\top$, where $\mu _s\in \mathbb {R}^{2m}$ is a stochastic vector for each s.

• (b) For any i, the entries $\varPhi _\mathcal {A}(k,s)_{ij}, j=1,\dots ,2m$, converge to the same limit $(\mu _s)_i$ as $k\rightarrow \infty$ with a geometric rate, i.e., for each i and $s\ge 0$,

  $$\begin{aligned} \left| \varPhi _\mathcal {A}(k,s)_{ij}-\left( \mu _s\right) _i\right| \le C\beta ^{k-s},\ \forall \, k\ge s, \end{aligned}$$

  where $C=2\frac{1+2\eta ^{-B_0}}{1-2\eta ^{B_0}}\left( 1-\eta \right) ^{1/{B_0}}$, $\beta =(1-\eta ^{B_0})\in (0,1)$, $\eta$ is the lower bound of Assumption 2.2, $B_0=(m-1)B$, ingeter B is defined by Assumption 2.1.

Lemma C.2

([39], Corollary 1) Under the assumptions of Lemma C.1, the sequence $\{\phi _k\}$ is an absolute probability sequence for the matrix sequence $\{\mathcal {A}_k\}$, where

$$\begin{aligned} \begin{aligned} \phi _k^\top&=\mu _s^\top ,\quad k=sB,\\ \phi _k^\top&=\mu _{s+1}^\top \mathcal {A}_{(s+1)B-1}\cdots \mathcal {A}_k,\quad k\in \left( sB,(s+1)B\right) ,\\ \text {for} \ \ s&=0,1,2,\cdots . \end{aligned} \end{aligned}$$

(C.1)

Lemma C.3

([34], Lemma 11, Chapter 2.2) Let $\{b_k\}, \{c_k\}, \{d_k\}$ be non-negative sequences. Suppose that $\sum _{k=0}^\infty c_k<\infty$ and

$$\begin{aligned} b_{k+1}\le b_k-d_k+c_k,\quad \forall \, k\ge 1, \end{aligned}$$

then the sequence $\{b_k\}$ converges and $\sum _{k=0}^\infty d_k<\infty$.

Lemma C.4

([30], Lemma 7) Let $\beta \in (0,1)$, and $\{\gamma _k\}$ be a positive scalar sequence.

• (a) If $\lim _{k\rightarrow \infty }\gamma _k=0$, then $\lim _{k\rightarrow \infty }\sum _{l=0}^k\beta ^{k-l}\gamma _l=0$.

• (b) Furthermore, if $\sum _{k=0}^\infty \gamma _k<\infty$, then $\sum _{k=0}^\infty \sum _{l=0}^k\beta ^{k-l}\gamma _l<\infty$.