Fully distributed ADMM design for constrained nonconvex optimization

Abstract

This paper investigates a distributed nonconvex optimization problem constrained by a global coupled linear equality and nonidentical local feasible sets. To solve this problem, we propose a fully distributed alternating direction method of multiplier (ADMM) with a distributed dynamic average consensus for dual variables. Moreover, we demonstrate that the proposed algorithm converges to an approximately stationary solution of the considered problem using Lyapunov theory. Finally, we illustrate the effectiveness of the proposed algorithm by two simulation examples.

Keywords

Distributed optimization nonconvex ADMM constraint Lyapunov function

Introduction

In recent years, optimization over a network system has emerged as a significant research field (Bi et al., 2025; Luo and Liu, 2025; Wen et al., 2022; Yang et al., 2019). As the data size of optimization problems increases, conventional centralized algorithms find it challenging to handle large-scale optimization due to the limited computation resources of a single machine. Since the distributed optimization method promotes collaboration by enabling nodes or agents to tackle small-scale problems (Hu et al., 2024; Yu et al., 2023), it is especially well suited for systems with large volumes of data and intricate structures of networks. Distributed methods also have more robustness and privacy compared with centralized methods. Distributed optimization methods have been applied to solve a diverse range of contemporary problems, including power distribution control within communication networks (Cong et al., 2025; Yi et al., 2016), unmanned aerial vehicle formation (Fu et al., 2020; Tian et al., 2024), machine learning (Afia et al., 2024; Wang and Li, 2019), multi-agent robotic networks (Xin et al., 2023; Zhi et al., 2023), and image processing (Yuan et al., 2018; Zhang et al., 2015).

In many circumstances, the decision variables of the optimization problem are constrained by some local or global feasible sets (Cheng et al., 2023; Han et al., 2024; Hou et al., 2022; Jiang et al., 2022). Therefore, it is of great significance to study distributed optimization problems with coupled equality constraints. For distributed optimization with coupled equality constraints, Liang et al. (2017) introduced a continuous-time algorithm based on gradient projection and average tracking, and theoretically proved the convergence of the algorithm. For distributed optimization with feasible set constraints and coupled equality constraints, Chang (2016) proposed an alternating direction method of multiplier (ADMM)-based proximal dual consensus algorithm and obtained the optimal solution with an $O (1 ∕ T)$ convergence rate. Aybat et al. (2017) designed a distributed proximal gradient algorithm to minimize the sum of local functions and a composite function with $O (1 ∕ T)$ and $O (1 ∕ \sqrt{T})$ traversal convergence on suboptimality error and consistency violation, respectively. Subsequently, using stochastic gradient methods with variance reduction techniques, Bai et al. (2022) proposed an inexact accelerated stochastic alternating direction multiplier and its variants, achieving a convergence rate of $O (1 ∕ T)$ . It is also worth mentioning that there are many excellent works on solving distributed optimization with coupled constraints, as seen in Falsone et al. (2020), Zeng et al. (2018), Zhang et al. (2025) and Wang and Liu (2025), to name a few. However, the above methods (Aybat et al., 2017; Bai et al., 2022; Chang, 2016; Falsone et al., 2020; Liang et al., 2017; Zeng et al., 2018) only designed for solving convex optimization problems.

In engineering and academia, nonconvex optimization problems are more general and practical than convex counterparts. Nonconvexity may arise from the nature of the objective function or penalties imposed by operational constraints. For nonconvex optimization problems satisfying either the Polyak-Łojasiewicz (PŁ) condition or weak convexity properties, global convergence to stationary solutions can be guaranteed under standard regularity assumptions. For centralized nonconvex optimization with coupled constraints, Chatzipanagiotis and Zavlanos (2017) established the local convergence of the method by assuming the existence of stationary points that satisfy the strong second-order optimality condition for the Lagrangian dual problem. Yang et al. (2022) adjusted the update of dual variables by introducing a discount factor $τ \in (0, 1]$ , and their approach updated the dual variables by applying a discounted restriction on the residuals during the iterative process, which effectively maintained the lower bound property of the Lyapunov function while controlling the dual variables and ultimately proved the convergence of the method. Houska et al. (2016) introduced an augmented Lagrangian method for nonconvex problems with coupled equality constraints and proved that the algorithm converged to a local optimum. In particular, the convergence of the nonconvex ADMM algorithm has been analyzed by Hong et al. (2016) and Wang et al. (2019), which developed a general framework demonstrating that the Gauss-Seidel ADMM converges to local optima or stationary points for nonconvex objective functions, with the framework comprising two main steps: (1) identifying a sufficiently decreasing Lyapunov function; (2) proving the lower bound property of the Lyapunov function. However, the distributed method remains absent for nonconvex optimization with coupled constraints. Due to the lack of effective methods for simultaneously addressing coupled constraints and nonconvex objectives, developing a distributed algorithm with guaranteed convergence is a challenging task. The results and the main process of solving the optimization problem with coupled constraints are summarized in Table 1.

Table 1.

Optimization with coupled constraints

Related works	Objective functions	Methods	Distributed	Convergence
Liang et al. (2017)	Convex	Gradient Projection	✓	Global convergence
		+Average Tracking		Optimal solution
Falsone et al. (2020)	Convex	Tracking-ADMM	✓	Global convergence Optimal solution
Chatzipanagiotis and Zavlanos (2017)	Nonconvex	ADAL	✗	Local convergence
				Local optima
Yang et al. (2022)	Nonconvex	Proximal-ADMM	✗	Global convergence Approximate stationary
				points
Houska et al. (2016)	Nonconvex	ALADIN	✗	Global convergence Local optima
Sun and Sun (2023)	Nonconvex	ALM +ADMM	✗	Global convergence
				Stationary points
This work	Nonconvex	Fully distributed ADMM	✓	Global convergence Approximate stationary
				points

Most existing publications with coupled constraints focus on distributed convex optimization or centralized nonconvex optimization. Building on existing research, we consider a nonconvex optimization problem with feasible set constraints and globally coupled equality constraints. We propose a fully distributed ADMM algorithm. There are three principal facets within our work, which constitute the core of our contributions:

Compared with the works on convex optimization problems in Cheng et al. (2023), Liu et al. (2025), Liang et al. (2017), Chang (2016), Falsone et al. (2020) and Huang et al. (2024) and unconstrained optimization problems in Pu and Nedić (2021), Lei and Chen (2019) and Yi et al. (2022), we designed a distributed algorithm for solving nonconvex optimization problems, which is more challenging in algorithm design and analysis because of the nonconvexity and coupled equality constraints.

Compared with the centralized algorithms proposed in Chatzipanagiotis and Zavlanos (2017), Yang et al. (2022) and Wang et al. (2019) to solve nonconvex optimization problems with coupled constraints, we develop a fully distributed ADMM algorithm, which overcomes the challenge of algorithm centralization when dealing with nonconvex optimization problems.

We design a distributed ADMM by integrating the classic ADMM with the dynamic average consensus mechanism to handle nonconvex objective functions and coupled constraints. Moreover, we demonstrate the global convergence of the algorithm to approximate stationary points, yielding the same results as the centralized algorithm (Sun and Sun, 2023; Wang et al., 2019; Yang et al., 2022).

The remainder is organized as follows. The “Preliminaries” section presents some notations and preliminaries of graph theory. The “Problem formulation and algorithm” section formulates the considered optimization problem and provides our algorithm, and the “Convergence of fully distributed ADMM” section analyzes the convergence of the proposed algorithm. The “Numerical experiments” section presents the simulation results, and the “Conclusion” section ends this paper. Last, the proofs of Propositions 1, 2, and 3 are given in “Appendix”.

Preliminaries

Notations

In this paper, let $ℝ^{m}$ and $ℝ^{m \times n}$ be the m-dimensional real vector space and the $m \times n$ matrices collection of real-valued space, respectively. Let $I_{m}$ or I be the identity matrix of order m or suitable size and $1_{m} = {[1, \dots, 1]}^{⊤} \in ℝ^{m}$ . Denote ⊗ as the Kronecker product. For a matrix $A \in ℝ^{m \times m}$ , $A^{⊤}$ denotes its transpose, $A ≻ 0$ denotes that A is positive definite, and $∥ A ∥$ denotes its spectral norm. We write $ζ = {[ζ_{1}^{⊤}, ζ_{2}^{⊤}, \dots, ζ_{N}^{⊤}]}^{⊤} \in ℝ^{m}$ to refer to the stack of sub-vector $ζ_{i} \in ℝ^{m_{i}}$ with $\sum_{i = 1}^{N} m_{i} = m$ , and $∥ ζ ∥_{A}^{2} = ζ^{⊤} Aζ$ is the weighted norm of ζ. Besides, we denote $⟨ ζ_{1}, ζ_{2} ⟩$ as the inner product of vectors $ζ_{1}, ζ_{2} \in ℝ^{m}$ and use $blkdiag (A_{1}, A_{2}, \dots, A_{N})$ as the diagonal matrix formed by the sub-matrices $A_{1}, A_{2}, \dots, A_{N}$ . We define $N_{X} (ζ^{*}) = {ν \in ℝ^{m} | ⟨ ν, ζ - ζ^{*} ⟩ \leq 0, \forall ζ \in X}$ as the normal cone to a convex set $X \subseteq ℝ^{m}$ at $ζ^{*}$ . Let $dist (ζ, X) = \min_{ζ_{1} \in X} ∥ ζ - ζ_{1} ∥$ be the distance from $ζ \in ℝ^{m}$ to $X \subseteq ℝ^{m}$ .

Graph theory

The graph of a network systems is denoted by $G = (V, E)$ , where $V = {1, \dots, N}$ and $E \subseteq V \times V$ are the sets of vertex and edge, respectively. Edge ${i, j} \in E$ means that agents i and j exchange information with each other. $N_{i} = {j \in V | {i, j} \in E}$ denotes the set of neighbors of agent i, and ${i, i} \in E$ for any $i \in V$ . For graph $G$ , we take $W \in ℝ^{N \times N}$ as its weighted adjacency matrix, where $w_{ij} \in [η, 1) \subset (0, 1)$ if and only if ${j, i} \in E$ and $w_{ij} = 0$ if ${i, j} \notin E$ .

Regarding the graph $G$ , we make the following mild assumption.

Assumption 1. The graph $G$ is undirected and connected.

Remark 1. Under Assumption 1, it is feasible to construct a doubly stochastic and symmetrical positive semidefinite matrix $W$ .

Problem formulation and algorithm

Problem formulation

This paper considers a class of constrained optimization problems in the form of

\begin{matrix} \min_{ζ} & ψ (ζ) : = \sum_{i = 1}^{N} ψ_{i} (ζ_{i}) \\ s . t . & \sum_{i = 1}^{N} C_{i} ζ_{i} = b, \\ ζ_{i} \in X_{i} \subseteq ℝ^{n_{i}}, i \in V \end{matrix}

(1)

where $ψ_{i} : ℝ^{n_{i}} \to ℝ$ , $ζ = {[ζ_{1}^{⊤}, ζ_{2}^{⊤}, \dots, ζ_{N}^{⊤}]}^{⊤} \in ℝ^{n}$ , and $\sum_{i = 1}^{N} n_{i}$ $= n$ . $C_{i} \in ℝ^{p \times n_{i}}$ and $b \in ℝ^{p}$ specify the coupled constraint. By defining $C = [C_{1}, C_{2}, \dots, C_{N}] \in ℝ^{p \times n}$ , the coupled linear constraints take on the form $Cζ = b$ .

For the considered problem in (1), we take the following mild assumptions.

Assumption 2. For any $i \in V$ , $X_{i}$ is convex and compact.

Assumption 3. For any $i \in V$ , local objective function $ψ_{i} (ζ_{i})$ is lower bounded and has Lipschitz continuous gradient over the set $X_{i}$ , that is

\begin{matrix} ψ_{i} (ζ_{1}) > - \infty, ∥ \nabla ψ_{i} (ζ_{1}) - \nabla ψ_{i} (ζ_{2}) ∥ \leq L ∥ ζ_{1} - ζ_{2} ∥, \forall ζ_{1}, \\ ζ_{2} \in X_{i} \end{matrix}

Algorithm design

For problem (1), we design the following fully distributed ADMM algorithm.

Algorithm 1 Fully distributed ADMM.
1: Initialization: $ζ_{i, 0} \in X_{i}$ , $r_{i, 0} = C_{i} ζ_{i, 0} - b_{i}$ , $s_{i, 0} \in ℝ^{p}$ , $\forall i \in V$ and $\sum_{i = 1}^{N} b_{i} = b$
2: for $k = 1$ , 2, $\dots$ do
3: $δ_{i, k} = \sum_{j \in N_{i}} w_{ij} r_{j, k}$
4: $ℓ_{i, k} = \sum_{j \in N_{i}} w_{ij} s_{j, k}$
5: $ζ_{i, k + 1} = \underset{ζ_{i} \in X_{i}}{argmin} {f_{i} (ζ_{i}) + ℓ_{i, k}^{⊤} C_{i} ζ_{i} + \frac{ρ}{2} ∥ ζ_{i} - ζ_{i, k} ∥^{2} + \frac{ρ}{2} ∥ C_{i} ζ_{i}$ $- C_{i} ζ_{i, k} + δ_{i, k} ∥^{2}}$
6: $r_{i, k + 1} = δ_{i, k} + C_{i} ζ_{i, k + 1} - C_{i} ζ_{i, k}$
7: $s_{i, k + 1} = ℓ_{i, k} + ρ r_{i, k + 1}$
8: End for

Remark 2. Algorithm 1 is fully distributed (refer to Steps 4-6) because agent i only uses local information from itself or receives information from neighbors (refer to Steps 2 and 3). Compared with centralized algorithms (Chatzipanagiotis and Zavlanos, 2017; Yang et al., 2022), each node of Algorithm 1 does not need to know the primal variables of all other agents as well as share the dual variables and only independently receives information from neighbors. In our proposed framework, the system does not enforce consensus on the primal variables. Instead, by employing a dynamic average consensus mechanism to update dual variables $s_{i}$ and constraint estimates $r_{i}$ , Algorithm 1 ensures their consistency.

Recalling $ζ_{k} = {[ζ_{1, k}^{⊤}, ζ_{2, k}^{⊤}, \dots, ζ_{N, k}^{⊤}]}^{⊤}$ , and similar notations for $δ_{k}, l_{k}, r_{k}, s_{k}$ , we rewritten Algorithm 1 in the following compact form

\begin{matrix} ζ_{k + 1} = \underset{ζ \in X}{argmin} {ψ (ζ) + {(W s_{k})}^{⊤} C_{r} ζ + \frac{ρ}{2} ∥ ζ - ζ_{k} ∥^{2} + \frac{ρ}{2} ∥ C_{r} ζ \\ - C_{r} ζ_{k} + W r_{k} ∥^{2}}, \end{matrix}

(2a)

\begin{matrix} r_{k + 1} = W r_{k} + C_{r} ζ_{k + 1} - C_{r} ζ_{k}, \end{matrix}

(2b)

\begin{matrix} s_{k + 1} = W s_{k} + ρ r_{k + 1} \end{matrix}

(2c)

where $C_{r} = blkdiag (C_{1}, C_{2}, \dots, C_{N}) \in ℝ^{Np \times n}$ and $W = W \otimes I_{p} \in ℝ^{Np \times Np}$ . $δ_{k} = W r_{k} \in ℝ^{Np \times n}$ and $l_{k} = W s_{k} \in ℝ^{Np \times n}$ represent the compact form of Steps 2 and 3 of Algorithm 1, respectively.

Convergence of fully distributed ADMM

This section analyzes the proposed ADMM and provides two theorems to show the convergence performance of Algorithm 1. First, if Assumptions 1 and 2 hold, the sequences ${ζ_{k}}_{k \geq 0}$ , ${z_{k}}_{k \geq 0}$ , ${e_{k}^{r}}_{k \geq 0}$ , and ${e_{k}^{s}}_{k \geq 0}$ are bounded. Second, we design a Lyapunov function that is sufficiently decreasing and has a lower bound. Finally, we state the main convergence results of fully distributed ADMM.

Useful lemmas

Since local variables $r_{i, k}$ and $s_{i, k}$ are updated using a dynamic average consistency mechanism, we analyze the average values ${\bar{r}}_{k} = \frac{1}{N} \sum_{i = 1}^{N} r_{i, k}$ and ${\bar{s}}_{k} = \frac{1}{N} \sum_{i = 1}^{N} s_{i, k}$ as well as consensus errors $e_{i, k}^{r} = r_{i, k} - {\bar{r}}_{k}$ and $e_{i, k}^{s} = s_{i, k} - {\bar{s}}_{k}$ .

For the convenience of the discussion hereinafter, we define

\begin{matrix} {\hat{r}}_{k} = 1_{N} \otimes {\bar{r}}_{k}, ŝ_{k} = 1_{N} \otimes {\bar{s}}_{k}, e_{k}^{r} = r_{k} - {\hat{r}}_{k}, e_{k}^{s} = s_{k} - ŝ_{k}, \\ z_{k} = C_{r} ζ_{k} - {\hat{r}}_{k} \end{matrix}

(3)

Lemma 1. (Li et al., 2021) If Assumption 1 holds, then ${\bar{r}}_{k} = \frac{1}{N} (\sum_{i = 1}^{N} C_{i} ζ_{i, k} - b)$ and ${\bar{s}}_{k + 1} = {\bar{s}}_{k} + ρ {\bar{r}}_{k + 1}$ .

Remark 3. Lemma 1 provides a complete characterization of the global iterative average pertaining to the local variables $r_{i, k}$ and $s_{i, k}$ . The analysis of ${\bar{r}}_{k}$ and ${\bar{s}}_{k}$ is helpful for studying the evolution of $r_{i, k}$ and $s_{i, k}$ . Specifically, as analyzed in consensus-based optimization methods, we examine the consensus error, which plays a pivotal role in the convergence of the proposed algorithm.

Lemma 2. (Falsone et al., 2020) If Assumption 1 holds, then

\begin{matrix} e_{k + 1}^{r} = \tilde{W} e_{k}^{r} + z_{k + 1} - z_{k}, \end{matrix}

(4a)

\begin{matrix} e_{k + 1}^{s} = \tilde{W} e_{k}^{s} + ρ e_{k + 1}^{r} \end{matrix}

(4b)

where $\tilde{W} = W - (J \otimes I_{p})$ with $J = \frac{1}{N} 1_{N} 1_{N}^{⊤}$ .

Lemma 3. (Falsone et al., 2020) If Assumptions 1 and 2 hold, then sequences ${ζ_{k}}_{k \geq 0}$ , ${z_{k}}_{k \geq 0}$ , ${e_{k}^{r}}_{k \geq 0}$ , and ${e_{k}^{s}}_{k \geq 0}$ generated by Algorithm 1 are bounded.

Main results

Two key steps should be addressed to guarantee the convergence of a distributed ADMM in nonconvex settings: (1) identifying a Lyapunov function that decreases sufficiently; (2) establishing the lower boundedness property of this Lyapunov function. To achieve the above two steps, we present the following 6 propositions.

Proposition 1. Under Assumptions 1-3, it holds that

\begin{matrix} ∥ s_{k + 1} - s_{k} ∥_{W^{2}}^{2} - ∥ s_{k} - s_{k - 1} ∥_{W^{2}}^{2} + ∥ s_{k + 1} ∥_{{(I - W)}^{2}}^{2} \\ - ∥ s_{k} ∥_{{(I - W)}^{2}}^{2} + ρ^{2} ∥ ζ_{k + 1} - ζ_{k} ∥^{2} \\ - ρ^{2} ∥ ζ_{k} - ζ_{k - 1} ∥^{2} + ρ^{2} ∥ h_{k} ∥^{2} \\ \leq 2 ρL ∥ ζ_{k + 1} - ζ_{k} ∥^{2} - ∥ s_{k + 1} - s_{k} ∥_{I + 2 W - 3 W^{2}}^{2} \end{matrix}

Proof. See Appendix A.

Remark 4. Proposition 1 unveils the relationships between variables during the evolution of Algorithm 1. To be specific, we leverage the Lipschitz gradient continuity property of ψ, which provides an upper bound on the combination of the primal and dual variables. This bound is crucial for designing the Lyapunov function.

For the augmented Lagrangian function of problem (1), we take $\bar{s}$ as the dual variable and define

L_{ρ} (ζ, \bar{s}) = ψ (ζ) + {\bar{s}}^{⊤} (Cζ - b) + \frac{ρ}{2} ∥ Cζ - b ∥^{2}

(5)

Moreover, we rewrite $e_{k}^{r}$ and $e_{k}^{s}$ in Lemma 2 as follows

e_{k + 1} = G e_{k} + F (u_{k + 1} - u_{k})

(6)

where $e_{k} = {[e_{k}^{s}, ρ e_{k}^{r}]}^{⊤}$ , $G = [\begin{matrix} \tilde{W} & \tilde{W} \\ 0 & \tilde{W} \end{matrix}]$ , $F = {[I, I]}^{⊤}$ , $u_{k} = ρ (z_{k} - C_{r} ζ^{*})$ , and $ζ^{*}$ is the stationary solution of problem (1).

Let $L_{ρ}^{+} (ζ_{k + 1}, {\bar{s}}_{k + 1}) = L_{ρ} (ζ_{k + 1}, {\bar{s}}_{k + 1}) - \frac{1}{2 ρ} ∥ u_{k + 1} ∥^{2} -$ $\frac{N^{2} - N}{2 ρ} ∥ {\bar{s}}_{k + 1} - {\bar{s}}_{k} ∥^{2}$ be the regularized augmented Lagrangian function at iteration k. Our following proposition quantifies the variation of the regularized augmented Lagrangian function in continuous iterations.

Proposition 2. Under Assumptions 1-3, for the regularized augmented Lagrangian function changed by Algorithm 1, it holds that

\begin{matrix} L_{ρ}^{+} (ζ_{k + 1}, {\bar{s}}_{k + 1}) - L_{ρ}^{+} (ζ_{k}, {\bar{s}}_{k}) \leq \frac{ρ}{2 ρ - 1} ∥ s_{k + 1} - s_{k} ∥^{2} - ∥ ζ_{k + 1} \\ - ζ_{k} ∥_{\frac{ρ}{2 N} C^{⊤} C + ρI - \frac{3 L}{2} I}^{2} \\ - \frac{1}{ρ} u_{k + 1}^{⊤} [I, 0] e_{k + 1} + \frac{1}{ρ} u_{k}^{⊤} [\tilde{W}, \tilde{W}] e_{k} \end{matrix}

Proof. See Appendix B.

Remark 5. Proposition 2 reveals the essential relationship of the evolution of the agents’ states. It provides an upper bound for the difference between the two terms of the regularized augmented Lagrangian function during the iteration process, and it is also a major component of the Lyapunov function.

Proposition 3. Under Assumptions 1-3, if $A = A^{⊤} ≻ 0$ then the following holds

\begin{matrix} L_{ρ}^{+} (ζ_{k + 1}, {\bar{s}}_{k + 1}) - L_{ρ}^{+} (ζ_{k}, {\bar{s}}_{k}) + a_{1} (∥ s_{k + 1} - s_{k} ∥_{W^{2}}^{2} - ∥ s_{k} \\ - s_{k - 1} ∥_{W^{2}}^{2} - ∥ s_{k} ∥_{{(I - W)}^{2}}^{2} \\ + ∥ s_{k + 1} ∥_{{(I - W)}^{2}}^{2} + ρ^{2} ∥ ζ_{k + 1} - ζ_{k} ∥^{2} - ρ^{2} ∥ ζ_{k} - ζ_{k - 1} ∥^{2}) \\ + a_{2} (∥ F u_{k + 1} - e_{k + 1} ∥_{A}^{2} - ∥ F u_{k} - e_{k} ∥_{A}^{2}) \\ \leq - ∥ ζ_{k + 1} - ζ_{k} ∥_{H_{ζ}}^{2} - ∥ s_{k + 1} - s_{k} ∥_{H_{s}}^{2} - \frac{1}{ρ} u_{k + 1}^{⊤} H e_{k + 1} + \frac{1}{ρ} u_{k}^{⊤} H e_{k} \\ - a_{1} ρ^{2} ∥ h_{k} ∥^{2} - a_{2} ∥ e_{k} ∥_{Q}^{2} \end{matrix}

where $H = [I, 0]$ , $H_{ζ} = \frac{ρ}{2 N} C^{⊤} C + ρI - \frac{3 L}{2} I - 2 a_{1} ρLI$ , $H_{s} = a_{1} I + 2 a_{1} W - 3 a_{1} W^{2} - \frac{ρ}{2 ρ - 1} I$ and $Q = A - G^{⊤} AG$ .

Proof. See Appendix C.

Remark 6. If the cross terms $u_{k + 1}^{⊤} [I, 0] e_{k + 1}$ and $u_{k}^{⊤} [\tilde{W}, \tilde{W}] e_{k}$ of the Proposition 2 are ignored, a decreasing Lyapunov function $T_{a}^{k + 1} = L_{ρ}^{+} (ζ_{k + 1}, {\bar{s}}_{k + 1}) + a (∥ s_{k + 1} - s_{k} ∥_{W^{2}}^{2} + ∥ s_{k + 1} ∥_{{(I - W)}^{2}}^{2}$ $+ ρ^{2} ∥ ζ_{k + 1} - ζ_{k} ∥^{2})$ can be constructed based on Propositions 1 and 2. In particular, the descending and ascending properties are exactly opposite in $∥ s_{k + 1} - s_{k} ∥^{2}$ and $∥ ζ_{k + 1} - ζ_{k} ∥^{2}$ from the term $∥ s_{k + 1} - s_{k} ∥_{W^{2}}^{2} + ∥ s_{k + 1} ∥_{{(I - W)}^{2}}^{2} + ρ^{2} ∥ ζ_{k + 1} - ζ_{k} ∥^{2}$ in Proposition 1 and the regularized augmented Lagrangian function $L_{ρ}^{+} (ζ_{k + 1}, {\bar{s}}_{k + 1})$ in Proposition 2. Proposition 3 utilizes the dynamics (6) to construct a paradigm equation that counteracts the effect of the cross terms, and it is also the basis for constructing the Lyapunov function.

Therefore, we design the Lyapunov function as

\begin{matrix} T_{a_{1}, a_{2}} (ζ_{k + 1}, ζ_{k}, {\bar{s}}_{k + 1}, s_{k + 1}, s_{k}, u_{k + 1}, e_{k + 1}) \\ = L_{ρ}^{+} (ζ_{k + 1}, {\bar{s}}_{k + 1}) + \frac{1}{ρ} u_{k + 1}^{⊤} H e_{k + 1} \\ + a_{1} (∥ s_{k + 1} - s_{k} ∥_{W^{2}}^{2} + ∥ s_{k + 1} ∥_{{(I - W)}^{2}}^{2} + ρ^{2} ∥ ζ_{k + 1} - ζ_{k} ∥^{2}) \\ + a_{2} ∥ F u_{k + 1} - e_{k + 1} ∥_{A}^{2} \end{matrix}

(7)

where $a_{1}$ and $a_{2}$ are constant parameters.

Let $T_{a_{1}, a_{2}}^{k + 1} = T_{a_{1}, a_{2}} (ζ_{k + 1}, ζ_{k}, {\bar{s}}_{k + 1}, s_{k + 1}, s_{k}, u_{k + 1}, e_{k + 1})$ . Based on Proposition 3 and (7), we can directly obtain the following proposition.

Proposition 4. If Assumptions 1-3 hold, then we have

\begin{matrix} T_{a_{1}, a_{2}}^{k + 1} - T_{a_{1}, a_{2}}^{k} \leq - ∥ ζ_{k + 1} - ζ_{k} ∥_{H_{ζ}}^{2} - a_{1} ρ^{2} ∥ h_{k} ∥^{2} \\ - ∥ s_{k + 1} - s_{k} ∥_{H_{s}}^{2} - a_{2} ∥ e_{k} ∥_{Q}^{2} \end{matrix}

Remark 7. Proposition 4 states that if $H_{ζ} ≻ 0, H_{s} ≻ 0, Q ≻ 0$ , $a_{1} \geq 0$ and $a_{2} \geq 0$ , then the designed Lyapunov function $T_{a_{1}, a_{2}}^{k}$ has sufficient decreasing properties. In fact, this can be realized by appropriately setting parameters $(a_{1}, a_{2}, ρ)$ , which will be discussed later.

As mentioned above, another crucial step in achieving convergence is to prove that the Lyapunov function has a lower bound. For this purpose, we propose the following proposition to explain the boundedness of ${u_{k}}_{k \geq 0}$ , ${s_{k}}_{k \geq 0}$ , and ${ŝ_{k}}_{k \geq 0}$ .

Proposition 5. Under Assumptions 1-3, the sequences ${u_{k}}_{k \geq 0}$ , ${s_{k}}_{k \geq 0}$ , and ${ŝ_{k}}_{k \geq 0}$ generated by Algorithm 1 are bounded.

Proof. According to ${\bar{r}}_{k} = (1 ∕ N) (Cζ - b)$ (see Lemma 1) and ${ζ_{k}}_{k \geq 0}$ is bounded (see Lemma 3). Therefore, ${{\bar{r}}_{k}}_{k \geq 0}$ and ${{\hat{r}}_{k}}_{k \geq 0}$ are bounded. Since $z_{k} = C_{r} ζ_{k} - {\hat{r}}_{k}$ and $u_{k} = ρ (z_{k} - C_{r} ζ^{*})$ , it follows that the sequence ${u_{k}}_{k \geq 0}$ is also bounded.

We define $ŝ_{k} = 0$ if $k < 0$ . Since ${e_{k}^{s}}_{k \geq 0}$ (see Lemma 3) and ${{\hat{r}}_{k}}_{k \geq 0}$ are bounded, we deduce that ${s_{k} - ŝ_{k}}_{k \geq 0}$ and ${ŝ_{k} - ŝ_{k - 1}}_{k \geq 0}$ are bounded. By adding ${s_{k} - ŝ_{k}}_{k \geq 0}$ and ${ŝ_{k} - ŝ_{k - 1}}_{k \geq 0}$ , we gain ${s_{k} - ŝ_{k - 1}}_{k \geq 0}$ is bounded. Extending ${ŝ_{k} - ŝ_{k - 1}}_{k \geq 0}$ to iteration $k - 1$ , we have ${ŝ_{k - 1} - ŝ_{k - 2}}_{k \geq 0}$ is bounded. By summing up ${s_{k} - ŝ_{k - 1}}_{k \geq 0}$ and ${ŝ_{k - 1} - ŝ_{k - 2}}_{k \geq 0}$ , then we have ${s_{k} - ŝ_{k - 2}}_{k \geq 0}$ is bounded. With the induction, we can obtain ${s_{k} - ŝ_{0}}_{k \geq 0}$ is bounded (i.e. ${s_{k}}_{k \geq 0}$ is bounded). Because ${e_{k}^{s}}_{k \geq 0}$ is bounded, this means that ${ŝ_{k}}_{k \geq 0}$ is bounded. □

Remark 8. Lemmas 1 and 3 and Proposition 5 show that sequences ${s_{k}}_{k \geq 0}$ and ${r_{k}}_{k \geq 0}$ remain within a (possibly large) neighborhood of $ŝ_{k}$ and ${\hat{r}}_{k}$ , respectively. Based on Lemma 3 and Proposition 5, we deduce the following boundedness result for the Lyapunov function.

Proposition 6. Under Assumptions 1-3, if the Lyapunov function is defined in (7), then $T_{a_{1}, a_{2}}^{k + 1} > - \infty$ , $\forall k \geq 0$ .

Proof. By Lemma 3 and Proposition 5, ${ζ_{k}}_{k \geq 0}$ , ${z_{k}}_{k \geq 0}$ , ${u_{k}}_{k \geq 0}$ , ${s_{k}}_{k \geq 0}$ , ${e_{k}^{r}}_{k \geq 0}$ , and ${e_{k}^{s}}_{k \geq 0}$ are bounded, which implies

\begin{matrix} a_{1} (∥ s_{k + 1} - s_{k} ∥_{W^{2}}^{2} + ∥ s_{k + 1} ∥_{{(I - W)}^{2}}^{2} + ρ^{2} ∥ ζ_{k + 1} - ζ_{k} ∥^{2}), \\ a_{2} ∥ F u_{k + 1} - e_{k + 1} ∥_{A}^{2}, \\ \frac{1}{ρ} u_{k + 1}^{⊤} H e_{k + 1}, \frac{1}{2 ρ} ∥ u_{k + 1} ∥^{2}, and \frac{N^{2} - N}{2 ρ} ∥ {\bar{s}}_{k + 1} - {\bar{s}}_{k} ∥^{2} \end{matrix}

are bounded for any $k \geq 0$ . From (7) and the definition of $L_{ρ}^{+} (ζ_{k + 1}, {\bar{s}}_{k + 1})$ , we only need to prove that $L_{ρ} (ζ, \bar{s}) = ψ (ζ) + {\bar{s}}^{⊤} (Cζ - b) + ρ ∕ 2 ∥ Cζ - b ∥^{2}$ has a lower boundness. Since $ψ (ζ_{k + 1}) > - \infty$ over $X$ (see Assumption 3) and $∥ Cζ - b ∥^{2} \geq 0$ , this means that we only need to prove the lower boundness for ${\bar{s}}^{⊤} (Cζ - b)$ . Based on Lemma 1, we have

\begin{matrix} {\bar{s}}^{⊤} (Cζ - b) & = N {\bar{s}}^{⊤} {\bar{r}}_{k + 1} \\ = \frac{N}{ρ} {\bar{s}}^{⊤} ({\bar{s}}_{k + 1} - {\bar{s}}_{k}) \\ \geq \frac{N}{2 ρ} (∥ {\bar{s}}_{k + 1} ∥^{2} - ∥ {\bar{s}}_{k} ∥^{2}) \end{matrix}

(8)

Owing to the sequence ${s_{k}}_{k \geq 0}$ is bounded (see Proposition 5), we have that ${\bar{s}}^{⊤} (Cζ - b) > - \infty$ . This completes the proof. □

We define the approximate stationary solution and provide the main results of the convergence of fully distributed ADMM.

Definition 1. (Yang et al., 2022) The tuple $(ζ^{*}, {\bar{s}}^{*})$ is called ϵ-stationary solution of problem (1), if it satisfies

dist (\nabla ψ (ζ^{*}) + C^{⊤} {\bar{s}}^{*} + N_{X} (ζ^{*}), 0) + ∥ C ζ^{*} - b ∥ \leq ϵ

where $ϵ \in ℝ$ .

For the convergence of problem (1) in Algorithm 1, we have the following results.

Theorem 1. Suppose that Assumptions 1-3 hold, if $a_{1}$ , $a_{2}$ , and ρ satisfy

\begin{matrix} a_{1} \geq 0, a_{2} \geq 0, \frac{ρ}{2 N} C^{⊤} C + ρI - \frac{3 L}{2} I - 2 a_{1} ρLI ≻ 0, \\ a_{1} I + 2 a_{1} W - 3 a_{1} W^{2} - \frac{ρ}{2 ρ - 1} I ≻ 0 \end{matrix}

then

\begin{matrix} \lim_{k \to \infty} ∥ ζ_{k + 1} - ζ_{k} ∥ = 0, \lim_{k \to \infty} ∥ s_{k + 1} - s_{k} ∥ = 0, \\ \lim_{k \to \infty} ∥ e_{k}^{r} ∥ = 0, \lim_{k \to \infty} ∥ e_{k}^{s} ∥ = 0 \end{matrix}

Proof. By Proposition 4, we obtain

\begin{matrix} \sum_{k = 1}^{K} (T_{a_{1}, a_{2}}^{k} - T_{a_{1}, a_{2}}^{k + 1}) \geq \\ \sum_{k = 1}^{K} ∥ ζ_{k + 1} - ζ_{k} ∥_{H_{ζ}}^{2} + a_{1} ρ^{2} \sum_{k = 1}^{K} ∥ h_{k} ∥^{2} + \sum_{k = 1}^{K} ∥ s_{k + 1} \\ - s_{k} ∥_{H_{s}}^{2} + a_{2} \sum_{k = 1}^{K} ∥ e_{k} ∥_{Q}^{2} \end{matrix}

Taking $K \to \infty$

\begin{matrix} T_{a_{1}, a_{2}}^{1} - \lim_{K \to \infty} T_{a_{1}, a_{2}}^{K + 1} \geq \sum_{k = 1}^{\infty} ∥ ζ_{k + 1} - ζ_{k} ∥_{H_{ζ}}^{2} \\ + a_{1} ρ^{2} \sum_{k = 1}^{\infty} ∥ h_{k} ∥^{2} + \sum_{k = 1}^{\infty} ∥ s_{k + 1} - s_{k} ∥_{H_{s}}^{2} + a_{2} \sum_{k = 1}^{\infty} ∥ e_{k} ∥_{Q}^{2} \end{matrix}

Since $T_{a_{1}, a_{2}}^{k + 1} > - \infty$ (see Proposition 6), it yields

\begin{matrix} \sum_{k = 1}^{\infty} ∥ ζ_{k + 1} - ζ_{k} ∥_{H_{ζ}}^{2} + \sum_{k = 1}^{\infty} ∥ s_{k + 1} - s_{k} ∥_{H_{s}}^{2} \\ + a_{1} ρ^{2} \sum_{k = 1}^{\infty} ∥ h_{k} ∥^{2} + a_{2} \sum_{k = 1}^{\infty} ∥ e_{k} ∥_{Q}^{2} < \infty \end{matrix}

Therefore, we conclude

\begin{matrix} \lim_{k \to \infty} ∥ ζ_{k + 1} - ζ_{k} ∥ = 0, \lim_{k \to \infty} ∥ s_{k + 1} - s_{k} ∥ = 0, \lim_{k \to \infty} ∥ e_{k}^{s} ∥ = 0, \\ \lim_{k \to \infty} ∥ e_{k}^{r} ∥ = 0, \\ \lim_{k \to \infty} ∥ h_{k} ∥ = \lim_{k \to \infty} ∥ (ζ_{k + 1} - ζ_{k}) - (ζ_{k} - ζ_{k - 1}) ∥ = 0 \end{matrix}

which completes the proof. □

Remark 9. Theorem 1 establishes that the sequences ${ζ_{k}}_{k \geq 0}$ and ${s_{k}}_{k \geq 0}$ generated by Algorithm 1 converge under specific conditions. The following theorem addresses the limiting behavior of the tuple generated by Algorithm 1.

Theorem 2. Under Assumptions 1-3. The tuple sequence ${(ζ_{k}, {\bar{s}}_{k})}_{k \geq 0}$ generated by Algorithm 1 converges to an approximate stationary solution of the problem (1).

Proof. By Theorem 1, it is easy to get that the tuple sequence $(ζ_{k}, s_{k})$ generated by Algorithm 1 converges to limit tuple $(ζ^{'}, s^{'})$ , that is, $\lim_{k \to \infty} ζ_{k} = ζ^{'}$ and $\lim_{k \to \infty} s_{k} = s^{'}$ .

By ${\bar{s}}_{k + 1} = {\bar{s}}_{k} - ρ {\bar{r}}_{k + 1}$ (see Lemma 1), we have $\lim_{k \to \infty} ∥ {\bar{r}}_{k + 1} ∥ = 1 ∕ ρ \lim_{k \to \infty} ∥ {\bar{s}}_{k + 1} - {\bar{s}}_{k} ∥ = 0$ , that is

∥ C ζ^{'} - b ∥ = 0

(9)

Taking limits on (15) as $k \to \infty$ , we have

⟨ \nabla ψ (ζ^{'}), ζ^{'} - ζ ⟩ + ⟨ s^{'}, C_{r} (ζ^{'} - ζ) ⟩ \leq 0, \forall ζ \in X

(10)

which can be equivalently rewritten as $⟨ \nabla ψ (ζ^{'}), ζ^{'} - ζ ⟩ + ⟨ {\bar{s}}^{'}, C (ζ^{'} - ζ) ⟩ \leq 0, \forall ζ \in X .$

Consequently

dist (\nabla ψ (ζ^{'}) + C^{⊤} {\bar{s}}^{'} + N_{X} (ζ^{'}), 0) = 0

(11)

By combining (9) and (11), we obtain $dist (\nabla ψ (ζ^{'}) + C^{⊤} {\bar{s}}^{'} + N_{X} (ζ^{'}), 0) + ∥ C ζ^{'} - b ∥ = 0$ , which completes the proof. □

Remark 10. Theorem 2 guarantees that the algorithm’s limit tuple $(ζ^{*}, {\bar{s}}^{*})$ is an approximate stationary solution of the problem (1). In contrast to the convergence performance in Yang et al. (2022), our algorithm achieves the same convergence rate under a fully distributed scheme.

Numerical experiments

Example 1. In this example, we implement the proposed algorithm to address the following optimization problem

\begin{matrix} \min_{ζ} & ψ (ζ) : = \cos (ζ_{1}) + \sin (ζ_{2}) + e^{ζ_{3}} + 0.1 ζ_{4}^{3} + 5 \\ s . t . & ζ_{1} + 2 ζ_{2} + ζ_{3} + 2 ζ_{4} = 2, \\ - 5 \leq ζ_{1} \leq 5, - 3 \leq ζ_{2} \leq 0, - 5 \leq ζ_{3} \leq 0, - 1 \leq ζ_{4} \leq 5 \end{matrix}

(12)

We consider three different settings for the parameter ρ in Algorithm 1. The other parameters are set as follows: $a_{1} = 20$ , $a_{2} = 5$ , $ζ_{0} = {[0.5 - 0.50.5 - 0.5]}^{⊤}$ , $r_{0} = {[0 - 1.50 - 1.5]}^{⊤}$ , and $s_{0} = {[1212]}^{⊤}$ . Moreover, a communication network and its corresponding weight matrix $W$ are randomly generated to satisfy Assumption 1. Under these conditions, we examine how varying values of ρ affect the convergence rate of the proposed method.

First, before operating Algorithm 1, we can readily verify the convergence condition outlined in Theorem 1. The simulations are conducted in MATLAB, utilizing the $fmincon$ command to address the centralized problem and the local subproblems (Step 4). The centralized method finds that the exact value of the stationary point of this problem is that $ζ_{1}^{*} = 3.3607$ , $ζ_{2}^{*} = - 1.1211$ , $ζ_{3}^{*} = - 1.5262$ , and $ζ_{4}^{*} = 1.2038$ . Next, as in the previous analysis, we study the convergence of the method through the Lyapunov function. Figure 1(a) illustrates the evolution of the Lyapunov function in relation to different parameters ρ. We also plot the convergence of primal variables with $ρ = 5$ in Figure 1(b). The Lyapunov functions consistently decrease over the iterations and eventually stabilize at a value close to the optimum $ψ^{*} = 3.5151$ , from Figure 1(a). By comparison, we observe that smaller values of ρ converge faster. Specifically, this is due to the dominance of $∥ ζ_{k + 1} - ζ_{k} ∥_{H_{ζ}}^{2}$ and $∥ s_{k + 1} - s_{k} ∥_{H_{s}}^{2}$ for changes in ρ, which can also be seen from Proposition 4. From Figures 1(a), 2(a) and 2(b), it can be seen that while the actual value of the penalty parameter ρ influences the convergence speed, the proposed algorithm has achieved feasibility while meeting the conditions of Theorem 1.

Figure 1.

Convergence of the Lyapunov function and primal variables in Example 1: (a) Evolution of the Lyapunov function $T_{a, b}^{k}$ . (b) Evolution of primal variables with $ρ = 5$ .

Figure 2.

Convergence of consensus error of dual variables and constraints residual in Example 1: (a) Evolution of consensus error of dual variables. (b) Evolution of the norm of constraints residual.

In Figure 2, we demonstrate the evolution of the consensus error $e_{k}^{s}$ and the constraints residual $∥ C ζ_{k} - b ∥$ for different values of ρ, respectively. From Figure 2, it is clear that the consensus error and the constraints residual asymptotically diminish, as predicted by Theorem 1.

Example 2. To investigate the algorithm’s performance in complex situations, we apply it to the following optimal consensus problem of the Rosenbrock function. The goal is to optimize the sum of the Rosenbrock functions of $N = 8$ agents. The problem is presented as

\begin{matrix} \min_{ζ} & ψ (ζ) : = \sum_{i = 1}^{N} {(v_{1, i} - ς_{i})}^{2} + v_{2, i} {(ξ_{i} - ς_{i}^{2})}^{2} \\ s . t . & ς_{i} = ς_{j}, ξ_{i} = ξ_{j}, ζ_{i} \in X_{i} j \in N_{i}, \forall i, j \in [N] \end{matrix}

(13)

where $ζ_{i} = {[ς_{i} ξ_{i}]}^{⊤}$ and $X_{i} = {[0.5 i - 8, i - 8]}^{⊤} \times {[0.5 i + 2, 0.5 i + 4]}^{⊤} .$

The coupled constraint can be expressed by $L_{m} \otimes I_{2} ζ = 0$ , where $L_{m}$ is the Laplacian matrix of the communication graph. Randomly generate a communication network and its corresponding weight matrix $W$ to satisfy Assumption 1. The ADMM algorithm is configured with $ρ = 100$ , $a_{1} = 20$ , $a_{2} = 5$ , and the parameters $v_{1} = {[12344321]}^{⊤}$ , $v_{2} = {[85 90 95 100 105 110 115 120]}^{⊤}$ . Moreover, for the ith agent, the entries for the initialization of $ς_{i}$ , $ξ_{i}$ , and the dual variables are chosen randomly using a uniform distribution supporting $[0, 5]$ and $[0, 10]$ , respectively.

First, a centralized algorithm is used to obtain the exact value of the stationary point the problem (13), which are $ς^{*} = 2.1215$ , $ξ^{*} = 4.5000$ , and $ψ^{*} = 11.1466$ (the stationary point is within the intersection of the constraint sets $X_{i}$ ). Next, the algorithm’s convergence is verified through the trajectories of the Lyapunov function and the primal variables, which are shown in Figure 3. It is noteworthy that the Lyapunov function exhibits a continuous decrease during the entire process of iterations, which is in accordance with the theoretical analysis. In addition, we demonstrate the behavior of the evolution of the primal variables for agents 2, 5, and 8, and their convergent solutions ς and ξ are very close to the optimal solutions $ς^{*} = 2.1215$ and $ξ^{*} = 4.5000$ obtained by the centralized method, which proves the feasibility of the proposed Algorithm 1 for such problems.

Figure 3.

Convergence of the Lyapunov function and trajectories of primal variables in Example 2: (a) Convergence of the Lyapunov function $T_{a, b}^{k}$ . (b) Trajectories of primal variables.

We compared Algorithm 1 with proximal ADMM in Yang et al. (2022) and the distributed augmented Lagrangian method in Houska et al. (2016). For systematic evaluation, the convergence trajectories of the norm of the constraints residual were analyzed across three methods—Algorithm 1, proximal ADMM, and the augmented Lagrangian method, in Figure 4(b). Simulation results reveal that Algorithm 1 achieves an excellent convergence speed comparable to that of a centralized method (proximal ADMM) while maintaining a smaller constraints residual.

Figure 4.

Convergence of consensus error of dual variables and constraints residual in Example 2: (a) Trajectory of consensus error of dual variables. (b) Trajectories of the norm of constraints residual.

Conclusion

In this paper, we investigated a distributed algorithm for solving a class of nonconvex-constrained optimization problems. Specifically, we considered noncontinuous set constraints and globally coupled linear constraints. We proposed a fully distributed ADMM algorithm that combines ADMM and the dynamic average consensus mechanism, and established its global convergence to an approximate stationary point under the general assumptions of nonconvex optimization methods. In future work, we will focus on designing more generalized distributed ADMM algorithms for nonconvex problems.

Footnotes

Appendix A

Appendix B

Appendix C

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China under Grants 62103003 and 61973002 and in part by the Anhui Provincial Natural Science Foundation under Grant 2008085J32 and in part by the Anhui Provincial Science and Technology Innovation Key Project 202423i08050033.

ORCID iDs

Weizhen Wang

Songsong Cheng

Data availability statement

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

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