Consensus-Based ADMM for Distributed Optimization

• Consensus-Based ADMM is a distributed optimization framework that enforces agreement among multiple agents through a primal-dual methodology with consensus constraints.
• The method employs parallel local (primal) updates and a global averaging step (z-update) combined with a dual update to drive the solution towards optimality.
• Its convergence theory, adaptive penalty tuning, and applications in multi-agent systems like robotics and MPC highlight its practical scalability and accuracy.

The Consensus-Based Alternating Direction Method of Multipliers (ADMM) extends the classical ADMM framework to distributed optimization settings, targeting problems where multiple agents (nodes, robots, processors) collectively seek agreement on shared variables while each holds private, possibly heterogeneous convex objectives. This paradigm is central in large-scale optimization for machine learning, control, robotics, and networked systems, providing a rigorous methodology for primal-dual distributed computation with provable convergence under consensus constraints.

1. Mathematical Definition of Consensus ADMM

Consider a network of $N$ agents, each having a private convex cost function $f_i:\mathbb{R}^n \rightarrow \mathbb{R}$. The consensus optimization problem is posed as

$$\min_{x_1,\dots,x_N, z}\; \sum_{i=1}^N f_i(x_i) \quad \text{s.t.} \quad x_i = z, \;\forall i,$$

where $x_i$ is the local copy of the global variable $z\in\mathbb{R}^n$ (Chen, 2 Oct 2024).

The augmented Lagrangian is

$$L_\rho(\{x_i\}, z, \{\lambda_i\}) = \sum_{i=1}^N \Big[ f_i(x_i) + \lambda_i^\top(x_i - z) + \frac{\rho}{2} \|x_i - z\|^2 \Big],$$

with dual multipliers $\lambda_i\in\mathbb{R}^n$ and penalty $\rho>0$.

Consensus ADMM alternates the following updates:

• Primal block update ($x_i$): in parallel, each agent solves

$$x_i^{k+1} = \arg\min_{x_i} f_i(x_i) + \lambda_i^{k\top}(x_i - z^k) + \frac{\rho}{2}\|x_i - z^k\|^2,$$

or equivalently,

$$x_i^{k+1} = \arg\min_{x_i} f_i(x_i) + \frac{\rho}{2}\left\|x_i - z^k + \frac{1}{\rho}\lambda_i^k\right\|^2.$$

• Consensus (global averaging) update ($z$):

$$z^{k+1} = \frac{1}{N}\sum_{i=1}^N \left(x_i^{k+1} + \frac{1}{\rho}\lambda_i^k\right).$$

In decentralized contexts, $z$-averaging may be local or distributed via message-passing (Chen, 2 Oct 2024).

• Dual update:

$$\lambda_i^{k+1} = \lambda_i^k + \rho(x_i^{k+1} - z^{k+1}).$$

This structure, a prototypical block-separable ADMM, enables distributed computation with rigorous consensus enforcement.

2. Underlying Primal-Dual Framework and Algorithmic Properties

Consensus ADMM arises from block-structured convex optimization using the augmented Lagrangian. The inclusion of quadratic penalty terms (with $\rho$) boosts numerical stability and enforces the consensus constraint more tightly as $\rho$ increases. Each agent operates independently on its private cost, with consensus coordination only via the $z$-averaging and dual variable exchange.

The algorithm guarantees—under convexity, proper closedness of $f_i$, and Slater's condition—the primal residuals $x_i^k - z^k$ and dual residuals $\rho(z^k - z^{k-1})$ both vanish as $k\rightarrow \infty$. The primal and dual iterates converge to an optimal solution and multiplier, with an ergodic convergence rate $O(1/k)$ in objective and feasibility (Chen, 2 Oct 2024).

Penalty parameter $\rho$ critically impacts practical performance:

• Too small: weak enforcement of consensus, slow primal convergence.
• Too large: potential overshoot and slow dual convergence. A standard heuristic is adaptive tuning via residual monitoring: $$r^k = \max_i \| x_i^k - z^k \|, \quad s^k = \rho \| z^k - z^{k-1} \|,$$ and adjusting $\rho$ upward if $r^k > 10 s^k$, downward if $s^k > 10 r^k$ (Chen, 2 Oct 2024).

3. Connections to Other Distributed and Parallel ADMM Variants

Consensus ADMM is foundational in distributed optimization (Liu et al., 2021), model predictive consensus (Summers et al., 2012), and parallel block frameworks. Extensions include:

• Parallel block-wise updates: Each agent updates its primal and dual states simultaneously; dual variables on edges drive local consensus over the network topology (Liu et al., 2021).
• Incremental and random-walk ADMM: Token-passing or Hamiltonian-cycle scheduling of updates enables communication-efficient asynchronous consensus, with privacy-preserving variants (step-size or primal perturbations) to prevent adversarial inference of private data (Ye et al., 2020).
• Model Predictive Control: In distributed MPC for multi-agent or robotic systems, consensus ADMM enables coordination of local trajectory planning with global agreement. The $z$ variable becomes a network-wide aggregated trajectory, with local QP subproblems incorporating inter-agent collision constraints (Chen, 2 Oct 2024, Summers et al., 2012).
• Generalizations: Flexible block-selection ADMM, asynchronous block-wise ADMM, and majorized/proximal extensions improve scalability to nonconvex problems, allow unbalanced participation, and accommodate composite constraints (Hong et al., 2014, Zhu et al., 2018, Kumar et al., 2016).

4. Convergence Theory and Scaling Behavior

Under standard convexity and regularity assumptions:

• Global optimality: ADMM converges to an optimal solution of the consensus constrained problem—both for objective and consensus error.
• Rate: $O(1/k)$ in the ergodic average of the final cost and feasibility (Chen, 2 Oct 2024, Liu et al., 2021).
• Scalability: Practical experiments in model predictive consensus and multi-robot trajectory optimization indicate convergence to millimeter-level feasibility within $20-50$ iterations for networks of $N=5$ drones (horizon $H=10$), achieving trajectories close to those from centralized solvers but at reduced per-agent cost (Chen, 2 Oct 2024).

5. Example Application: Distributed Multi-Robot Trajectory Optimization

In distributed model predictive trajectory planning for $N$ robots:

• Each agent $i$ controls a trajectory $$x_i = \{x_i(0),\dots,x_i(H), u_i(0), \dots, u_i(H-1)\}$$ over a horizon $H$, with $x_i(k)\in\mathbb{R}^{12}$ (state) and $u_i(k)\in\mathbb{R}^4$ (input).
• Local cost: quadratic tracking and control regularization,

$$f_i(x_i) = \sum_{k=0}^{H-1} \|x_i(k) - x_i^{\text{ref}}(k)\|_Q^2 + \|u_i(k)\|_R^2,$$

with $Q \succeq 0$, $R \succ 0$.

• Local constraints: nonlinear dynamics, collision avoidance ($\|p_i(k) - p_j(k)\| \ge d_{\min}$), state/input bounds.
• Consensus variable $z$ aggregates all agents’ trajectories, enforcing global prediction agreements needed for collision avoidance.

At each ADMM iteration:

• Each agent $i$ solves, in parallel,

$$\tilde\theta_i^{k+1} = \arg\min_{\tilde\theta_i \in \Theta_i}\; f_i(\tilde\theta_i) + \lambda_i^{k\top}(\tilde\theta_i - z^k) + \frac{\rho}{2}\|\tilde\theta_i - z^k\|^2,$$

subject to local constraints.

• All agents average:

$$z^{k+1} = \frac{1}{N} \sum_{i=1}^N \left(\tilde\theta_i^{k+1} + \frac{1}{\rho} \lambda_i^k \right),$$

and update multipliers:

$$\lambda_i^{k+1} = \lambda_i^k + \rho (\tilde\theta_i^{k+1} - z^{k+1}).$$

Agents communicate only their latest trajectory estimates to neighbors. The decentralized implementation thus matches centralized accuracy—millimeter-level feasibility—while reducing per-agent computational and communication burden (Chen, 2 Oct 2024).

6. Practical Implementation Considerations

For practitioners:

• Local problem solution: Each agent’s primal update typically reduces to a small convex QP, solvable via standard numerical routines.
• Communication model: In fully distributed networks, consensus-averaging for $z$ can be implemented via peer-to-peer protocols (gossip, consensus subroutines), or by averaging with immediate neighbors (Chen, 2 Oct 2024).
• Computational efficiency: Practical systems effectively exploit parallelism at the agent level, with per-step solution times typically in the millisecond regime for small QPs.
• Penalty parameter tuning: Use adaptive heuristics based on primal/dual residuals for dynamic adjustment of $\rho$.
• Extension to nonconvex settings: For smooth nonconvex problems, consensus ADMM can maintain convergence to stationary solutions provided penalty parameters are large enough to dominate local curvature (Hong et al., 2014).

7. Impact and Research Directions

Consensus-based ADMM has become a foundational tool in distributed convex and nonconvex optimization across:

• Multi-agent control and robotics (MPC, trajectory consensus).
• Machine learning (distributed model fitting, federated learning).
• Large-scale resource allocation and networked systems.
• Signal processing, smart grids, and sensor fusion.

Continuing directions include improving communication efficiency (incremental and random-walk ADMMs), privacy-preserving variants (step-size or primal perturbations), support for asynchronous and unreliable networks, and extension to more complex coupling constraints (e.g., composite regularization, generalized consensus over heterogeneous graphs) (Chen, 2 Oct 2024, Ye et al., 2020, Zhu et al., 2018).

Consensus ADMM provides a robust and scalable framework for coordinated optimization, enabling high-precision agreement and efficient decentralized computation in demanding multi-agent environments.