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A Partition-Based Implementation of the Relaxed ADMM for Distributed Convex Optimization over Lossy Networks

Nicola Bastianello, Marco Todescato, Ruggero Carli, Luca Schenato

Abstract

In this paper we propose a distributed implementation of the relaxed Alternating Direction Method of Multipliers algorithm (R-ADMM) for optimization of a separable convex cost function, whose terms are stored by a set of interacting agents, one for each agent. Specifically the local cost stored by each node is in general a function of both the state of the node and the states of its neighbors, a framework that we refer to as `partition-based' optimization. This framework presents a great flexibility and can be adapted to a large number of different applications. We show that the partition-based R-ADMM algorithm we introduce is linked to the relaxed Peaceman-Rachford Splitting (R-PRS) operator which, historically, has been introduced in the literature to find the zeros of sum of functions. Interestingly, making use of non expansive operator theory, the proposed algorithm is shown to be provably robust against random packet losses that might occur in the communication between neighboring nodes. Finally, the effectiveness of the proposed algorithm is confirmed by a set of compelling numerical simulations run over random geometric graphs subject to i.i.d. random packet losses.

A Partition-Based Implementation of the Relaxed ADMM for Distributed Convex Optimization over Lossy Networks

Abstract

In this paper we propose a distributed implementation of the relaxed Alternating Direction Method of Multipliers algorithm (R-ADMM) for optimization of a separable convex cost function, whose terms are stored by a set of interacting agents, one for each agent. Specifically the local cost stored by each node is in general a function of both the state of the node and the states of its neighbors, a framework that we refer to as `partition-based' optimization. This framework presents a great flexibility and can be adapted to a large number of different applications. We show that the partition-based R-ADMM algorithm we introduce is linked to the relaxed Peaceman-Rachford Splitting (R-PRS) operator which, historically, has been introduced in the literature to find the zeros of sum of functions. Interestingly, making use of non expansive operator theory, the proposed algorithm is shown to be provably robust against random packet losses that might occur in the communication between neighboring nodes. Finally, the effectiveness of the proposed algorithm is confirmed by a set of compelling numerical simulations run over random geometric graphs subject to i.i.d. random packet losses.

Paper Structure

This paper contains 10 sections, 3 theorems, 34 equations, 4 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. The Relaxed-ADMM algorithm
  3. Distributed Partition-Based Convex Optimization
  4. Problem Formulation
  5. R-ADMM for Partition-Based Optimization
  6. Partition-based R-ADMM over lossy networks
  7. Simulations
  8. Conclusions and Future Directions
  9. Proof of Proposition \ref{['prop:distributed-r-admm']}
  10. Proof of Propositions \ref{['prop:convergence']} and \ref{['prop:convergence_lossy']}

Key Result

Proposition 1

The trajectories generated by the variables $\mathbf{x}^{(i)}$, $i \in V$, obtained by applying the R-ADMM algorithm in eq:r-admm-1, eq:r-admm-2, eq:r-admm-3, to the problem in eq:primal-indicator-f, starting from a given initial condition $\mathbf{x}^{(i)}(0)$, $\mathbf{w}^{(i)}(0)$, $\mathbf{y}^{( for all $i \in V$, and for all $j\in\mathcal{N}_i$, where the auxiliary variables are initialized

Figures (4)

  • Figure 1: Evolution, in log-scale, of the relative error of Alg. \ref{['alg:robust-smart-distributed-r-admm']} computed w.r.t. the unique optimal solution $\mathbf{x}^*$ as function of different values of packet loss probability $p$ for step size $\alpha=0.75$ and penalty $\rho=3$. Average over 100 Monte Carlo runs.
  • Figure 2: Stability boundaries of Alg. \ref{['alg:robust-smart-distributed-r-admm']} as function of the step size $\alpha$ and the penalty $\rho$ for different values of loss probability $p$. Average over 100 Monte Carlo runs.
  • Figure 3: Evolution, in log-scale, of the relative error of Alg. \ref{['alg:robust-smart-distributed-r-admm']} computed w.r.t. the unique optimal solution $\mathbf{x}^*$ as function of different values of the step size $\alpha$, with fixed packet loss probability $p=0.2$ and penalty $\rho=3$. Average over 100 Monte Carlo runs.
  • Figure 4: Evolution, in log-scale, of the relative error of Alg. \ref{['alg:robust-smart-distributed-r-admm']} computed w.r.t. the unique optimal solution $\mathbf{x}^*$ as function of different values of the penalty $\rho$, with fixed packet loss probability $p=0.2$ and step size $\alpha=0.75$. Average over 100 Monte Carlo runs.

Theorems & Definitions (7)

  • Proposition 1
  • Proposition 2
  • Remark 1
  • Proposition 3
  • Remark 2
  • Remark 3
  • Remark 4