Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Distributed Prediction-Correction ADMM for Time-Varying Convex Optimization

Nicola Bastianello, Andrea Simonetto, Ruggero Carli

Abstract

This paper introduces a dual-regularized ADMM approach to distributed, time-varying optimization. The proposed algorithm is designed in a prediction-correction framework, in which the computing nodes predict the future local costs based on past observations, and exploit this information to solve the time-varying problem more effectively. In order to guarantee linear convergence of the algorithm, a regularization is applied to the dual, yielding a dual-regularized ADMM. We analyze the convergence properties of the time-varying algorithm, as well as the regularization error of the dual-regularized ADMM. Numerical results show that in time-varying settings, despite the regularization error, the performance of the dual-regularized ADMM can outperform inexact gradient-based methods, as well as exact dual decomposition techniques, in terms of asymptotical error and consensus constraint violation.

Distributed Prediction-Correction ADMM for Time-Varying Convex Optimization

Abstract

This paper introduces a dual-regularized ADMM approach to distributed, time-varying optimization. The proposed algorithm is designed in a prediction-correction framework, in which the computing nodes predict the future local costs based on past observations, and exploit this information to solve the time-varying problem more effectively. In order to guarantee linear convergence of the algorithm, a regularization is applied to the dual, yielding a dual-regularized ADMM. We analyze the convergence properties of the time-varying algorithm, as well as the regularization error of the dual-regularized ADMM. Numerical results show that in time-varying settings, despite the regularization error, the performance of the dual-regularized ADMM can outperform inexact gradient-based methods, as well as exact dual decomposition techniques, in terms of asymptotical error and consensus constraint violation.

Paper Structure

This paper contains 15 sections, 5 theorems, 54 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. An ADMM Reformulation
  3. Consensus problem formulation
  4. Dual-regularized Problem
  5. Distributed Dual-regularized ADMM
  6. Distributed Prediction-Correction ADMM
  7. Convergence analysis
  8. Trade-offs
  9. Communication complexity
  10. Numerical results
  11. Conclusions
  12. Regularized dual problem
  13. Proof of Lemma \ref{['lem:regularization-error']}
  14. Regularized ADMM
  15. Distributed ADMM

Key Result

Lemma 1

Let the cost $f(\cdot; t_k)$ be in $\mathcal{S}_{\mu,L}(\mathbb{R}^{nN})$ uniformly in time. Let $\mathbold{w}^*(\epsilon; t_k)$ be the optimal solution of the regularized dual problem eq:dual-reg-problem, and $\mathbold{w}^*(t_k)$ be a solution to the original dual problem eq:dual-problem. Then the

Figures (2)

  • Figure 1: Error trajectory comparison, with $N_\mathrm{P}, N_\mathrm{C} = 5$.
  • Figure 2: Distance from consensus comparison, with $N_\mathrm{P}, N_\mathrm{C} = 5$.

Theorems & Definitions (11)

  • Lemma 1: Regularization error
  • proof
  • Lemma 2
  • proof
  • Remark 1
  • Corollary 1: Convergence to $\mathbold{w}^*(\epsilon; t_k)$
  • proof
  • Proposition 1: Convergence to $\mathbold{w}^*(t_k)$
  • proof
  • Corollary 2: Convergence to $\mathbold{x}^*(t_k)$
  • ...and 1 more