A fully distributed dual gradient method with linear convergence for large-scale separable convex problems

In this paper we propose a distributed dual gradient algorithm for minimizing linearly constrained separable convex problems and analyze its rate of convergence. In particular, we show that under the assumption that the Hessian of the primal objective function is bounded we have a global error bound type property for the dual problem. Using this error bound property we devise a fully distributed dual gradient scheme for which we derive global linear rate of convergence. The proposed dual gradient method is fully distributed, requiring only local information, since is based on a weighted stepsize. Our method can be applied in many applications, e.g. distributed model predictive control, network utility maximization or optimal power flow.

[1]  Yurii Nesterov,et al.  Efficiency of Coordinate Descent Methods on Huge-Scale Optimization Problems , 2012, SIAM J. Optim..

[2]  Ion Necoara,et al.  On linear convergence of a distributed dual gradient algorithm for linearly constrained separable convex problems , 2014, Autom..

[3]  A. Ozdaglar,et al.  Optimal Distributed Gradient Methods for Network Resource Allocation Problems , 2013 .

[4]  Stephen J. Wright,et al.  Cooperative distributed model predictive control , 2010, Syst. Control. Lett..

[5]  Lucas Barcelos de Oliveira,et al.  Distributed Optimization for Model Predictive Control of Linear-Dynamic Networks , 2009, IEEE Transactions on Systems, Man, and Cybernetics - Part A: Systems and Humans.

[6]  Paul Tseng,et al.  On the Convergence Rate of Dual Ascent Methods for Linearly Constrained Convex Minimization , 1993, Math. Oper. Res..

[7]  Ion Necoara,et al.  Parallel and distributed optimization methods for estimation and control in networks , 2011, 1302.3103.

[8]  Bart De Schutter,et al.  Accelerated gradient methods and dual decomposition in distributed model predictive control , 2013, Autom..

[9]  Ion Necoara,et al.  Computational Complexity of Inexact Gradient Augmented Lagrangian Methods: Application to Constrained MPC , 2013, SIAM J. Control. Optim..

[10]  Alberto Bemporad,et al.  An Accelerated Dual Gradient-Projection Algorithm for Embedded Linear Model Predictive Control , 2014, IEEE Transactions on Automatic Control.

[11]  Ion Necoara,et al.  Rate Analysis of Inexact Dual First-Order Methods Application to Dual Decomposition , 2014, IEEE Transactions on Automatic Control.

[12]  Chih-Jen Lin,et al.  Iteration complexity of feasible descent methods for convex optimization , 2014, J. Mach. Learn. Res..

[13]  Bart De Schutter,et al.  A distributed optimization-based approach for hierarchical MPC of large-scale systems with coupled dynamics and constraints , 2011, IEEE Conference on Decision and Control and European Control Conference.

[14]  Johan A. K. Suykens,et al.  Application of a Smoothing Technique to Decomposition in Convex Optimization , 2008, IEEE Transactions on Automatic Control.

[15]  R. Tyrrell Rockafellar,et al.  Variational Analysis , 1998, Grundlehren der mathematischen Wissenschaften.

[16]  A. Bakirtzis,et al.  A decentralized solution to the DC-OPF of interconnected power systems , 2003 .

[17]  Jacques Gauvin,et al.  A necessary and sufficient regularity condition to have bounded multipliers in nonconvex programming , 1977, Math. Program..

[18]  Marcello Farina,et al.  Distributed predictive control: A non-cooperative algorithm with neighbor-to-neighbor communication for linear systems , 2012, Autom..

[19]  Asuman E. Ozdaglar,et al.  Approximate Primal Solutions and Rate Analysis for Dual Subgradient Methods , 2008, SIAM J. Optim..

[20]  Hanif D. Sherali,et al.  A class of convergent primal-dual subgradient algorithms for decomposable convex programs , 1986, Math. Program..

[21]  Yurii Nesterov,et al.  Introductory Lectures on Convex Optimization - A Basic Course , 2014, Applied Optimization.

[22]  Marc Teboulle,et al.  An $O(1/k)$ Gradient Method for Network Resource Allocation Problems , 2014, IEEE Transactions on Control of Network Systems.

[23]  Michael Ulbrich,et al.  A class of distributed optimization methods with event-triggered communication , 2013, Computational Optimization and Applications.