Decentralized Composite Optimization in Stochastic Networks: A Dual Averaging Approach with Linear Convergence

Decentralized optimization, particularly the class of decentralized composite convex optimization (DCCO) problems, has found many applications. Due to ubiquitous communication congestion and random dropouts in practice, it is highly desirable to design decentralized algorithms that can handle stochastic communication networks. However, most existing algorithms for DCCO only work in time-invariant networks and cannot be extended to stochastic networks because they inherently need knowledge of network topology $\textit{a priori}$. In this paper, we propose a new decentralized dual averaging (DDA) algorithm that can solve DCCO in stochastic networks. Under a rather mild condition on stochastic networks, we show that the proposed algorithm attains $\textit{global linear convergence}$ if each local objective function is strongly convex. Our algorithm substantially improves the existing DDA-type algorithms as the latter were only known to converge $\textit{sublinearly}$ prior to our work. The key to achieving the improved rate is the design of a novel dynamic averaging consensus protocol for DDA, which intuitively leads to more accurate local estimates of the global dual variable. To the best of our knowledge, this is the first linearly convergent DDA-type decentralized algorithm and also the first algorithm that attains global linear convergence for solving DCCO in stochastic networks. Numerical results are also presented to support our design and analysis.Comment: 22 pages, 2 figure

A Distributed Primal Decomposition Scheme for Nonconvex Optimization

In this paper, we deal with large-scale nonconvex optimization problems, typically arising in distributed nonlinear optimal control, that must be solved by agents in a network. Each agent is equipped with a local cost function, depending only on a local variable. The variables must satisfy private nonconvex constraints and global coupling constraints. We propose a distributed algorithm for the fast computation of a feasible solution of the nonconvex problem in finite time, through a distributed primal decomposition framework. The method exploits the solution of a convexified version of the problem, with restricted coupling constraints, to compute a feasible solution of the original problem. Numerical computations corroborate the results. Copyright (C) 2019. The Authors. Published by Elsevier Ltd. All rights reserved