Distributed Constraint-Coupled Optimization over Random Time-Varying Graphs via Primal Decomposition and Block Subgradient Approaches

Abstract

In this paper, we consider a network of processors that want to cooperatively solve a large-scale, convex optimization problem. Each processor has knowledge of a local cost function that depends only on a local variable. The goal is to minimize the sum of the local costs, while making the variables satisfy both local constraints and a global coupling constraint. We propose a simple, fully distributed algorithm, that works in a random, time-varying communication model, where at each iteration multiple edges are randomly drawn from an underlying graph. The algorithm is interpreted as a primal decomposition scheme applied to an equivalent problem reformulation. Almost sure convergence to the optimal cost of the original problem is proven by resorting to approaches from block subgradient methods. Specifically, the communication structure is mapped to a block structure, where the blocks correspond to the graph edges and are randomly selected at each iteration. Moreover, an almost sure asymptotic primal recovery property, with no averaging mechanisms, is shown. A numerical example corroborates the theoretical analysis.

The Distributed Dual Ascent Algorithm is Robust to Asynchrony

“…Lemma 1: For all j ∈ I, the mapping x j in ( 16) is θ j ρ j -Lipschitz continuous, with θ j as in (10).…”

“…Block-coordinate versions of the dual ascent, where only part of the variables is updated at each iteration, are also explored in the literature [9]. More generally, a variety of distributed algorithms has been proposed to solve constraintcoupled optimization problems, possibly with block-updates and time-varying communication [10], [11], [12]. Nonetheless, in all the cited works, a common clock is employed to synchronize the communication and update frequencies.…”

The distributed dual ascent algorithm is robust to asynchrony

“…Lemma 1: For all j ∈ I, the mapping x ⋆ j in ( 16) is θj ρj -Lipschitz continuous, with θ j as in (10).…”

“…This work was partially supported by NWO under research project OMEGA (grant n. 613.001.702) and by the ERC under research project COSMOS (802348). and time-varying communication [10], [11], [12]. Nonetheless, in all the cited works, a common clock is employed to synchronize the communication and update frequencies.…”

Constraint-coupled Optimization with Unknown Costs: A Distributed Primal Decomposition Approach

“…Under the assumptions that the agents are able to build more and more refined estimates of their cost functions, we propose a novel, distributed algorithm to solve the problem exactly. The algorithm is inspired to a distributed primal decomposition approach for constraint-coupled optimization [14], [22], where the algorithm has been suitably modified to account for the online cost estimation mechanism. The resulting scheme is a three-step procedure where each agent first obtains an updated version of the cost estimation, then it solves a local version of the original problem with the true cost function replaced by an estimated version, and finally updates a local state after exchanging dual information with neighbors.…”