Newton-Raphson consensus for distributed convex optimization

Abstract

Abstract-We study the problem of unconstrained distributed optimization in the context of multi-agents systems subject to limited communication connectivity. In particular we focus on the minimization of a sum of convex cost functions, where each component of the global function is available only to a specific agent and can thus be seen as a private local cost. The agents need to cooperate to compute the minimizer of the sum of all costs. We propose a consensus-like strategy to estimate a Newton-Raphson descending update for the local estimates of the global minimizer at each agent. In particular, the algorithm is based on the separation of time-scales principle and it is proved to converge to the global minimizer if a specific parameter that tunes the rate of convergence is chosen sufficiently small. We also provide numerical simulations and compare them with alternative distributed optimization strategies like the Alternating Direction Method of Multipliers and the Distributed Subgradient Method.

A comparative analysis of the Fast-Lipschitz convergence speed

“…In this subsection we characterize the best convergence speed of the Lagrangian method for problem (5). We then compare such a bound to the Fast-Lipschitz method in the following subsection.…”

“…Since these approaches rely on subgradients or Lagrangian methods, they usually have slow rates of convergence. Recent works have attempted to speed up convergence by considering higher order methods [5], [6]. However, updating the dual variables requires a broadcast and collect iterative message exchanging from all nodes of the network.…”

Distributed optimal in-network resource allocation algorithm design via a control theoretic approach

“…convex functions, each representing a private local cost only available to a single agent, subject to some convex constraints. Some of the recent literature on distributed optimization algorithm design includes distributed algorithms implemented both in discretetime [5][6][7][8][9] and continuous-time [10][11][12][13][14]. Although some of these algorithms can solve the optimal resource allocation problem (1), they require each agent to keep and evolve a copy of the global decision variable of the problem which is of order N, where N is the size of network.…”

“…[17]). Singularly perturbed distributed algorithms are used in [12] for unconstrained in-network convex optimization, and in [18] for dynamic consensus problem over networked systems.…”

Extensions of Fast-Lipschitz Optimization for Convex and Non-convex Problems

“…Dual decomposition and subgradient methods can then be applied, recent works are Nedic and Ozdaglar (2009);Rantzer (2009);Boyd (2010); Sundhar Ram et al (2010). As these methods typically have slow rates of convergence, recent papers explores higher order methods, see Zanella et al (2011);Wei et al (2011). Other methods are based on the consensus iterations, where nodes update their decision variables as a weighted sum of other decision variables in the neighborhood, see, e.g., Notarstefano and Bullo (2009) ;Olshevsky and Tsitsiklis (2009) ;Nedic et al (2010); Chiuso et al (2011).…”