Fully Distributed Algorithms for Convex Optimization Problems

  • Mosk-Aoyama D
  • Roughgarden T
  • Shah D
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Abstract

We design and analyze a fully distributed algorithm for convex constrained opti-mization in networks without any consistent naming infrastructure. The algorithm produces an approximately feasible and near-optimal solution in time polynomial in the network size, the inverse of the permitted error, and a measure of curvature variation in the dual optimization problem. It blends, in a novel way, gossip-based information spreading, iterative gradient ascent, and the barrier method from the design of interior-point algorithms. 1. Introduction. The development of modern networks, such as sensor and peer-to-peer networks, has stimulated interest in decentralized approaches to compu-tational problems. These networks often have unreliable nodes with limited power, computation, and communication constraints. Frequent changes in the network topol-ogy make it hard to establish infrastructure for coordinated centralized computation. However, efficient use of network resources requires solving global optimization prob-lems. This motivates the study of fully distributed algorithms for global optimization problems that do not rely on any form of network infrastructure. Informally, we call an algorithm fully distributed with respect to a network con-nectivity graph G if each node of G operates without using any information beyond that in its local neighborhood in G. More concretely, we assume that each node in the network knows only its neighbors in the network, and that nodes do not have unique identifiers that can be attached to the messages that they send. This constraint is natural in networks that lack infrastructure (such as IP addresses or static GPS loca-tions), including ad-hoc and mobile networks. It also severely limits how a node can aggregate information from beyond its local neighborhood, thereby providing a clean way to differentiate between distributed algorithms that are " truly local " and those which gather large amounts of global information at all of the nodes and subsequently perform centralized computations. Previous work [8] observed that when every network node possesses a positive real number, the minimum of these can be efficiently computed by a fully distributed algorithm, and leveraged this fact to design fully distributed algorithms for evaluating various separable functions, including the summation function. This paper studies the significantly more difficult task of constrained optimization for a class of problems that capture many key operational network problems such as routing and congestion

Author-supplied keywords

  • 080743706
  • 1
  • 10
  • 1137
  • 68w15
  • 90c25
  • ams subject classifications
  • convex optimization
  • distributed algorithms
  • doi
  • gradient ascent
  • introduction
  • networks
  • such as sensor and
  • the development of modern

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Authors

  • Damon Mosk-Aoyama

  • Tim Roughgarden

  • Devavrat Shah

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