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Large-Scale and Distributed Optimization pp 247–287Cite as

Decentralized Consensus Optimization and Resource Allocation

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Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2227))

Abstract

We consider the problems of consensus optimization and resource allocation, and we discuss decentralized algorithms for solving such problems. By “decentralized”, we mean the algorithms are to be implemented in a set of networked agents, whereby each agent is able to communicate with its neighboring agents. For both problems, every agent in the network wants to collaboratively minimize a function that involves global information, while having access to only partial information. Specifically, we will first introduce the two problems in the context of distributed optimization, review the related literature, and discuss an interesting “mirror relation” between the problems. Afterwards, we will discuss some of the state-of-the-art algorithms for solving the decentralized consensus optimization problem and, based on the “mirror relationship”, we then develop some algorithms for solving the decentralized resource allocation problem. We also provide some numerical experiments to demonstrate the efficacy of the algorithms and validate the methodology of using the “mirror relation”.

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Notes

  1. 1.

    Even for special topologies such as a “star” shaped network, it is considered as decentralized since the agent at the “center” has a similar/same computational ability as the others in the network and is not responsible for any extra work of coordination.

  2. 2.

    Suppose that a sequence {x(k)} converges to x ∗ in some norm ∥⋅∥. The convergence is: (i) Q-linear if there is λ ∈ (0,  1) such that \(\frac {\|x(k+1)-x^*\|}{\|x(k)-x^*\|} \leq \lambda \) for all k; (ii) R-linear if there is λ ∈ (0,  1) and some positive constant C such that ∥x(k) − x ∗∥≤ Cλ k for all k. Both of these rates are geometric. They are often referred to as global rates to distinguish them from the case when the given relations are valid for some sufficiently large k. The difference between these two types of geometric rate is in that Q-linear rate implies monotonic decrease of ∥x(k) − x ∗∥, while R-linear rate does not. We will use “geometric(ally)” and “linear(ly)” interchangeably when it does not cause confusion.

  3. 3.

    A nonnegative sequence {a k} is said to be convergent to 0 at an O(1∕k) rate if \({\lim \sup }_{k\rightarrow \infty } ka_k<+\infty \). In contrast, it is said to have an o(1∕k) rate if \({\lim \sup }_{k\rightarrow \infty } ka_k=0\).

  4. 4.

    When an algorithm has convergence rate of O(θ(k)), we say that the rate is sublinear if \(\lim _{k\rightarrow +\infty }\frac {\lambda ^k}{\theta (k)}=0\) for any constant λ ∈ (0,  1). A typical sublinear rates include O(1∕k p) with p > 0.

  5. 5.

    See Ref. [78] for the definition of “dual friendly”.

  6. 6.

    A description of the range for δ can be found in [49].

  7. 7.

    A more specific description of the range for δ can be found in [49].

  8. 8.

    Copyright Ⓒ 2017 Society for Industrial and Applied Mathematics. Reprinted with permission. All rights reserved.

  9. 9.

    To guarantee connectedness of the graph, we first grow a random tree; then uniformly randomly add edges into the graph to reach the specified connectivity ratio.

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Acknowledgements

The work of A.N. and A.O. was supported by the Office of Naval Research grant N000014-16-1-2245. The work of A.O. was also supported by the NSF award CMMI-1463262 and AFOSR award FA-95501510394.

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Authors and Affiliations

  1. School of Electrical, Computer, and Energy Engineering, Arizona State University, Tempe, AZ, USA

    Angelia Nedić & Wei Shi

  2. Department of Electrical Engineering, Boston University, Boston, MA, USA

    Alexander Olshevsky

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  1. Angelia Nedić
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Correspondence to Angelia Nedić .

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  1. Department of Automatic Control, Lund University, Lund, Sweden

    Pontus Giselsson

  2. Department of Automatic Control, Lund University, Lund, Sweden

    Anders Rantzer

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Nedić, A., Olshevsky, A., Shi, W. (2018). Decentralized Consensus Optimization and Resource Allocation. In: Giselsson, P., Rantzer, A. (eds) Large-Scale and Distributed Optimization. Lecture Notes in Mathematics, vol 2227. Springer, Cham. https://doi.org/10.1007/978-3-319-97478-1_10

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