Journal
IEEE TRANSACTIONS ON AUTOMATIC CONTROL
Volume 61, Issue 4, Pages 892-904Publisher
IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC
DOI: 10.1109/TAC.2015.2448011
Keywords
Alternating direction method of multipliers (ADMM); consensus algorithms; convergence rate; distributed optimization; linear convergence
Funding
- French Defense Agency (DGA)
- Telecom/Eurecom Carnot Institute
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Consider a set of N agents seeking to solve distributively the minimization problem inf(x) Sigma(N)(n=1) f(n)(x) where the convex functions f(n) are local to the agents. The popular Alternating Direction Method of Multipliers has the potential to handle distributed optimization problems of this kind. We provide a general reformulation of the problem and obtain a class of distributed algorithms which encompass various network architectures. The rate of convergence of our method is considered. It is assumed that the infimum of the problem is reached at a point x(star), the functions fn are twice differentiable at this point and Sigma del(2) f(n)(x(star)) > 0 in the positive definite ordering of symmetric matrices. With these assumptions, it is shown that the convergence to the consensus x(star) is linear and the exact rate is provided. Application examples where this rate can be optimized with respect to the ADMM free parameter rho are also given.
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