Linear Convergence of First- and Zeroth-Order Primal-Dual Algorithms for Distributed Nonconvex Optimization

This article considers the distributed nonconvex optimization problem of minimizing a global cost function formed by a sum of local cost functions by using local information exchange. We first consider a distributed first-order primal-dual algorithm. We show that it converges sublinearly to a statio...

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Bibliographic Details
Published in:IEEE transactions on automatic control Vol. 67; no. 8; pp. 4194 - 4201
Main Authors: Yi, Xinlei, Zhang, Shengjun, Yang, Tao, Chai, Tianyou, Johansson, Karl H.
Format: Journal Article
Language:English
Published: New York IEEE 01.08.2022
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subjects:
Algorithms
Convergence
Convex functions
Convexity
Cost function
Costs
Distributed nonconvex optimization
First order algorithms
first-order algorithm
Laplace equations
linear convergence
Lyapunov methods
Optimization
primal-dual algorithm
Technological innovation
zeroth-order algorithm
zerothorder algorithm
ISSN:0018-9286, 1558-2523, 1558-2523
Online Access:Get full text
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Summary:This article considers the distributed nonconvex optimization problem of minimizing a global cost function formed by a sum of local cost functions by using local information exchange. We first consider a distributed first-order primal-dual algorithm. We show that it converges sublinearly to a stationary point if each local cost function is smooth and linearly to a global optimum under an additional condition that the global cost function satisfies the Polyak-Łojasiewicz condition. This condition is weaker than strong convexity, which is a standard condition for proving linear convergence of distributed optimization algorithms, and the global minimizer is not necessarily unique. Motivated by the situations where the gradients are unavailable, we then propose a distributed zeroth-order algorithm, derived from the considered first-order algorithm by using a deterministic gradient estimator, and show that it has the same convergence properties as the considered first-order algorithm under the same conditions. The theoretical results are illustrated by numerical simulations.
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ISSN:0018-9286
1558-2523
1558-2523
DOI:10.1109/TAC.2021.3108501