Distributed algorithms for convex problems with linear coupling constraints

Distributed and parallel algorithms have been frequently investigated in the recent years, in particular in applications like machine learning. Nonetheless, only a small subclass of the optimization algorithms in the literature can be easily distributed, for the presence, e.g., of coupling constrain...

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Bibliographic Details
Published in:Journal of global optimization Vol. 77; no. 1; pp. 53 - 73
Main Authors: Colombo, Tommaso, Sagratella, Simone
Format: Journal Article
Language:English
Published: New York Springer US 01.05.2020
Springer
Springer Nature B.V
Subjects:
Algorithms
Analysis
Computer Science
Coupling
Dependent variables
Iterative methods
Machine learning
Mathematical optimization
Mathematics
Mathematics and Statistics
Nonlinear programming
Operations Research/Decision Theory
Optimization
Parameters
Production scheduling
Real Functions
Lagrangian methods
Parallel algorithms
Distributed algorithms
Nonlinear optimization
ISSN:0925-5001, 1573-2916
Online Access:Get full text
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Summary:Distributed and parallel algorithms have been frequently investigated in the recent years, in particular in applications like machine learning. Nonetheless, only a small subclass of the optimization algorithms in the literature can be easily distributed, for the presence, e.g., of coupling constraints that make all the variables dependent from each other with respect to the feasible set. Augmented Lagrangian methods are among the most used techniques to get rid of the coupling constraints issue, namely by moving such constraints to the objective function in a structured, well-studied manner. Unfortunately, standard augmented Lagrangian methods need the solution of a nested problem by needing to (at least inexactly) solve a subproblem at each iteration, therefore leading to potential inefficiency of the algorithm. To fill this gap, we propose an augmented Lagrangian method to solve convex problems with linear coupling constraints that can be distributed and requires a single gradient projection step at every iteration. We give a formal convergence proof to at least ε -approximate solutions of the problem and a detailed analysis of how the parameters of the algorithm influence the value of the approximating parameter ε . Furthermore, we introduce a distributed version of the algorithm allowing to partition the data and perform the distribution of the computation in a parallel fashion.
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ISSN:0925-5001
1573-2916
DOI:10.1007/s10898-019-00792-z