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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Vydáno v:	Journal of global optimization Ročník 77; číslo 1; s. 53 - 73
Hlavní autoři:	Colombo, Tommaso, Sagratella, Simone
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	New York Springer US 01.05.2020 Springer Springer Nature B.V
Témata:	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
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Abstract	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.
AbstractList	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. 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 [Formula omitted]-approximate solutions of the problem and a detailed analysis of how the parameters of the algorithm influence the value of the approximating parameter [Formula omitted]. 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. 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.
Audience	Academic
Author	Sagratella, Simone Colombo, Tommaso
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Cites_doi	10.1007/BF00939685 10.1109/TSP.2016.2637314 10.1109/TSP.2015.2399858 10.1007/978-3-642-99789-1_13 10.1137/S1052623497321894 10.1007/978-1-4899-0289-4_7 10.1007/s10288-007-0054-4 10.1016/j.ejor.2013.05.049 10.1007/s10107-016-1034-2 10.1007/s10589-014-9686-4 10.1137/1.9781611973365 10.1137/0312021 10.1007/s11590-010-0218-6 10.1007/s10957-008-9489-9 10.1007/s10589-018-9987-0 10.1007/s10589-017-9927-4 10.1007/s10898-014-0236-5 10.1007/s10589-007-9044-x 10.1007/s00186-016-0565-x 10.1109/TSP.2016.2637317 10.1007/s10288-018-0378-2 10.1137/1.9781611971309 10.1007/s10287-008-0090-3 10.1080/0233193031000079856 10.1109/CAMSAP.2017.8313161 10.1109/ICASSP.2014.6853715 10.1109/IJCNN.2016.7727590 10.1137/1.9781611972801.22
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Snippet	Distributed and parallel algorithms have been frequently investigated in the recent years, in particular in applications like machine learning. Nonetheless,...
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SubjectTerms	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
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Title	Distributed algorithms for convex problems with linear coupling constraints
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