Random Coordinate Descent Algorithms for Multi-Agent Convex Optimization Over Networks

In this paper, we develop randomized block-coordinate descent methods for minimizing multi-agent convex optimization problems with linearly coupled constraints over networks and prove that they obtain in expectation an ε accurate solution in at most O(1/λ 2 (Q)ϵ) iterations, where λ 2 (Q) is the sec...

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Vydáno v:	IEEE transactions on automatic control Ročník 58; číslo 8; s. 2001 - 2012
Hlavní autor:	Necoara, Ion
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	IEEE 01.08.2013
Témata:	Algorithm design and analysis Convergence Convex functions Distributed control Eigenvalues and eigenfunctions Linear programming linearly coupled constraints Optimization random coordinate descent rate of convergence resource allocation problems smooth convex optimization Vectors
ISSN:	0018-9286, 1558-2523
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Abstract	In this paper, we develop randomized block-coordinate descent methods for minimizing multi-agent convex optimization problems with linearly coupled constraints over networks and prove that they obtain in expectation an ε accurate solution in at most O(1/λ 2 (Q)ϵ) iterations, where λ 2 (Q) is the second smallest eigenvalue of a matrix Q that is defined in terms of the probabilities and the number of blocks. However, the computational complexity per iteration of our methods is much simpler than the one of a method based on full gradient information and each iteration can be computed in a completely distributed way. We focus on how to choose the probabilities to make these randomized algorithms to converge as fast as possible and we arrive at solving a sparse SDP. Analysis for rate of convergence in probability is also provided. For strongly convex functions our distributed algorithms converge linearly. We also extend the main algorithm to a more general random coordinate descent method and to problems with more general linearly coupled constraints. Preliminary numerical tests confirm that on very large optimization problems our method is much more numerically efficient than methods based on full gradient.
AbstractList	In this paper, we develop randomized block-coordinate descent methods for minimizing multi-agent convex optimization problems with linearly coupled constraints over networks and prove that they obtain in expectation an ε accurate solution in at most O(1/λ 2 (Q)ϵ) iterations, where λ 2 (Q) is the second smallest eigenvalue of a matrix Q that is defined in terms of the probabilities and the number of blocks. However, the computational complexity per iteration of our methods is much simpler than the one of a method based on full gradient information and each iteration can be computed in a completely distributed way. We focus on how to choose the probabilities to make these randomized algorithms to converge as fast as possible and we arrive at solving a sparse SDP. Analysis for rate of convergence in probability is also provided. For strongly convex functions our distributed algorithms converge linearly. We also extend the main algorithm to a more general random coordinate descent method and to problems with more general linearly coupled constraints. Preliminary numerical tests confirm that on very large optimization problems our method is much more numerically efficient than methods based on full gradient.
Author	Necoara, Ion
Author_xml	– sequence: 1 givenname: Ion surname: Necoara fullname: Necoara, Ion email: ion.necoara@acse.pub.ro organization: Dept. of Autom. Control & Syst. Eng., Univ. Politeh. Bucharest, Bucharest, Romania
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SubjectTerms	Algorithm design and analysis Convergence Convex functions Distributed control Eigenvalues and eigenfunctions Linear programming linearly coupled constraints Optimization random coordinate descent rate of convergence resource allocation problems smooth convex optimization Vectors
Title	Random Coordinate Descent Algorithms for Multi-Agent Convex Optimization Over Networks
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