NDLI: Fast alternating linearization methods for minimizing the sum of two convex functions

Content Provider	Springer Nature Link
Author	Ma, Shiqian Goldfarb, Donald Scheinberg, Katya
Copyright Year	2012
Abstract	We present in this paper alternating linearization algorithms based on an alternating direction augmented Lagrangian approach for minimizing the sum of two convex functions. Our basic methods require at most $${O(1/\epsilon)}$$ iterations to obtain an $${\epsilon}$$ -optimal solution, while our accelerated (i.e., fast) versions of them require at most $${O(1/\sqrt{\epsilon})}$$ iterations, with little change in the computational effort required at each iteration. For both types of methods, we present one algorithm that requires both functions to be smooth with Lipschitz continuous gradients and one algorithm that needs only one of the functions to be so. Algorithms in this paper are Gauss-Seidel type methods, in contrast to the ones proposed by Goldfarb and Ma in (Fast multiple splitting algorithms for convex optimization, Columbia University, 2009) where the algorithms are Jacobi type methods. Numerical results are reported to support our theoretical conclusions and demonstrate the practical potential of our algorithms.
Ending Page	382
Page Count	34
Starting Page	349
File Format	PDF
ISSN	00255610
e-ISSN	14364646
Journal	Mathematical Programming
Issue Number	1-2
Volume Number	141
Language	English
Publisher	Springer Berlin Heidelberg
Publisher Date	2012-03-24
Publisher Place	Berlin, Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Alternating direction method Augmented Lagrangian method Optimal gradient method Variable splitting Gauss-Seidel method Theoretical, Mathematical and Computational Physics Alternating linearization method Convex programming Analysis of algorithms and problem complexity Mathematical Methods in Physics Convex optimization Calculus of Variations and Optimal Control; Optimization Mathematics of Computing Numerical Analysis Mathematical programming methods Peaceman-Rachford method Combinatorics
Content Type	Text
Resource Type	Article
Subject	Mathematics Software

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