Minimax Theory and its Applications 01 (2016), No. 1, 029--049
Copyright Heldermann Verlag 2016
An Inertial Alternating Direction Method of Multipliers
Radu Ioan Bot
Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria
radu.bot@univie.ac.at
Ernö Robert Csetnek
Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria
ernoe.robert.csetnek@univie.ac.at
In the context of convex optimization problems in Hilbert spaces, we induce inertial effects into the classical ADMM numerical scheme and obtain in this way so-called inertial ADMM algorithms, the convergence properties of which we investigate into detail. To this aim we make use of the inertial version of the Douglas-Rachford splitting method for monotone inclusion problems recently introduced by R. I. Bot, E. R. Csetnek and C. Hendrich [Inertial Douglas-Rachford splitting for monotone inclusion problems, arXiv:1403.3330v2 (2014)], in the context of concomitantly solving a convex minimization problem and its Fenchel dual. The convergence of both sequences of the generated iterates and of the objective function values is addressed. We also show how the obtained results can be extended to the treating of convex minimization problems having as objective a finite sum of convex functions.
Keywords: Inertial ADMM algorithm, inertial Douglas-Rachford splitting, maximally monotone operator, resolvent, subdifferential, convex optimization, Fenchel duality.
MSC: 47H05, 65K05, 90C25
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