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Infeasibility detection in the alternating direction method of multipliers for convex optimization

Goran Banjac (goran.banjac***at***eng.ox.ac.uk)
Paul Goulart (paul.goulart***at***eng.ox.ac.uk)
Bartolomeo Stellato (bartolomeo.stellato***at***eng.ox.ac.uk)
Stephen Boyd (boyd***at***stanford.edu)

Abstract: The alternating direction method of multipliers (ADMM) is a powerful operator splitting technique for solving structured optimization problems. For convex optimization problems, it is well known that the iterates generated by ADMM converge to a solution provided that it exists. If a solution does not exist, then the ADMM iterates do not converge. Nevertheless, we show that the ADMM iterates yield conclusive information regarding problem infeasibility for a wide class of convex optimization problems including both quadratic and conic programs. In particular, we show that in the limit the ADMM iterates either satisfy a set of first-order optimality conditions or produce a certificate of either primal or dual infeasibility. Based on these results, we propose termination criteria for detecting primal and dual infeasibility in ADMM.

Keywords: convex optimization, alternating direction method of multipliers, infeasibility detection, conic programming

Category 1: Convex and Nonsmooth Optimization (Convex Optimization )

Category 2: Linear, Cone and Semidefinite Programming

Citation:

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Entry Submitted: 06/02/2017
Entry Accepted: 06/02/2017
Entry Last Modified: 06/05/2017

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