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Infeasibility detection in the alternating direction method of multipliers for convex optimization. (English) Zbl 1429.90050
Summary: The alternating direction method of multipliers is a powerful operator splitting technique for solving structured optimization problems. For convex optimization problems, it is well known that the algorithm generates iterates that converge to a solution, provided that it exists. If a solution does not exist, then the iterates diverge. Nevertheless, we show that they yield conclusive information regarding problem infeasibility for optimization problems with linear or quadratic objective functions and conic constraints, which includes quadratic, second-order cone, and semidefinite programs. In particular, we show that in the limit the 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.

90C25 Convex programming
90C20 Quadratic programming
90C06 Large-scale problems in mathematical programming
90C22 Semidefinite programming
68Q25 Analysis of algorithms and problem complexity
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