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We consider a class of linear-programming based estimators in reconstructinga sparse signal from linear measurements. Specific formulations of thereconstruction problem considered here include Dantzig selector, basis pursuit(for the case in which the measurements contain no errors), and the fusedDantzig selector (for the case in which the underlying signal is piecewiseconstant). In spite of being estimators central to sparse signal processing andmachine learning, solving these linear programming problems for large scaleinstances remains a challenging task, thereby limiting their usage in practice.We show that classic constraint- and column-generation techniques from largescale linear programming, when used in conjunction with a commercialimplementation of the simplex method, and initialized with the solution from aclosely-related Lasso formulation, yields solutions with high efficiency inmany settings.

Computing Estimators of Dantzig Selector type via Column and Constraint Generation