ℓ1 solution of linear inequalities

The numerical solution of a possibly inconsistent system of linear inequalities in the ℓ1 sense is considered. The non-differentiable ℓ1 norm minimization problem is approximated by a piecewise quadratic Huber smooth function. A continuation algorithm is designed to find an ℓ1 solution of the inequality system. In the case where the linear inequality system is consistent, a solution is obtained by solving any smoothed problem. Otherwise, the algorithm is shown to terminate in a finite number of iterations. We also consider an alternative smoothing scheme which shares similar properties with the first one, but results in an improved computational performance of the continuation algorithm on inconsistent systems. Numerical experiments are conducted to test the efficiency of the algorithm