A penalty continuation method for the ∓∞ solution of overdetermined linear systems

A new algorithm for the ∓_∞ solution of overdetermined linear systems is given. The algorithm is based on the application of quadratic penalty functions to a primal linear programming formulation of the ∓_∞ problem. The minimizers of the quadratic penalty function generate piecewise-linear non-interior paths to the set of ∓_∞ solutions. It is shown that the entire set of ∓_∞ solutions is obtained from the paths for sufficiently small values of a scalar parameter. This leads to a finite penalty/continuation algorithm for ∓_∞ problems. The algorithm is implemented and extensively tested using random and function approximation problems. Comparisons with the Barrodale-Phillips simplex based algorithm and the more recent predictor-corrector primal-dual interior point algorithm are given. The results indicate that the new algorithm shows a promising performance on random (non-function approximation) problems.