Browsing by Subject "Overdetermined linear systems"

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Open Access
New characterizations of ℓ1 solutions to overdetermined systems of linear equations
(Elsevier, 1994) Madsen, K.; Nielsen H. B.; Pınar, M. Ç.
New characterizations of the ℓ1 solutions to overdetermined system of linear equations are given. The first is a polyhedral characterization of the solution set in terms of a special sign vector using a simple property of the ℓ1 solutions. The second characterization is based on a smooth approximation of the ℓ1 function using a "Huber" function. This allows a description of the solution set of the ℓ1 problem from any solution to the approximating problem for sufficiently small positive values of an approximation parameter. A sign approximation property of the Huber problem is also considered and a characterization of this property is given. © 1994.
Open Access
A penalty continuation method for the ℓ∞ solution of overdetermined linear systems
(Springer Netherlands, 1998) Pınar, M. Ç.; Elhedhli, S.
A new algorithm for the ℓ∞ 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 ℓ∞ problem. The minimizers of the quadratic penalty function generate piecewise-linear non-interior paths to the set of ℓ∞ solutions. It is shown that the entire set of ℓ∞ solutions is obtained from the paths for sufficiently small values of a scalar parameter. This leads to a finite penalty/continuation algorithm for ℓ∞ 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.