Two linear programming algorithms for the linear discrete 𝐿₁ norm problem

Armstrong, Ronald D.; Godfrey, James P.

doi:10.1090/S0025-5718-1979-0525398-2

Two linear programming algorithms for the linear discrete $L_{1}$ norm problem

Authors: Ronald D. Armstrong and James P. Godfrey
Journal: Math. Comp. 33 (1979), 289-300
MSC: Primary 90C05
DOI: https://doi.org/10.1090/S0025-5718-1979-0525398-2
MathSciNet review: 0525398
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Computational studies by several authors have indicated that linear programming is currently the most efficient procedure for obtaining ${L_1}$, norm estimates for a discrete linear problem. However, there are several linear programming algorithms, and the "best" approach may depend on the problem’s structure (e.g., sparsity, triangularity, stability). In this paper we shall compare two published simplex algorithms, one referred to as primal and the other referred to as dual, and show that they are conceptually equivalent.

Nabih N. Abdelmalek, On the discrete linear $L_{1}$ approximation and $L_{1}$ solutions of overdetermined linear equations, J. Approximation Theory 11 (1974), 38–53. MR 388750, DOI https://doi.org/10.1016/0021-9045%2874%2990037-9
Nabih N. Abdelmalek, An efficient method for the discrete linear $L_{1}$ approximation problem, Math. Comp. 29 (1975), 844–850. MR 378354, DOI https://doi.org/10.1090/S0025-5718-1975-0378354-8

R. D. ARMSTRONG & E. L. FROME, "Least absolute value estimators for one-way and two-way tables," Naval Res. Logist. Quart. (To appear.)

Ronald D. Armstrong and John W. Hultz, An algorithm for a restricted discrete approximation problem in the $L_{1}$ norm, SIAM J. Numer. Anal. 14 (1977), no. 3, 555–565. MR 438660, DOI https://doi.org/10.1137/0714034
I. Barrodale and F. D. K. Roberts, An improved algorithm for discrete $l_{1}$ linear approximation, SIAM J. Numer. Anal. 10 (1973), 839–848. MR 339449, DOI https://doi.org/10.1137/0710069
A. Charnes, Optimality and degeneracy in linear programming, Econometrica 20 (1952), 160–170. MR 56264, DOI https://doi.org/10.2307/1907845
A. Charnes and W. W. Cooper, Management models and industrial applications of linear programming, John Wiley & Sons, Inc., New York-London, 1961. MR 0157773
A. Charnes and W. W. Cooper, Optimal estimation of executive compensation by linear programming, Management Sci. 1 (1955), 138–151. MR 73914, DOI https://doi.org/10.1287/mnsc.1.2.138

J. GILSINN, K. HOFFMAN, R. H. F. JACKSON, E. LEYENDECKER, P. SAUNDERS & D. SHIER, "Methodology and analysis for comparing discrete linear ${L_1}$ approximation codes," Comm. Statist., v. B6(4), 1977.

G. Hadley, Linear programming, Addison-Wesley Series in Industrial Management, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1962. MR 0135622
Otto J. Karst, Linear curve fitting using least deviations, J. Amer. Statist. Assoc. 53 (1958), 118–132. MR 134453
G. F. McCormick and V. A. Sposito, Using the $L_{2}$-estimator in $L_{1}$-estimation, SIAM J. Numer. Anal. 13 (1976), no. 3, 337–343. MR 448808, DOI https://doi.org/10.1137/0713030
Michel Simonnard, Linear programming, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1966. Translated from the French by William S. Jewell. MR 0201195
Harvey M. Wagner, The dual simplex algorithm for bounded variables, Naval Res. Logist. Quart. 5 (1958), 257–261. MR 101168, DOI https://doi.org/10.1002/nav.3800050306
Harvey M. Wagner, Linear programming techniques for regression analysis, J. Amer. Statist. Assoc. 54 (1959), 206–212. MR 130753