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

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- by Ronald D. Armstrong and James P. Godfrey PDF
- Math. Comp.
**33**(1979), 289-300 Request permission

## 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 problemβs 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.## References

- Nabih N. Abdelmalek,
*On the discrete linear $L_{1}$ approximation and $L_{1}$ solutions of overdetermined linear equations*, J. Approximation Theory**11**(1974), 38β53. MR**388750**, DOI 10.1016/0021-9045(74)90037-9 - Nabih N. Abdelmalek,
*An efficient method for the discrete linear $L_{1}$ approximation problem*, Math. Comp.**29**(1975), 844β850. MR**378354**, DOI 10.1090/S0025-5718-1975-0378354-8
R. D. ARMSTRONG & E. L. FROME, "Least absolute value estimators for one-way and two-way tables," - 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.Β 3, 555β565. MR**438660**, DOI 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), 839β848. MR**339449**, DOI 10.1137/0710069 - A. Charnes,
*Optimality and degeneracy in linear programming*, Econometrica**20**(1952), 160β170. MR**56264**, DOI 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), 138β151. MR**73914**, DOI 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," - 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), 118β132. MR**134453**, DOI 10.1080/01621459.1958.10501430 - G. F. McCormick and V. A. Sposito,
*Using the $L_{2}$-estimator in $L_{1}$-estimation*, SIAM J. Numer. Anal.**13**(1976), no.Β 3, 337β343. MR**448808**, DOI 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), 257β261. MR**101168**, DOI 10.1002/nav.3800050306 - Harvey M. Wagner,
*Linear programming techniques for regression analysis*, J. Amer. Statist. Assoc.**54**(1959), 206β212. MR**130753**, DOI 10.1080/01621459.1959.10501506

*Naval Res. Logist. Quart*. (To appear.)

*Comm. Statist.*, v. B6(4), 1977.

## Additional Information

- © Copyright 1979 American Mathematical Society
- Journal: Math. Comp.
**33**(1979), 289-300 - MSC: Primary 90C05
- DOI: https://doi.org/10.1090/S0025-5718-1979-0525398-2
- MathSciNet review: 0525398