On scaling linear programs—some experimental results

We consider the possibility of preprocessing linear programs by row and column scaling in order to reduce the number of simplex iterations required for solving the programs when Dantzig's original entering variable rule is used. It is conjectured that an appropriate scaling objective should be to try to give all nonzero coefficients of the linear program the same magnitude. This objective is quantified by the introduction of different optimal scaling models. Optimality conditions are derived for each model, and based on these a number of scaling methods are developed. The scaling methods have been applied to two classes of small-size randomly generated linear programs, one dense class and one sparse. The results of these experiments are promising, and motivate further research in this direction

[1]  J. C. Dickson,et al.  A decision rule for improved efficiency in solving linear programming problems with the simplex algorithm , 1960, CACM.

[2]  D. R. Fulkerson,et al.  An algorithm for scaling matrices. , 1962 .

[3]  George B. Dantzig,et al.  Linear programming and extensions , 1965 .

[4]  L. Bregman.  The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming , 1967 .

[5]  William Orchard-Hays,et al.  Advanced Linear-Programming Computing Techniques , 1968 .

[6]  S. Kullback,et al.  Contingency tables with given marginals. , 1968, Biometrika.

[7]  A. Sluis.  Condition numbers and equilibration of matrices , 1969 .

[8]  Richard W. Hamming,et al.  Introduction to Applied Numerical Analysis. , 1971 .

[9]  J. Reid,et al.  On the Automatic Scaling of Matrices for Gaussian Elimination , 1972 .

[10]  Paula M. J. Harris.  Pivot selection methods of the Devex LP code , 1973, Math. Program..

[11]  Robert G. Jeroslow,et al.  The simplex algorithm with the pivot rule of maximizing criterion improvement , 1973, Discret. Math..

[12]  Alfred V. Aho,et al.  The Design and Analysis of Computer Algorithms , 1974 .

[13]  Åke Björck,et al.  Numerical Methods , 1995, Handbook of Marine Craft Hydrodynamics and Motion Control.

[14]  H. Crowder,et al.  Partially normalized pivot selection in linear programming , 1975 .

[15]  J. A. Tomlin,et al.  On scaling linear programming problems , 1975 .

[16]  Michel Bénichou,et al.  The efficient solution of large-scale linear programming problems—some algorithmic techniques and computational results , 1977, Math. Program..

[17]  Donald Goldfarb,et al.  A practicable steepest-edge simplex algorithm , 1977, Math. Program..

[18]  William Y. Sit,et al.  Worst case behavior of the steepest edge simplex method , 1979, Discret. Appl. Math..

[19]  M. Golitschek.  An algorithm for scaling matrices and computing the minimum cycle mean in a digraph , 1980 .

[20]  Uriel G. Rothblum,et al.  Characterizations of optimal scalings of matrices , 1980, Math. Program..

[21]  T. Elfving.  On some methods for entropy maximization and matrix scaling , 1980 .

[22]  Sven Erlander,et al.  Entropy in linear programs , 1981, Math. Program..

[23]  N. F. Stewart,et al.  Bregman's balancing method , 1981 .

[24]  Robert Fourer,et al.  Solving staircase linear programs by the simplex method, 1: Inversion , 1982, Math. Program..

[25]  Uriel G. Rothblum,et al.  A conforming decomposition theorem, a piecewise linear theorem of the alternative, and scalings of matrices satisfying lower and upper bounds , 1983, Math. Program..

[26]  Uriel G. Rothblum,et al.  Computing optimal scalings by parametric network algorithms , 1985, Math. Program..