Hacijan's algorithm in VAXIMA: improvements and difficulties
Improvements to Hacijan's polynomial-time algorithm in linear programming are described. New iteration formulas are given and proved. The algorithm has been implemented in a version of the MACSYMA system on the VAX-11 known as VAXIMA. The implementation is described. Data from computer experiments are included and they indicate (a) the substantial improvement over the original algorithm and (b) the difficulty in making any Hacijan-type algorithm competitive with the simplex algorithm.
KeywordsVolume Reduction Feasible Region Feasible Point Simplex Algorithm Decimal Digit
Unable to display preview. Download preview PDF.
- 1.N. Anderson and P. Wang, "MACSYMA experiments with a modified Khachiyan-type algorithm in linear programming", ACM SIGSAM Bulletin, February 1980, pp. 8–13.Google Scholar
- 3.G. B. Dantzig, "Khachiyan's Algorithm: a comment by George Dantzig", SIAM NEWS, Vol. 13, No. 5, October 1980.Google Scholar
- 4.G. B. Dantzig, "Expected Number of Steps of the Simplex Method for a Linear Program with a Convexity Constraint", Technical Report SOL 80-3, Systems Optimization Laboratory, Department of Operations Research, Stanford University, March 1980.Google Scholar
- 5.R. J. Fateman, "The Macsyma ‘Big-Floating-Point" Arithmetic System", Proceedings, ACM SYMSAC '76 Conference, Yorktown Heights, N.Y., August 10–12, 1976, pp. 209–213.Google Scholar
- 6.J. K. Foderaro and R. J. Fateman, "Characterization of VAX Macsyma", Proceedings, ACM SYMSAC '81 Conference, Snowbird, Utah, August 5–7, 1981, pp. 14–19.Google Scholar
- 7.P. Gill, W. Murray, M. Saunders and M. Wright, "A Numerical Investigation of Ellipsoid Algorithms for Large-Scale Linear Programming", Technical Report, SOL 80-27, Department of Operations Research, Stanford University, 1980.Google Scholar
- 8.L. G. Hacijan, "A Polynomial Algorithm in Linear Programming", Soviet Math. Dokl. Vol. 20(1979), No. 1.Google Scholar
- 9.J. Hopcraft, "Recent Directions in Algorithmic Research", Springer Lecture Notes in Computer Science, Vol. 104, 1981, pp. 123–134.Google Scholar
- 10.R. M. Karp, "Complexity of Computer Computations", (Proceedings, Sympos., IBM Thomas J. Watson Research Ctr., Yorktown Heights, N.Y., 1972) Plenum Press, New York, 1972, p. 85.Google Scholar
- 11.V. Klee and G. J. Minty, "How Good is the Simplex Algorithm?". Inequalities III, ed. O. Shisha, Academic Press, N.Y., 1972.Google Scholar
- 12.P. Wolfe, "A Bibliography for the Ellipsoid Algorithm", IBM Research Center Report No. 8237 (Apr. 1980).Google Scholar
- 13.Math Lab Group, MACSYMA Reference Manual, version 9, LCS, MIT, Cambridge, Mass. 1977.Google Scholar