Mathematical Programming

, Volume 36, Issue 2, pp 183–209 | Cite as

On projected newton barrier methods for linear programming and an equivalence to Karmarkar’s projective method

  • Philip E. Gill
  • Walter Murray
  • Michael A. Saunders
  • J. A. Tomlin
  • Margaret H. Wright


Interest in linear programming has been intensified recently by Karmarkar’s publication in 1984 of an algorithm that is claimed to be much faster than the simplex method for practical problems. We review classical barrier-function methods for nonlinear programming based on applying a logarithmic transformation to inequality constraints. For the special case of linear programming, the transformed problem can be solved by a “projected Newton barrier” method. This method is shown to be equivalent to Karmarkar’s projective method for a particular choice of the barrier parameter. We then present details of a specific barrier algorithm and its practical implementation. Numerical results are given for several non-trivial test problems, and the implications for future developments in linear programming are discussed.

Key words

Linear programming Karmarkar’s method barrier methods 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. M. Benichou, J.M. Gauthier, G. Hentges and G. Ribière, “The efficient solution of large-scale linear programming problems—some algorithmic techniques and computational results,”Mathematical Programming. 13 (1977) 280–322.zbMATHCrossRefMathSciNetGoogle Scholar
  2. J.L. Bentley,Writing Efficient Programs (Prentice-Hall, Englewood Cliffs, NJ, 1982).zbMATHGoogle Scholar
  3. R.P. Brent,Algorithms for Minimization without Derivatives (Prentice-Hall, Englewood Cliffs, NJ, 1973).zbMATHGoogle Scholar
  4. G.B. Dantzig,Linear Programming and Extensions (Princeton University Press, Princeton, NJ, 1963).zbMATHGoogle Scholar
  5. J.J. Dongarra and E. Grosse, “Distribution of mathematical software via electronic mail,”SIGNUM Newsletter 20 (1985) 45–47.Google Scholar
  6. I.S. Duff and J.K. Reid, “MA27—a set of Fortran subroutines for solving sparse symmetric sets of linear equations,” Report AERE R-10533, Computer Science and Systems Division, AERE Harwell, (Harwell, England, 1982).Google Scholar
  7. I.S. Duff and J.K. Reid, “The multifrontal solution of indefinite sparse symmetric linear equations,”ACM Transactions on Mathematical Software 9 (1983) 302–325.zbMATHCrossRefMathSciNetGoogle Scholar
  8. S.C. Eisenstat, M.C. Gursky, M.H. Schultz and A.H. Sherman, “Yale sparse matrix package I: The symmetric codes,”International Journal of Numerical Methods in Engineering 18 (1982) 1145–1151.zbMATHCrossRefGoogle Scholar
  9. J. Eriksson, “A note on solution of large sparse maximum entropy problems with linear equality constraints,”Mathematical Programming 18 (1980) 146–154.zbMATHCrossRefMathSciNetGoogle Scholar
  10. J. Eriksson, “Algorithms for entropy and mathematical programming,” Ph.D. Thesis, Linköping University, (Linköping, Sweden, 1981).Google Scholar
  11. J. Eriksson, “An iterative primal-dual algorithm, for linear programming,” Report LiTH-MAT-R-1985-10, Department of Mathematics, Linköping University (Linköping, Sweden, 1985).Google Scholar
  12. S. Erlander, “Entropy in linear programs—an approach to planning,” Report LiTH-MAT-R-77-3, Department of Mathematics, Linköping University (Linköping, Sweden, 1977).Google Scholar
  13. A.V. Fiacco, “Barrier methods for nonlinear programming,” in: A. Holzman, ed.,Operations Research Support Methodology (Marcel Dekker, New York, NY, 1979) pp. 377–440.Google Scholar
  14. A.V. Fiacco and G.P. McCormick,Nonlinear Programming: Sequential Unconstrained Minimization Techniques (John Wiley and Sons, New York, 1968).zbMATHGoogle Scholar
  15. R. Fletcher,Practical Methods of Optimization, Volume 2 (John Wiley and Sons, Chichester, 1981).Google Scholar
  16. R. Fletcher and A.P. McCann, “Acceleration techniques for nonlinear programming,” in: R. Fletcher, ed.,Optimization (Academic Press, London, 1969) pp. 203–213.Google Scholar
  17. R. Fourer, “Solving staircase linear programs by the simplex method, 1: Inversion,”Mathematical Programming 23 (1982) 274–313.zbMATHCrossRefMathSciNetGoogle Scholar
  18. K.R. Frisch, “The logarithmic potential method of convex programming,” University Institute of Economics (Oslo, Norway, 1955).Google Scholar
  19. K.R. Frisch, “Linear dependencies and a mechanized form of the multiplex method for linear programming,” University Institute of Economics (Oslo, Norway, 1957).Google Scholar
  20. O. Garcia, “FOLPI, a forestry-oriented linear programming interpreter,” Reprint 1728, New Zealand Forest Service (Christchurch, New Zealand, 1984).Google Scholar
  21. D.M. Gay, “Solving sparse least-squares problems,” Presentation, Department of Operations Research, Stanford University (Stanford, CA, 1985).Google Scholar
  22. J.A. George and J.W.H. Liu,Computer Solution of Large Sparse Positive Definite Systems (Prentice-Hall, Englewood Cliffs, NJ, 1981).zbMATHGoogle Scholar
  23. J.A. George and E. Ng, “A new release of SPARSPAK—the Waterloo sparse matrix package,”SIGNUM Newsletter 19 (1984) 9–13.Google Scholar
  24. P.E. Gill, W. Murray, M.A. Saunders and M.H. Wright, “Sparse matrix methods in optimization,”SIAM Journal on Scientific and Statistical Computing 5 (1984) 562–589.zbMATHCrossRefMathSciNetGoogle Scholar
  25. P.E. Gill, W. Murray, M.A. Saunders and M.H. Wright, “A note on nonlinear approaches to linear programming,” Report SOL 86-7, Department of Operations Research, Stanford University (Stanford, CA, 1986a).Google Scholar
  26. P.E. Gill, W. Murray, M.A. Saunders and M.H. Wright, “Maintaining LU factors of a general sparse matrix,” Report SOL 86-8, Department of Operations Research, Stanford University (Stanford, CA, 1986b). [To appear inLinear Algebra and its Applications.]Google Scholar
  27. P.E. Gill, W. Murray and M.H. Wright,Practical Optimization (Academic Press, London, 1981).zbMATHGoogle Scholar
  28. M.T. Heath, “Numerical methods for large sparse linear least squares problems,”SIAM Journal on Scientific and Statistical Computing 5 (1984) 497–513.zbMATHCrossRefMathSciNetGoogle Scholar
  29. J.K. Ho and E. Loute, “A set of staircase linear programming test problems,”Mathematical Programming 20 (1981) 245–250.zbMATHCrossRefMathSciNetGoogle Scholar
  30. A.J. Hoffman, M. Mannos, D. Sokolowsky, and N. Wiegmann, “Computational experience in solving linear programs,”Journal of the Society for Industrial and Applied Mathematics 1 (1953) 17–33.zbMATHCrossRefMathSciNetGoogle Scholar
  31. P. Huard, “Resolution of mathematical programming with nonlinear constraints by the method of centres,” in: J. Abadie, ed.,Nonlinear Programming (North-Holland, Amsterdam, 1967) pp. 207–219.Google Scholar
  32. K. Jittorntrum,Sequential Algorithms in Nonlinear Programming, Ph.D. Thesis, Australian National University (Canberra, Australia, 1978).Google Scholar
  33. K. Jittorntrum and M.R. Osborne, “Trajectory analysis and extrapolation in barrier function methods,”Journal of Australian Mathematical Society Series B 20 (1978) 352–369.zbMATHMathSciNetCrossRefGoogle Scholar
  34. N. Karmarkar, “A new polynomial-time algorithm for linear programming,”Proceedings of the 16th Annual ACM Symposium on the Theory of Computing (1984a) 302–311.Google Scholar
  35. N. Karmarkar, “A new polynomial-time algorithm for linear programming,”Combinatorica 4 (1984b) 373–395.zbMATHCrossRefMathSciNetGoogle Scholar
  36. L.G. Khachiyan, “A polynomial algorithm in linear programming,”Doklady Akademiia Nauk SSSR Novaia Seriia 244 (1979) 1093–1096. [English translation inSoviet Mathematics Doklady 20 (1979) 191–194.]zbMATHMathSciNetGoogle Scholar
  37. J.W.H. Liu, “Modification of the minimum-degree algorithm by multiple elimination,”ACM Transactions on Mathematical Software 11 (1985) 141–153.zbMATHCrossRefGoogle Scholar
  38. I.J. Lustig, “A practical approach to karmarkar’s algorithm,” Report SOL 85-5, Department of Operations Research, Stanford University (Stanford, CA, 1985).Google Scholar
  39. R. Mifflin, “On the convergence of the logarithmic barrier function method,” in: F. Lootsma, ed.,Numerical Methods for Non-Linear Optimization (Academic Press, London, 1972) pp. 367–369.Google Scholar
  40. R. Mifflin, “Convergence bounds for nonlinear programming algorithms,”Mathematical Programming 8 (1975) 251–271.zbMATHCrossRefMathSciNetGoogle Scholar
  41. C.B. Moler, Private communication (1985).Google Scholar
  42. W. Murray and M.H. Wright, “Efficient linear search algorithms for the logarithmic barrier function,” Report SOL 76-18, Department of Operations Research, Stanford University (Stanford, CA, 1976).Google Scholar
  43. B.A. Murtagh and M.A. Saunders, “MINOS 5.0 user’s guide,” Report SOL 83-20, Department of Operations Research, Stanford University (Stanford, CA, 1983).Google Scholar
  44. J.M. Ortega and W.C. Rheinboldt,Iterative Solution of Nonlinear Equations in Several Variables (Academic Press, New York, NY, 1970).zbMATHGoogle Scholar
  45. C.C. Paige and M.A. Saunders, “LSQR: An algorithm for sparse linear equations and sparse least-squares,”ACM Transactions on Mathematical Software 8 (1982a) 43–71.zbMATHCrossRefMathSciNetGoogle Scholar
  46. C.C. Paige and M.A. Saunders, “Algorithm 583. LSQR: Sparse linear equations and least squares problems,”ACM Transactions on Mathematical Software 8 (1982b) 195–209.CrossRefMathSciNetGoogle Scholar
  47. M.J. Todd and B.P. Burrell, “An extension of Karmarkar’s algorithm for linear programming using dual variables,” Report 648, School of Operations Research and Industrial Engineering, Cornell University (Ithaca, NY, 1985).Google Scholar
  48. J.A. Tomlin, “An experimental approach to Karmarkar’s projective method for linear programming,” Manuscript, Ketron Inc. (Mountain, View, CA, 1985). [To appear inMathematical Programming Studies.]Google Scholar
  49. J.A. Tomlin and J.S. Welch, “Formal optimization of some reduced linear programming problems,”Mathematical Programming 27 (1983) 232–240.zbMATHMathSciNetGoogle Scholar
  50. C.B. Tompkins, “Projection methods in calculation,” in: H.A. Antosiewicz, ed.,Proceedings of the Second Symposium in Linear Programming (United States Air Force, Washington, DC, 1955) pp. 425–448.Google Scholar
  51. C.B. Tompkins, “Some methods of computational attack on programming problems, other than the simplex method,”Naval Research Logistics Quarterly 4 (1957) 95–96.MathSciNetGoogle Scholar
  52. R.J. Vanderbei, M.S. Meketon and B.A. Freedman, “A modification of Karmarkar’s linear programming algorithm,” Manuscript, AT&T Bell Laboratories (Holmdel, NJ, 1985).Google Scholar
  53. J. von Neumann, “On a maximization problem,” Manuscript, Institute for Advanced Study (Princeton, NJ, 1947).Google Scholar

Copyright information

© The Mathematical Programming Society, Inc. 1986

Authors and Affiliations

  • Philip E. Gill
    • 1
  • Walter Murray
    • 1
  • Michael A. Saunders
    • 1
  • J. A. Tomlin
    • 2
  • Margaret H. Wright
    • 3
  1. 1.Department of Operations ResearchStanford UniversityStanfordUSA
  2. 2.Ketron IncorporatedMountain ViewUSA
  3. 3.Department of Operations ResearchStanford UniversityStanfordUSA

Personalised recommendations