Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content
Springer Nature Link
Log in

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

  • Published:
Mathematical Programming Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • 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.

    Article  MATH  MathSciNet  Google Scholar 

  • J.L. Bentley,Writing Efficient Programs (Prentice-Hall, Englewood Cliffs, NJ, 1982).

    MATH  Google Scholar 

  • R.P. Brent,Algorithms for Minimization without Derivatives (Prentice-Hall, Englewood Cliffs, NJ, 1973).

    MATH  Google Scholar 

  • G.B. Dantzig,Linear Programming and Extensions (Princeton University Press, Princeton, NJ, 1963).

    MATH  Google Scholar 

  • J.J. Dongarra and E. Grosse, “Distribution of mathematical software via electronic mail,”SIGNUM Newsletter 20 (1985) 45–47.

    Google Scholar 

  • 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 

  • 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.

    Article  MATH  MathSciNet  Google Scholar 

  • 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.

    Article  MATH  Google Scholar 

  • J. Eriksson, “A note on solution of large sparse maximum entropy problems with linear equality constraints,”Mathematical Programming 18 (1980) 146–154.

    Article  MATH  MathSciNet  Google Scholar 

  • J. Eriksson, “Algorithms for entropy and mathematical programming,” Ph.D. Thesis, Linköping University, (Linköping, Sweden, 1981).

    Google Scholar 

  • 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 

  • 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 

  • 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 

  • A.V. Fiacco and G.P. McCormick,Nonlinear Programming: Sequential Unconstrained Minimization Techniques (John Wiley and Sons, New York, 1968).

    MATH  Google Scholar 

  • R. Fletcher,Practical Methods of Optimization, Volume 2 (John Wiley and Sons, Chichester, 1981).

    Google Scholar 

  • 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 

  • R. Fourer, “Solving staircase linear programs by the simplex method, 1: Inversion,”Mathematical Programming 23 (1982) 274–313.

    Article  MATH  MathSciNet  Google Scholar 

  • K.R. Frisch, “The logarithmic potential method of convex programming,” University Institute of Economics (Oslo, Norway, 1955).

    Google Scholar 

  • 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 

  • O. Garcia, “FOLPI, a forestry-oriented linear programming interpreter,” Reprint 1728, New Zealand Forest Service (Christchurch, New Zealand, 1984).

    Google Scholar 

  • D.M. Gay, “Solving sparse least-squares problems,” Presentation, Department of Operations Research, Stanford University (Stanford, CA, 1985).

    Google Scholar 

  • J.A. George and J.W.H. Liu,Computer Solution of Large Sparse Positive Definite Systems (Prentice-Hall, Englewood Cliffs, NJ, 1981).

    MATH  Google Scholar 

  • J.A. George and E. Ng, “A new release of SPARSPAK—the Waterloo sparse matrix package,”SIGNUM Newsletter 19 (1984) 9–13.

    Google Scholar 

  • 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.

    Article  MATH  MathSciNet  Google Scholar 

  • 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 

  • 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 

  • P.E. Gill, W. Murray and M.H. Wright,Practical Optimization (Academic Press, London, 1981).

    MATH  Google Scholar 

  • M.T. Heath, “Numerical methods for large sparse linear least squares problems,”SIAM Journal on Scientific and Statistical Computing 5 (1984) 497–513.

    Article  MATH  MathSciNet  Google Scholar 

  • J.K. Ho and E. Loute, “A set of staircase linear programming test problems,”Mathematical Programming 20 (1981) 245–250.

    Article  MATH  MathSciNet  Google Scholar 

  • 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.

    Article  MATH  MathSciNet  Google Scholar 

  • 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 

  • K. Jittorntrum,Sequential Algorithms in Nonlinear Programming, Ph.D. Thesis, Australian National University (Canberra, Australia, 1978).

    Google Scholar 

  • 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.

    Article  MATH  MathSciNet  Google Scholar 

  • 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.

  • N. Karmarkar, “A new polynomial-time algorithm for linear programming,”Combinatorica 4 (1984b) 373–395.

    Article  MATH  MathSciNet  Google Scholar 

  • 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.]

    MATH  MathSciNet  Google Scholar 

  • J.W.H. Liu, “Modification of the minimum-degree algorithm by multiple elimination,”ACM Transactions on Mathematical Software 11 (1985) 141–153.

    Article  MATH  Google Scholar 

  • 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 

  • 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 

  • R. Mifflin, “Convergence bounds for nonlinear programming algorithms,”Mathematical Programming 8 (1975) 251–271.

    Article  MATH  MathSciNet  Google Scholar 

  • C.B. Moler, Private communication (1985).

  • 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 

  • 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 

  • J.M. Ortega and W.C. Rheinboldt,Iterative Solution of Nonlinear Equations in Several Variables (Academic Press, New York, NY, 1970).

    MATH  Google Scholar 

  • 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.

    Article  MATH  MathSciNet  Google Scholar 

  • 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.

    Article  MathSciNet  Google Scholar 

  • 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 

  • 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 

  • J.A. Tomlin and J.S. Welch, “Formal optimization of some reduced linear programming problems,”Mathematical Programming 27 (1983) 232–240.

    MATH  MathSciNet  Google Scholar 

  • 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.

  • C.B. Tompkins, “Some methods of computational attack on programming problems, other than the simplex method,”Naval Research Logistics Quarterly 4 (1957) 95–96.

    MathSciNet  Google Scholar 

  • 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 

  • J. von Neumann, “On a maximization problem,” Manuscript, Institute for Advanced Study (Princeton, NJ, 1947).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Operations Research, Stanford University, 94305, Stanford, CA, USA

    Philip E. Gill, Walter Murray & Michael A. Saunders

  2. Ketron Incorporated, 94040, Mountain View, California, USA

    J. A. Tomlin

  3. Department of Operations Research, Stanford University, 94305, Stanford, CA, USA

    Margaret H. Wright

Authors
  1. Philip E. Gill
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Walter Murray
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Michael A. Saunders
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. J. A. Tomlin
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Margaret H. Wright
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

The research of the Stanford authors was supported by the U.S. Department of Energy Contract DE-AA03-76SF00326, PA No. DE-AS03-76ER72018; National Science Foundation Grants DCR-8413211 and ECS-8312142; the Office of Naval Research Contract N00014-85-K-0343; and the U.S. Army Research Office Contract DAAG29-84-K-0156.

The research of J.A. Tomlin was supported by Ketron, Inc. and the Office of Naval Research Contract N00014-85-C-0338.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gill, P.E., Murray, W., Saunders, M.A. et al. On projected newton barrier methods for linear programming and an equivalence to Karmarkar’s projective method. Mathematical Programming 36, 183–209 (1986). https://doi.org/10.1007/BF02592025

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02592025

Key words

Search

Navigation