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.
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.
J.L. Bentley,Writing Efficient Programs (Prentice-Hall, Englewood Cliffs, NJ, 1982).
R.P. Brent,Algorithms for Minimization without Derivatives (Prentice-Hall, Englewood Cliffs, NJ, 1973).
G.B. Dantzig,Linear Programming and Extensions (Princeton University Press, Princeton, NJ, 1963).
J.J. Dongarra and E. Grosse, “Distribution of mathematical software via electronic mail,”SIGNUM Newsletter 20 (1985) 45–47.
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).
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.
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.
J. Eriksson, “A note on solution of large sparse maximum entropy problems with linear equality constraints,”Mathematical Programming 18 (1980) 146–154.
J. Eriksson, “Algorithms for entropy and mathematical programming,” Ph.D. Thesis, Linköping University, (Linköping, Sweden, 1981).
J. Eriksson, “An iterative primal-dual algorithm, for linear programming,” Report LiTH-MAT-R-1985-10, Department of Mathematics, Linköping University (Linköping, Sweden, 1985).
S. Erlander, “Entropy in linear programs—an approach to planning,” Report LiTH-MAT-R-77-3, Department of Mathematics, Linköping University (Linköping, Sweden, 1977).
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.
A.V. Fiacco and G.P. McCormick,Nonlinear Programming: Sequential Unconstrained Minimization Techniques (John Wiley and Sons, New York, 1968).
R. Fletcher,Practical Methods of Optimization, Volume 2 (John Wiley and Sons, Chichester, 1981).
R. Fletcher and A.P. McCann, “Acceleration techniques for nonlinear programming,” in: R. Fletcher, ed.,Optimization (Academic Press, London, 1969) pp. 203–213.
R. Fourer, “Solving staircase linear programs by the simplex method, 1: Inversion,”Mathematical Programming 23 (1982) 274–313.
K.R. Frisch, “The logarithmic potential method of convex programming,” University Institute of Economics (Oslo, Norway, 1955).
K.R. Frisch, “Linear dependencies and a mechanized form of the multiplex method for linear programming,” University Institute of Economics (Oslo, Norway, 1957).
O. Garcia, “FOLPI, a forestry-oriented linear programming interpreter,” Reprint 1728, New Zealand Forest Service (Christchurch, New Zealand, 1984).
D.M. Gay, “Solving sparse least-squares problems,” Presentation, Department of Operations Research, Stanford University (Stanford, CA, 1985).
J.A. George and J.W.H. Liu,Computer Solution of Large Sparse Positive Definite Systems (Prentice-Hall, Englewood Cliffs, NJ, 1981).
J.A. George and E. Ng, “A new release of SPARSPAK—the Waterloo sparse matrix package,”SIGNUM Newsletter 19 (1984) 9–13.
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.
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).
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.]
P.E. Gill, W. Murray and M.H. Wright,Practical Optimization (Academic Press, London, 1981).
M.T. Heath, “Numerical methods for large sparse linear least squares problems,”SIAM Journal on Scientific and Statistical Computing 5 (1984) 497–513.
J.K. Ho and E. Loute, “A set of staircase linear programming test problems,”Mathematical Programming 20 (1981) 245–250.
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.
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.
K. Jittorntrum,Sequential Algorithms in Nonlinear Programming, Ph.D. Thesis, Australian National University (Canberra, Australia, 1978).
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.
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.
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.]
J.W.H. Liu, “Modification of the minimum-degree algorithm by multiple elimination,”ACM Transactions on Mathematical Software 11 (1985) 141–153.
I.J. Lustig, “A practical approach to karmarkar’s algorithm,” Report SOL 85-5, Department of Operations Research, Stanford University (Stanford, CA, 1985).
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.
R. Mifflin, “Convergence bounds for nonlinear programming algorithms,”Mathematical Programming 8 (1975) 251–271.
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).
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).
J.M. Ortega and W.C. Rheinboldt,Iterative Solution of Nonlinear Equations in Several Variables (Academic Press, New York, NY, 1970).
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.
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.
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).
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.]
J.A. Tomlin and J.S. Welch, “Formal optimization of some reduced linear programming problems,”Mathematical Programming 27 (1983) 232–240.
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.
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).
J. von Neumann, “On a maximization problem,” Manuscript, Institute for Advanced Study (Princeton, NJ, 1947).
Author information
Authors and Affiliations
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
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
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02592025