### 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.

Original language | English (US) |
---|---|

Pages (from-to) | 183-209 |

Number of pages | 27 |

Journal | Mathematical Programming |

Volume | 36 |

Issue number | 2 |

DOIs | |

State | Published - Jun 1986 |

### Fingerprint

### Keywords

- barrier methods
- Karmarkar's method
- Linear programming

### ASJC Scopus subject areas

- Mathematics(all)
- Software
- Engineering(all)

### Cite this

*Mathematical Programming*,

*36*(2), 183-209. https://doi.org/10.1007/BF02592025

**On projected newton barrier methods for linear programming and an equivalence to Karmarkar's projective method.** / Gill, Philip E.; Murray, Walter; Saunders, Michael A.; Tomlin, J. A.; Wright, Margaret.

Research output: Contribution to journal › Article

*Mathematical Programming*, vol. 36, no. 2, pp. 183-209. https://doi.org/10.1007/BF02592025

}

TY - JOUR

T1 - On projected newton barrier methods for linear programming and an equivalence to Karmarkar's projective method

AU - Gill, Philip E.

AU - Murray, Walter

AU - Saunders, Michael A.

AU - Tomlin, J. A.

AU - Wright, Margaret

PY - 1986/6

Y1 - 1986/6

N2 - 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.

AB - 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.

KW - barrier methods

KW - Karmarkar's method

KW - Linear programming

UR - http://www.scopus.com/inward/record.url?scp=0022808235&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0022808235&partnerID=8YFLogxK

U2 - 10.1007/BF02592025

DO - 10.1007/BF02592025

M3 - Article

VL - 36

SP - 183

EP - 209

JO - Mathematical Programming

JF - Mathematical Programming

SN - 0025-5610

IS - 2

ER -