### Abstract

Adler and Beling considered the linear programming problem over the real algebraic numbers. They obtained the necessary bounds in terms of a notion of size and dimension needed to justify its polynomial-time solvability, using the ellipsoid method and under some models of computation. Based on a better notion of size than that used by Adler and Beling, we first reduce the feasibility problem in linear programming to some canonical problems preserving its size and its constraint-matrix dimensions. For these canonical problems as well as for the matrix scaling problem, shown to be a more general problem than linear programming, we obtain the necessary bounds; demonstrate simple rounding schemes; justify the applicability of two polynomial-time interior-point algorithms under some models of computation; describe a method for solving a system of linear equations over the algebraic numbers which is a subroutine within these interior-point algorithms under an input model; and give an alternative method to the traditional duality-based approach for the conversion of a general linear programming problem into a feasibility problem.

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

Pages (from-to) | 283-306 |

Number of pages | 24 |

Journal | Linear Algebra and Its Applications |

Volume | 262 |

Issue number | 1-3 |

State | Published - Sep 1 1997 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory
- Numerical Analysis
- Geometry and Topology
- Discrete Mathematics and Combinatorics

### Cite this

*Linear Algebra and Its Applications*,

*262*(1-3), 283-306.

}

*Linear Algebra and Its Applications*, vol. 262, no. 1-3, pp. 283-306.

**On linear programming and matrix scaling over the algebraic numbers.** / Kalantari, Bahman; Emamy-K, M. R.

Research output: Contribution to journal › Article

TY - JOUR

T1 - On linear programming and matrix scaling over the algebraic numbers

AU - Kalantari, Bahman

AU - Emamy-K, M. R.

PY - 1997/9/1

Y1 - 1997/9/1

N2 - Adler and Beling considered the linear programming problem over the real algebraic numbers. They obtained the necessary bounds in terms of a notion of size and dimension needed to justify its polynomial-time solvability, using the ellipsoid method and under some models of computation. Based on a better notion of size than that used by Adler and Beling, we first reduce the feasibility problem in linear programming to some canonical problems preserving its size and its constraint-matrix dimensions. For these canonical problems as well as for the matrix scaling problem, shown to be a more general problem than linear programming, we obtain the necessary bounds; demonstrate simple rounding schemes; justify the applicability of two polynomial-time interior-point algorithms under some models of computation; describe a method for solving a system of linear equations over the algebraic numbers which is a subroutine within these interior-point algorithms under an input model; and give an alternative method to the traditional duality-based approach for the conversion of a general linear programming problem into a feasibility problem.

AB - Adler and Beling considered the linear programming problem over the real algebraic numbers. They obtained the necessary bounds in terms of a notion of size and dimension needed to justify its polynomial-time solvability, using the ellipsoid method and under some models of computation. Based on a better notion of size than that used by Adler and Beling, we first reduce the feasibility problem in linear programming to some canonical problems preserving its size and its constraint-matrix dimensions. For these canonical problems as well as for the matrix scaling problem, shown to be a more general problem than linear programming, we obtain the necessary bounds; demonstrate simple rounding schemes; justify the applicability of two polynomial-time interior-point algorithms under some models of computation; describe a method for solving a system of linear equations over the algebraic numbers which is a subroutine within these interior-point algorithms under an input model; and give an alternative method to the traditional duality-based approach for the conversion of a general linear programming problem into a feasibility problem.

UR - http://www.scopus.com/inward/record.url?scp=0038913455&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0038913455&partnerID=8YFLogxK

M3 - Article

VL - 262

SP - 283

EP - 306

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

IS - 1-3

ER -