Dadush, Daniel, Végh, László A. and Zambelli, Giacomo
(2019)
*Rescaling algorithms for linear conic feasibility.*
Mathematics of Operations Research.
ISSN 0364-765X
(In Press)

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## Abstract

We propose simple polynomial-time algorithms for two linear conic feasibility problems. For a matrix A ∈ Rm×n, the kernel problem requires a positive vector in the kernel of A, and the image problem requires a positive vector in the image of AT. Both algorithms iterate between simple first order steps and rescaling steps. These rescalings improve natural geometric potentials. If Goffin’s condition measure ρA is negative, then the kernel problem is feasible and the worst-case complexity of the kernel algorithm is O ﰁ(m3n + mn2) log |ρA|−1ﰂ; if ρA > 0, then the image problem is feasible and the image algorithm runs in time O ﰁm2n2 log ρ−1ﰂ. We also extend the image algorithm to A the oracle setting. We address the degenerate case ρA = 0 by extending our algorithms to find maximum support nonnegative vectors in the kernel of A and in the image of A⊤. In this case the running time bounds are expressed in the bit-size model of computation: for an input matrix A with integer entries and total encoding length L, the maximum support kernel algorithm runs in time O ﰁ(m3n + mn2)Lﰂ, while the maximum support image algorithm runs in time Oﰁm2n2Lﰂ. The standard linear pro- gramming feasibility problem can be easily reduced to either maximum support problems, yielding polynomial-time algorithms for Linear Programming.

Item Type: | Article |
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Additional Information: | © 2019 INFORMS |

Divisions: | Mathematics |

Subjects: | Q Science > QA Mathematics |

Date Deposited: | 13 May 2019 13:24 |

Last Modified: | 20 Jun 2019 23:10 |

URI: | http://eprints.lse.ac.uk/id/eprint/100778 |

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