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A strongly polynomial-time algorithm for the strict homogeneous linear-inequality feasibility problem

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Abstract

A strongly polynomial-time algorithm is proposed for the strict homogeneous linear-inequality feasibility problem in the positive orthant, that is, to obtain \(x\in \mathbb {R}^n\), such that \(Ax > 0\), \(x> 0\), for an \(m\times n\) matrix \(A\), \(m\ge n\). This algorithm requires \(O(p)\) iterations and \(O(m^2(n+p))\) arithmetical operations to ensure that the distance between the solution and the iteration is \(10^{-p}\). No matrix inversion is needed. An extension to the non-homogeneous linear feasibility problem is presented.

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Acknowledgments

I appreciate my wife Sandra and Mestre Gonçalves immensely for non-technical help and I would like to thank the anonymous referees for their helpful comments.

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  1. PESC/COPPE - UFRJ, Centro de Tecnologia, Cidade Universitária, Ilha do Fundão, C. P. 68511, Rio de Janeiro, 21941-972, Brazil

    Paulo Roberto Oliveira

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  1. Paulo Roberto Oliveira
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Correspondence to Paulo Roberto Oliveira.

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Paulo Roberto Oliveira was partially supported by CNPq, Brazil.

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Oliveira, P.R. A strongly polynomial-time algorithm for the strict homogeneous linear-inequality feasibility problem. Math Meth Oper Res 80, 267–284 (2014). https://doi.org/10.1007/s00186-014-0480-y

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  • DOI: https://doi.org/10.1007/s00186-014-0480-y

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