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A polynomial projection algorithm for linear feasibility problems

Abstract

We propose a polynomial algorithm for linear feasibility problems. The algorithm represents a linear problem in the form of a system of linear equations and non-negativity constraints. Then it uses a procedure which either finds a solution for the respective homogeneous system or provides the information based on which the algorithm rescales the homogeneous system so that its feasible solutions in the unit cube get closer to the vector of all ones. In a polynomial number of calls to the procedure the algorithm either proves that the original system is infeasible or finds a solution in the relative interior of the feasible set.

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Acknowledgments

I wish to thank two anonymous referees and Kees Roos (the Delft University of Technology) for their detailed comments and for suggesting several important improvements. This work is supported by the DFG (Deutsche Forschungsgemeinschaft) research grant CH 1133/1-1.

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Correspondence to Sergei Chubanov.

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Chubanov, S. A polynomial projection algorithm for linear feasibility problems. Math. Program. 153, 687–713 (2015). https://doi.org/10.1007/s10107-014-0823-8

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  • DOI: https://doi.org/10.1007/s10107-014-0823-8

Keywords

  • Linear programming
  • Systems of linear inequalities
  • Polynomial algorithm

Mathematics Subject Classification

  • 90C05 Linear programming