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A polynomial projection algorithm for linear feasibility problems

Mathematical Programming volume 153, pages 687–713 (2015)Cite this article

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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References

  1. Agmon, Sh.: The relaxation method for linear inequalities. Can. J. Math. 6, 382–392 (1954)

  2. Chubanov, S.: A strongly polynomial algorithm for linear systems having a binary solution. Math. Program. 134, 533–570 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  3. Dantzig, G.: An \(\varepsilon \)-precise feasible solution to a linear program with a convexity constraint in \(1/\varepsilon ^2\) iterations independent of problem size. Tech. rep. SOL 92–5 (1992)

  4. Edmonds, J.: Systems of distinct representatives and linear algebra. J. Res. Nat. Bur. Stand. 71 B, 241–245 (1967)

    MathSciNet  Article  Google Scholar 

  5. Goffin, J.L.: The relaxation method for solving systems of linear inequalities. Math. Oper. Res. 5, 388–414 (1980)

    MathSciNet  Article  MATH  Google Scholar 

  6. Goffin, J.L.: On the non-polynomiality of the relaxation method for systems of linear inequalities. Math. Program. 22, 93–103 (1982)

    MathSciNet  Article  MATH  Google Scholar 

  7. Hager, W.W.: Updating the inverse of a matrix. SIAM Rev. 31, 221–239 (1989)

    MathSciNet  Article  MATH  Google Scholar 

  8. Karmarkar, N.: A new polynomial-time algorithm for linear programming. Combinatorica 4, 353–395 (1984)

    MathSciNet  Article  Google Scholar 

  9. Khachiyan, L.G.: A polynomial algorithm in linear programming. Dokl. Akad. Nauk SSSR 244 (English translation: Soviet Math. Dokl. 20, 191–194) (1979)

  10. Mehrotra, S.: On the implementation of a primal-dual interior point method. SIAM J. Optim. 2, 575–601 (1992)

    MathSciNet  Article  MATH  Google Scholar 

  11. Motzkin, Th., Schoenberg, I.J.: The relaxation method for linear inequalities. Can. J. Math. 6, 393–404 (1954)

  12. Renegar, J.: A polynomial-time algorithm, based on Newton’s method, for linear programming. Math. Program. 40, 59–93 (1988)

    MathSciNet  Article  MATH  Google Scholar 

  13. Roos, C.: Speeding up Chubanov’s method for solving a homogeneous inequality system. Optimization Online, Report 2013–100, TU Delft, NL, September (2013)

  14. Stiemke, E.: Über positive Lösungen homogener linearer Gleichungen. Math. Ann. 76, 340–342 (1915)

    MathSciNet  Article  MATH  Google Scholar 

  15. Todd, M.: The many facets of linear programming. Math. Program. 91, 417–436 (2002)

    MathSciNet  Article  MATH  Google Scholar 

  16. Zhang, Y.: Solving Large-Scale Linear Programs by Interior-point Methods Under the MATLAB Environment. Technical Report TR96-01, Department of Mathematics and Statistics, University of Maryland, Baltimore County, Baltimore, MD (1995)

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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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Authors and Affiliations

  1. Institute of Information Systems at the University of Siegen, Siegen, Germany

    Sergei Chubanov

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  1. Sergei Chubanov
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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