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On the non-polynomiality of the relaxation method for systems of linear inequalities

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Abstract

A class of linear programs is given in which the relaxation method for inequalities, under the same operating rules as Khacian's method, is not polynomial in the length of the input. This result holds for any value of the relaxation parameter.

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References

  1. S. Agmon, “The relaxation method for linear inequalities,Canadian Journal of Mathematics 6 (1954) 382–392.

    Google Scholar 

  2. P. Gacs and L. Lovasz, “Khacian's algorithm for linear programming”, Report C.S. 750, Computer Science Department, Stanford University (Stanford, CA, 1979).

    Google Scholar 

  3. J.L. Goffin, “The relaxation method for solving systems of linear inequalities”,Mathematics of Operation Research 5 (1980) 388–414.

    Google Scholar 

  4. J.L. Goffin, “Acceleration in the relaxation method for linear inequalities and subgradient optimization”, working paper 79-10; Faculty of Management, McGill University, April 1979, to appear in the proceedings of a task force on nondifferentiable optimization held at IIASA, Laxenburg, Austria, December 1978.

  5. L.G. Khacian, “A polynomial algorithm for linear programming”,Doklady Akademii Nauk SSSR, 244 (5) (1979) 1093–1095, translated inSoviet Mathematics Doklady 20 (1) (1979) 191–194.

    Google Scholar 

  6. T. Motzkin and I.Y. Schoenberg, “The relaxation method for linear inequalities”,Canadian Journal of Mathematics 6 (1954) 393–404.

    Google Scholar 

  7. A.M. Ostrowski, “On over and under relaxation in the theory of the cyclic single step iteration”,Mathematical Tables and other Aids to Computation 7 (1953) 152–159.

    Google Scholar 

  8. C.J. Papadimitriou, “Efficient search for rationals”,Information Processing Letters 8 (1979) 1–4.

    Google Scholar 

  9. N.Z. Shor, “Convergence rate of the gradient descent method with dilatation of the space”,Kibernetika 6 (2) (1970) 80–85, translated inCybernetics 6 (2) (1970) 102–108.

    Google Scholar 

  10. N.Z. Shor, “Cut-off method with space extension on convex programming problems”,Kibernetika 13 (6) (1977) 94–95, translated inCybernetics 13 (6) (1977) 94–96.

    Google Scholar 

  11. V.A. Skokov, “Note on minimization methods using space dilatation”,Kibernetika 10 (4) (1974) 115–117, translated inCybernetics 10 (4) (1974) 689–692.

    Google Scholar 

  12. D.B. Yudin and A.S. Nemirovskii, “Informational complexity and effective methods for the solution of convex extremal problems”,Ekonomika I Mathematicheskie Metody 12 (2) (1976) 357–369, translated inMatekon 13 (3) (1977) 25–45.

    Google Scholar 

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Authors and Affiliations

  1. Faculty of Management, McGill University, Montreal, Quebec, Canada

    J. L. Goffin

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  1. J. L. Goffin
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Additional information

This research was supported in part by the D.G.E.S. (Quebec), the N.S.E.R.C. of Canada under grant A 4152, and the S.S.H.R.C. of Canada.

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Goffin, J.L. On the non-polynomiality of the relaxation method for systems of linear inequalities. Mathematical Programming 22, 93–103 (1982). https://doi.org/10.1007/BF01581028

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  • DOI: https://doi.org/10.1007/BF01581028

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