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New complexity results for the Iri-Imai method

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Abstract

In this paper, we show that the number of main iterations required by the Iri-Imai algorithm to solve a linear programming problem isO(nL). Moreover, we show that a modification of this algorithm requires only\(\mathcal{O}(\sqrt {nL} )\) main iterations. In this modification, we measure progress by means of a primal-dual potential function.

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References

  1. K.M. Anstreicher, A combined phase I-phase II scaled potential algorithm for linear programming, Mathematical Programming 52(1991)429–439.

    Google Scholar 

  2. I.I. Dikin, Iterative solution of problems of linear and quadratic programming, Soviet Mathematics Doklady 8(1967)674–675.

    Google Scholar 

  3. R.M. Freund, Polynomial-time algorithms for linear programming based only on primal scaling and projected gradients of a potential function, Mathematical Programming 51(1991)203–222.

    Google Scholar 

  4. R.M. Freund, A potential-function reduction algorithm for solving a linear program directly from an infeasible “warm start”, Mathematical Programming 52(1991)441–466.

    Google Scholar 

  5. C.C. Gonzaga, Polynomial affine algorithms for linear programming, Mathematical Programming 49(1990)7–21.

    Google Scholar 

  6. C.C. Gonzaga, Interior point algorithms for linear programming problems with inequality constraints, Mathematical Programming 52(1991)209–226.

    Google Scholar 

  7. C.C. Gonzaga, Path-following methods for linear programming, SIAM Review 34(1992)167–224.

    Google Scholar 

  8. P. Huard, Resolution of mathematical programming with nonlinear constraints by the method of centres, in:Nonlinear Programming, ed. J. Abadie (North-Holland, Amsterdam, 1967) pp. 207–219.

    Google Scholar 

  9. H. Imai, On the polynomiality of the multiplicative penalty function method for linear programming and related inscribed ellipsoids, IEICE Transactions E74(1991)669–671.

    Google Scholar 

  10. M. Iri and H. Imai, A multiplicative barrier function method for linear programming, Algorithmica 1(1986)455–482.

    Google Scholar 

  11. M. Iri, A Proof of the polynomiality of the Iri-Imai Method, Journal of Complexity 9(1993) 269–290.

    Google Scholar 

  12. N. Karmarkar, A new polynomial-time algorithm for linear programming, Combinatorica 4 (1984)373–395.

    Google Scholar 

  13. J. Renegar, A polynomial-time algorithm, based on Newton's method, for linear programming, Mathematical Programming 40(1988)59–93.

    Google Scholar 

  14. Gy. Sonnevend, An “analytical centre” for polyhedrons and new classes of global algorithms for linear (smooth, convex) programming, in:Lecture Notes in Control and Information Sciences 84, eds. A. Prekopa, J. Szelezsan and B. Strazicky (Springer, 1986) pp. 866–875.

  15. J.F. Sturm, The scaling potential reduction method, Master's Thesis. Department of Econometrics, University of Groningen, The Netherlands (1993).

    Google Scholar 

  16. J.F. Sturm and S. Zhang, Potential reduction method for harmonically convex programming. Journal of Optimization Theory and Applications 84(1995)181–205.

    Google Scholar 

  17. K. Tanabe, Centered Newton method for mathematical programming, in:Proceedings of the 13th IFIP Conference, eds. M. Iri and K. Yajima, Tokyo (1987) pp. 197–206.

  18. M.J. Todd and Y. Ye, A centered projective algorithm for linear programming, Mathematics of Operations Research 15(1990)508–529.

    Google Scholar 

  19. T. Tsuchiya and M. Muramutsu, Global convergence of the long-step affine scaling algorithm for degenerate linear programming problems, Research Memorandum 423, The Institute of Statistical Mathematics, Tokyo, Japan (1992).

    Google Scholar 

  20. H. Yamashita, A polynomially and quadratically convergent method for linear programming, Manuscript, Mathematical Systems Institute, Inc., Tokyo, Japan (1986).

    Google Scholar 

  21. Y. Ye, AnO(n 3 L) potential reduction algorithm for linear programming, Mathematical Programming 50(1991)239–258.

    Google Scholar 

  22. S. Zhang and M. Shi, On the polynomiality of Iri and Imai's new algorithm for linear programming, Journal of Qinhua University 28(1988)121–126.

    Google Scholar 

  23. S. Zhang, Convergence property of the Iri-Imai algorithm for some smooth convex programming problems, Journal of Optimization Theory and Applications 82(1994)121–138.

    Google Scholar 

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Authors and Affiliations

  1. Tinbergen Institute, Erasmus University, Rotterdam, The Netherlands

    Jos F. Sturm

  2. Econometric Institute, Erasmus University, Rotterdam, The Netherlands

    Shuzhong Zhang

Authors
  1. Jos F. Sturm
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  2. Shuzhong Zhang
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Sturm, J.F., Zhang, S. New complexity results for the Iri-Imai method. Ann Oper Res 62, 539–564 (1996). https://doi.org/10.1007/BF02206829

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