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Superlinear convergence of the affine scaling algorithm

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Abstract

In this paper we show that a variant of the long-step affine scaling algorithm (with variable stepsizes) is two-step superlinearly convergent when applied to general linear programming (LP) problems. Superlinear convergence of the sequence of dual estimates is also established. For homogeneous LP problems having the origin as the unique optimal solution, we also show that 2/3 is a sharp upper bound on the (fixed) stepsize that provably guarantees that the sequence of primal iterates converge to the optimal solution along a unique direction of approach. Since the point to which the sequence of dual estimates converge depend on the direction of approach of the sequence of primal iterates, this result gives a plausible (but not accurate) theoretical explanation for why 2/3 is a sharp upper bound on the (fixed) stepsize that guarantees the convergence of the dual estimates.

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References

  1. I. Adler, N.K. Karmarkar, M.G.C. Resende and G. Veiga, “Data structures and programming techniques for the implementation of Karmarkar's algorithm,”ORSA Journal on Computing 1 (1989) 84–106.

    MATH  Google Scholar 

  2. I. Adler, N.K. Karmarkar, M.G.C. Resende and G. Veiga, “An implementation of Karmarkar's algorithm for linear programming,”Mathematical Programming 44 (1989) 297–335. (Errata inMathematical Programming 50 (1991) 415.)

    Article  MATH  MathSciNet  Google Scholar 

  3. I. Adler and R.D.C. Monteiro, “Limiting behavior of the affine scaling continuous trajectories for linear programming problems,”Mathematical Programming 50 (1991) 29–51.

    Article  MATH  MathSciNet  Google Scholar 

  4. E.R. Barnes, “A variation on Karmarkar's algorithm for solving linear programming problems,”Mathematical Programming 36 (1986) 174–182.

    MATH  MathSciNet  Google Scholar 

  5. Y.-C. Cheng, D.J. Houck, J.-M. Liu, M.S. Meketon, L. Slutsman, R.J. Vanderbei and P. Wang, “The AT& T KORBX System”,AT&T Technical Journal 68 (3) (1989) 7–19.

    MathSciNet  Google Scholar 

  6. I.I. Dikin, “Iterative solution of problems of linear and quadratic programming,”Doklady Akademii Nauk SSSR 174 (1967) 747–748. (Translated in:Soviet Mathematics Doklady 8 (1967) 674–675.)

    MathSciNet  Google Scholar 

  7. I.I. Dikin, “On the convergence of an iterative process,”Upravlyaemye Sistemi 12 (1974) 54–60 (In Russian).

    Google Scholar 

  8. I.I. Dikin, “The convergence of dual variables,” Technical Report Siberian Energy Institute (Irkutsk, Russia, December 1991).

  9. I.I. Dikin, “Determination of the interior point of one system of linear inequalities,”Kibernetica and System Analysis 1 (1992).

  10. I.I. Dikin and V.I. Zorkaltsev,Iterative Solutions of Mathematical Programming Problems (Nauka, Novosibirsk, USSR, 1980).

    Google Scholar 

  11. R. Fletcher,Practical Methods of Optimization, 2nd edition (Wiley, New York, 1987).

    MATH  Google Scholar 

  12. C.C. Gonzaga, “Convergence of the large step, primal affine-scaling algorithm for primal nondegenerate linear programs,” Technical Report, ES-230/90, Dept. of Systems Engineering and Computer Science, COPPE Federal University of Rio de Janeiro (Rio de Janeiro, Brazil, Sept. 1990).

    Google Scholar 

  13. L.A. Hall and R.J. Vanderbei, “Two-thirds is sharp for affine scaling,”Operations Research Letters 13 (1993) 197–201.

    Article  MATH  MathSciNet  Google Scholar 

  14. A.J. Hoffman, “On approximate solutions of systems of linear inequalities,”Journal of Research of the National Bureau of Standards 49 (1952) 263–265.

    MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  16. N. Megiddo and M. Shub, “Boundary behavior of interior point algorithms in linear programming,”Mathematics of Operations Research 14 (1989) 97–114.

    Article  MATH  MathSciNet  Google Scholar 

  17. S. Mehrotra, “Quadratic convergence in a primal-dual method,”Mathematics of Operations Research 18 (1993) 741–751.

    MATH  MathSciNet  Google Scholar 

  18. C.L. Monma and A.J. Morton, “Computational experience with the dual affine variant of Karmarkar's method for linear programming,”Operations Research Letters 6 (1987) 261–267.

    Article  MATH  MathSciNet  Google Scholar 

  19. R.D.C. Monteiro and T. Tsuchiya and Y. Wang, “A simplified global convergence proof of the affine scaling algorithm,”Annals of Operations Research 47 (1993) 443–482.

    Article  MathSciNet  Google Scholar 

  20. M. Muramatsu and T. Tsuchiya, “Convergence analysis of the projective scaling algorithm, based on a long-step homogeneous affine scaling algorithm,”Mathematical Programming 72, (1996) 291–305.

    Article  MathSciNet  Google Scholar 

  21. M.G.C. Resende and G. Veiga, “An efficient implementation of a network interior point method.” in: D.S. Johnson and C.C. McGeoch, eds.,Network Flows and Matching: First DIMACS Implementation Challenge, DIMACS Series on Discrete Mathematics and Theoretical Computer Science, Vol. 12 (American Mathematical Society, 1993) pp. 299–348.

  22. L. Sinha and B. Freedman and N. Karmarkar and A. Putcha and K. Ramakrishnan, “Overseas network planning,”Proc. Third Internat. Network Planning Sysmposium—Networks'86, (IEEE Communications Society, held on June 1–6, 1986, Tarpon Springs, FL, 1986) 121–124.

  23. M. Todd, “A low complexity interior-point algorithm for linear programming,”SIAM Journal on Optimization 2 (1992) 198–209.

    Article  MATH  MathSciNet  Google Scholar 

  24. M. Todd and J.-P. Vial, “Todd's complexity interior-point algorithm is a predictor-corrector path-following algorithm,”Operations Research Letters 11 (1992) 199–207.

    Article  MATH  MathSciNet  Google Scholar 

  25. P. Tseng and Z.Q. Luo, “On the convergence of the affine-scaling algorithm,”Mathematical Programming 56 (1992) 301–319.

    Article  MathSciNet  Google Scholar 

  26. T. Tsuchiya, “Global convergence of the affine-scaling methods for degenerate linear programming problems,”Mathematical Programming 52 (1991) 377–404.

    Article  MATH  MathSciNet  Google Scholar 

  27. T. Tsuchiya, “Quadratic convergence of Iri and Imai's method for degenerate linear programming problems,”Journal of Optimization Theory and Applications 87 (3) (1995) 703–726.

    Article  MATH  MathSciNet  Google Scholar 

  28. T. Tsuchiya, “Global convergence property of the affine scaling method for primal degenerate linear programming problems,”Mathematics of Operations Research 17 (3) (1992) 527–557.

    MATH  MathSciNet  Google Scholar 

  29. T. Tsuchiya and M. Muramatsu, “Global convergence of a long-step affine scaling algorithm for degenerate linear programming problems,”SIAM Journal on Optimization 5 (3) (1995) 525–551.

    Article  MATH  MathSciNet  Google Scholar 

  30. R.J. Vanderbei and J.C. Lagarias, “I.I. Dikin's convergence result for the affine-scaling algorithm,”Contemporary Mathematics 114 (1990) 109–119.

    MathSciNet  Google Scholar 

  31. R.J. Vanderbei, M.S. Meketon and B.A. Freedman, “A modification of Karmarkar's linear programming algorithm,”Algorithmica 1 (4) (1986) 395–407.

    Article  MATH  MathSciNet  Google Scholar 

  32. C. Witzgall, P.T. Boggs and P.D. Domich, “On the convergence behavior of trajectories for linear programming,”Contemporary Mathematics 114 (1990) 109–119.

    MathSciNet  Google Scholar 

  33. Y. Ye, O. Güler, R.A. Tapia and Y. Zhang, “A quadratically convergent\(O(\sqrt n L)\)-iteration algorithm for linear programming,”Mathematical Programming 59 (1993) 151–162.

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. The Institute of Statistical Mathematics, 4-6-7 Minami-Azabu, Minato-Ku, 106, Tokyo, Japan

    T. Tsuchiya

  2. School of Industrial and Systems Engineering, Georgia Institute of Technology, 30332, Atlanta, GA, USA

    R. D. C. Monteiro

Authors
  1. T. Tsuchiya
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  2. R. D. C. Monteiro
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Correspondence to T. Tsuchiya.

Additional information

The work of this author was based on research supported by the Overseas Research Scholars of the Ministry of Education, Science and Culture of Japan, 1992.

The work of this author was based on research supported by the National Science Foundation (NSF) under grant DDM-9109404 and the Office of Naval Research (ONR) under grant N00014-93-1-0234. This work was done while the second author was a faculty member of the Systems and Industrial Engineering Department at the University of Arizona.

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Tsuchiya, T., Monteiro, R.D.C. Superlinear convergence of the affine scaling algorithm. Mathematical Programming 75, 77–110 (1996). https://doi.org/10.1007/BF02592206

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  • DOI: https://doi.org/10.1007/BF02592206

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