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Superlinear and quadratic convergence of primal-dual interior-point methods for linear programming revisited

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Abstract

Recently, Zhang, Tapia, and Dennis (Ref. 1) produced a superlinear and quadratic convergence theory for the duality gap sequence in primal-dual interior-point methods for linear programming. In this theory, a basic assumption for superlinear convergence is the convergence of the iteration sequence; and a basic assumption for quadratic convergence is nondegeneracy. Several recent research projects have either used or built on this theory under one or both of the above-mentioned assumptions. In this paper, we remove both assumptions from the Zhang-Tapia-Dennis theory.

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References

  1. Zhang, Y., Tapia, R. A., andDennis, J. E.,On the Superlinear and Quadratic Convergence of Primal-Dual Interior-Point Linear Programming Algorithms, Rice University, Department of Mathematical Sciences, Technical Report TR90-6, 1990.

  2. Megiddo, N.,Pathways to the Optimal Set in Linear Programming, Progress in Mathematical Programming, Interior-Point and Related Methods, Edited by N. Megiddo, Springer-Verlag, New York, New York, pp. 131–158, 1989.

    Google Scholar 

  3. McShane, K. A., Monma, C. L., andShanno, D. F.,An Implementation of a Primal-Dual Interior-Point Method for Linear Programming, ORSA Journal of Computing, Vol. 1, pp. 70–83, 1989.

    Google Scholar 

  4. Lustig, I. J., Marsten, R. E., andShanno, D. F.,Computational Experience with a Primal-Dual Interior-Point Method for Linear Programming, Princeton University, Department of Civil Engineering and Operations Research, Technical Report SOR 89-17, 1989.

  5. Kojima, M., Mizuno, S., andYoshise, A.,A Primal-Dual Interior-Point Method for Linear Programming, Progress in Mathematical Programming, Interior-Point and Related Methods, Edited by N. Megiddo, Springer-Verlag, New York, New York, pp. 29–47, 1989.

    Google Scholar 

  6. Monteiro, R. C., andAdler, I.,Interior Path-Following Primal-Dual Algorithms, Part 1: Linear Programming, Mathematical Programming, Vol. 44, pp. 27–41, 1989.

    Google Scholar 

  7. Todd, M. J., andYe, Y.,A Centered Projective Algorithm for Linear Programming, Mathematics of Operations Research, Vol. 15, pp. 508–529, 1990.

    Google Scholar 

  8. Zhang, Y., andTapia, R. A.,A Quadratically Convergent Polynomial Primal-Dual Interior-Point Algorithm for Linear Programming, Rice University, Department of Mathematical Sciences, Technical Report TR90-40, 1990.

  9. Ye, Y., Tapia, R. A., andZhang, Y.,A Superlinearly Convergent O(√nL)-Iteration Algorithm for Linear Programming, Rice University, Department of Mathematical Sciences, Technical Report TR91-22, 1991.

  10. McShane, K.,A Superlinearly Convergent O(√nL)-Iteration Primal-Dual Linear Programming Algorithm, Unpublished Manuscript, 2534 Villanova Drive, Vienna, Virginia, 1991.

  11. Ye, Y., Güler, O., Tapia, R. A., andZhang, Y.,A Quadratically Convergent O(√nL)-Iteration Algorithm for Linear Programming, Rice University, Department of Mathematical Sciences, Technical Report TR91-26, 1991.

  12. Zhang, Y., Tapia, R. A., andPotra, F.,On the Superlinear Convergence of Interior-Point Algorithms for a General Class of Problems, Rice University, Department of Mathematical Sciences, Technical Report TR90-9, 1990.

  13. Kojima, M., Kurita, Y., andMizuno, S.,Large-Step Interior-Point Algorithms for Linear Complementarity Problems, Tokyo Institute of Technology, Department of Information Sciences, Research Report on Information Sciences B-243, 1991.

  14. Potra, F. A., Tapia, R. A., andZhang, Y.,An Interior-Point Method for Linear Complementarity Problems with Polynomial Complexity and Superlinear Convergence, Rice University, Department of Mathematical Sciences, Technical Report TR91-23, 1991.

  15. Tapia, R. A., Zhang, Y., andYe, Y.,On the Convergence of the Iteration Sequence in Primal-Dual Interior-Point Methods, Rice University, Department of Mathematical Sciences, Technical Report TR91-24, 1991.

  16. Güler, O., andYe, Y.,Convergence Behavior of Some Interior-Point Algorithms, University of Iowa, Department of Management Sciences, Working Paper 91-4, 1991.

  17. El-Bakry, A., Tapia, R. A., andZhang, Y.,On the Use of Indicators in Identifying Zero Variables for Interior-Point Methods, Rice University, Department of Mathematical Sciences, Technical Report TR91-15, 1991.

  18. Mizuno, S., Todd, M. J., andYe, Y.,On Adaptive-Step Primal-Dual Interior-Point Algorithms for Linear Programming, Cornell University, School of Operations Research and Industrial Engineering, Technical Report 944, 1990.

  19. Ye, Y.,Improving the Asymptotic Convergence of Interior-Point Algorithms for Linear Programming, University of Iowa, Department of Management Sciences, Working Paper 91-15, 1991.

  20. Mehrotra, S., andYe, Y.,On Finding the Optimal Facet of Linear Programs, Northwestern University, Department of Industrial Engineering and Management Science, Technical Report 91-10, 1991.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Maryland, Baltimore County Campus, Baltimore, Maryland

    Y. Zhang (Assistant Professor)

  2. Department of Mathematical Sciences and Center for Research in Parallel Computation, Rice University, Houston, Texas

    R. A. Tapia (Professor)

Authors
  1. Y. Zhang
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  2. R. A. Tapia
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Additional information

Dedicated to the Memory of Magnus R. Hestenes, 1906–1991

This research was supported in part by NSF Cooperative Agreement CCR-88-09615 and was initiated while the first author was at Rice University as a Visiting Member of the Center for Research in Parallel Computation.

The authors thank Yinyu Ye for constructive comments and discussions concerning this material.

This author was supported in part by NSF Grant DMS-91-02761 and DOE Grant DE-FG05-91-ER25100.

This author was supported in part by AFOSR Grant 89-0363, DOE Grant DE-FG05-86-ER25017, and ARO Grant 9DAAL03-90-G-0093.

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Zhang, Y., Tapia, R.A. Superlinear and quadratic convergence of primal-dual interior-point methods for linear programming revisited. J Optim Theory Appl 73, 229–242 (1992). https://doi.org/10.1007/BF00940179

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