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Limiting behavior of weighted central paths in linear programming

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Abstract

We study the limiting behavior of the weighted central paths{(x(μ), s(μ))} μ > 0 in linear programming at bothμ = 0 andμ = ∞. We establish the existence of a partition (B ∞,N ∞) of the index set { 1, ⋯,n } such thatx i(μ) → ∞ ands j (μ) → ∞ asμ → ∞ fori ∈ B ∞, andj ∈ N ∞, andx N∞ (μ),s B∞ (μ) converge to weighted analytic centers of certain polytopes. For allk ⩾ 1, we show that thekth order derivativesx (k) (μ) ands (k) (μ) converge whenμ → 0 andμ → ∞. Consequently, the derivatives of each order are bounded in the interval (0, ∞). We calculate the limiting derivatives explicitly, and establish the surprising result that all higher order derivatives (k ⩾ 2) converge to zero whenμ → ∞.

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References

  1. 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.

    Google Scholar 

  2. K.M. Anstreicher, “Linear programming and the Newton barrier flow,”Mathematical Programming 41 (1988) 363–373.

    Google Scholar 

  3. D.A. Bayer and J.C. Lagarias, “The nonlinear geometry of linear programming, I. affine and projective scaling trajectories, II. Legendre transform coordinates and central trajectories,”Transactions of the American Mathematical Society 314 (1989) 499–581.

    Google Scholar 

  4. A. Ben-Israel and T.N.E. Greville,Generalized Inverses: Theory and Applications (John Wiley and Sons, New York, 1974).

    Google Scholar 

  5. A.V. Fiacco and G.P. McCormick,Nonlinear Programming: Sequential Unconstrained Minimization Techniques (John Wiley and Sons, New York, 1968). Republished by SIAM (Philadelphia) 1990.

    Google Scholar 

  6. A.J. Goldman and A.W. Tucker, “Theory of linear programming,” in: H.W. Kuhn and A.W. Tucker, eds.,Linear Inequalities and Related Systems, Annals of Mathematical Studies 38 (Princeton University Press, Princeton, New Jersey, 1956) pp 53–97.

    Google Scholar 

  7. I.S. Gradshteyn and I.M. Ryzhik,Table of Integrals, Series, and Products, corrected and enlargened edition (Academic Press, New York, 1980).

    Google Scholar 

  8. O. Güler, “Existence of interior points and interior paths in nonlinear monotone complementarity problems,”Mathematics of Operations Research 18 (1993) 128–147.

    Google Scholar 

  9. O. Güler, C. Roos, T. Terlaky and J.-Ph. Vial, “Interior point approach to the theory of linear programming,” Technical Report 1992.3, Faculté des SES, Université de Genève (Genève, Switzerland, 1992).

    Google Scholar 

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

    Google Scholar 

  11. M. Kojima, S. Mizuno and A. Yoshise, “A polynomial-time algorithm for a class of linear complementarity problems,”Mathematical Programming 44 (1989) 1–26.

    Google Scholar 

  12. M. Kojima, S. Mizuno and A. Yoshise, “An O\((\sqrt {nL} )\) iteration potential reduction algorithm for linear complementarity problem,”Mathematical Programming 50 (1991) 331–342.

    Google Scholar 

  13. L. McLinden, “An analogue of Moreau's proximation theorem, with applications to the nonlinear complementarity problem,”Pacific Journal of Mathematics 88 (1980) 101–161.

    Google Scholar 

  14. L. McLinden, “The complementarity problem for maximal monotone multifunctions,” in: R.W. Cottle, F. Giannessi and J.L. Lions, eds.,Variational Inequalities and Complementarity Problems (John Wiley and Sons, New York, 1980) pp 251–270.

    Google Scholar 

  15. L. McLinden, “Polyhedral extensions of some theorems of linear programming,”Mathematical Programming 22 (1982) 162–176.

    Google Scholar 

  16. N. Megiddo, “Pathways to the optimal set in linear programming,” in: N. Megiddo, ed.,Progress in Mathematical Programming — Interior Point and Related Methods (Springer Verlag, New York, 1989) pp. 131–158.

    Google Scholar 

  17. N. Megiddo and M. Shub, “Boundary behavior of interior point algorithms in linear programming,”Mathematics of Operations Research 14 (1989) 97–146.

    Google Scholar 

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

    Google Scholar 

  19. S. Mehrotra, “Asymptotic convergence in the generalized predictor—corrector method,” Technical Report, Department of IE and MS, Northwestern University (Evanston, Illinois, 1992).

    Google Scholar 

  20. S. Mizuno, M.J. Todd and Y. Ye, “On adaptive-step primal-dual interior-point algorithms for linear programming,”Mathematics of Operations Research 18 (1993) 964–981.

    Google Scholar 

  21. R.D.C. Monteiro and I. Adler, “Interior path following primal-dual algorithms. Part I: Linear programming,”Mathematical Programming 44 (1989) 27–42.

    Google Scholar 

  22. Gy. Sonnevend, “An ‘analytic centre’ for polyhedrons and new classes of global algorithms for linear (smooth, convex) programming,” in:Lecture Notes in Control and Information Sciences (Springer Verlag, New York, 1985) pp. 866–876.

    Google Scholar 

  23. A.C. Williams, “Complementarity theorems in linear programming,”SIAM Review 12 (1970) 135–137.

    Google Scholar 

  24. C. Witzgall, P.T. Boggs and P.D. Domich, “On the convergence behavior of trajectories for linear programming,”Contemporary Mathematics 114 (1990) 161–187.

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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Author notes

  1. Osman Güler

    Present address: Department of Mathematics and Statistics, University of Maryland Baltimore County, 21228, Baltimore, MD, USA

Authors and Affiliations

  1. Faculty of Technical Mathematics and Computer Science, Delft University of Technology, Mekelweg 4, 2628 CD, Delft, The Netherlands

    Osman Güler

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  1. Osman Güler
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Güler, O. Limiting behavior of weighted central paths in linear programming. Mathematical Programming 65, 347–363 (1994). https://doi.org/10.1007/BF01581702

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  • DOI: https://doi.org/10.1007/BF01581702

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