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Interior-point algorithms for semi-infinite programming

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Abstract

In order to study the behavior of interior-point methods on very large-scale linear programming problems, we consider the application of such methods to continuous semi-infinite linear programming problems in both primal and dual form. By considering different discretizations of such problems we are led to a certain invariance property for (finite-dimensional) interior-point methods. We find that while many methods are invariant, several, including all those with the currently best complexity bound, are not. We then devise natural extensions of invariant methods to the semi-infinite case. Our motivation comes from our belief that for a method to work well on large-scale linear programming problems, it should be effective on fine discretizations of a semi-infinite problem and it should have a natural extension to the limiting semi-infinite case.

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References

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

    Google Scholar 

  2. E.L. Allgower, K. Böhmer, F.A. Potra and W.C. Rheinboldt, “A mesh-independence principle for operator equations and their discretizations,”SIAM Journal on Numerical Analysis 23 (1986) 160–169.

    Google Scholar 

  3. E.J. Anderson and P. Nash,Linear Programming in Infinite-Dimensional Spaces (Wiley, Chichester, 1987).

    Google Scholar 

  4. K.M. Anstreicher, “A monotonic projective algorithm for fractional linear programming,”Algorithmica 1 (1986) 483–498.

    Google Scholar 

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

    Google Scholar 

  6. R.E. Bixby, J.W. Gregory, I.J. Lustig, R.E. Marsten and D.F. Shanno, “Very large-scale linear programming: a case study in combining interior point and simplex methods,”Operations Research 40 (1992) 885–897.

    Google Scholar 

  7. T.M. Cavalier and A.L. Soyster, “Some computational experience and a modification of the Karmarkar algorithm,” Working Paper 85-105, Dept. of Industrial and Management Systems Engineering, Pennsylvania State University, University Park, PA, 1985.

    Google Scholar 

  8. E. Christiansen and K.O. Kortanek, “Computing material collapse displacement fields on a Cray X-MP/48 by the LP primal affine scaling algorithm,”Annals of Operations Research 22 (1990) 355–376.

    Google Scholar 

  9. I.I. Dikin, “Iterative solution of problems of linear and quadratic programming,”Doklady Academiia Nauk SSSR 174 (1967) 747–748.

    Google Scholar 

  10. M.C. Ferris and A.B. Philpott, “An interior point algorithm for semi-infinite linear programming,”Mathematical Programming 43 (1989) 257–276.

    Google Scholar 

  11. M.C. Ferris and A.B. Philpott, “On affine scaling and semi-infinite programming,”Mathematical Programming 56 (1992) 361–364.

    Google Scholar 

  12. R.M. Freund, “An analog of Karmarkar's algorithm for inequality constrained linear programs, with a ‘new’ class of projective transformations for centering a polytope,”Operations Research Letters 7 (1988) 9–13.

    Google Scholar 

  13. 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 

  14. D. Gay, “A variant of Karmarkar's linear programming algorithm for problems in standard form,”Mathematical Programming 37 (1987) 81–90.

    Google Scholar 

  15. G. de Ghellinck and J.-Ph. Vial, “A polynomial Newton method for linear programming,”Algorithmica 1 (1986) 425–453.

    Google Scholar 

  16. K. Glashoff and S.-A. Gustafson,Linear Optimization and Approximation (Springer-Verlag, New York, 1983).

    Google Scholar 

  17. D. Goldfarb and M.J. Todd, “Linear programming,” in: G.L. Nemhauser, A.H.G. Rinnooy Kan, and M.J. Todd, eds.,Handbooks in Operations Research and Management Science, Vol. 1, Optimization (North-Holland, Amsterdam, 1989), pp. 73–170.

    Google Scholar 

  18. C.C. Gonzaga, “Conical projection algorithms for linear programming,”Mathematical Programming 43 (1989) 151–173.

    Google Scholar 

  19. C.C. Gonzaga, “An algorithm for solving linear programming programs in O(n 3 L) operations,” in: N. Megiddo, ed.,Progress in Mathematical Programming, Interior Point and Related Methods (Springer-Verlag, New York, 1989), pp. 1–28.

    Google Scholar 

  20. C.C. Gonzaga, “Search directions for interior linear-programming methods,”Algorithmica 6 (1991) 153–181.

    Google Scholar 

  21. C.C. Gonzaga, “Interior point algorithms for linear programming problems with inequality constraints,”Mathematical Programming 52 (1991) 209–225.

    Google Scholar 

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

    Google Scholar 

  23. C.C. Gonzaga, “Large-steps path-following methods for linear programming, Part II: Potential reduction method,”SIAM Journal on Optimization 1 (1991) 280–292.

    Google Scholar 

  24. D. den Hertog and C. Roos, “A survey of search directions in interior point methods for linear programming,”Mathematical Programming 52 (1991) 481–509.

    Google Scholar 

  25. R. Hettich and K.O. Kortanek, “Semi-infinite programming: theory, method and applications,”SIAM Review 35 (1993) 380–429.

    Google Scholar 

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

    Google Scholar 

  27. M. Kojima, S. Mizuno and A. Yoshise, “A primal—dual interior point algorithm for linear programming,” in: N. Megiddo, ed.,Progress in Mathematical Programming, Interior Point and Related Methods (Springer-Verlag, New York, 1989), pp. 29–47.

    Google Scholar 

  28. 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 

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

    Google Scholar 

  30. K.O. Kortanek and M. Shi, “Convergence results and numerical experiments on a linear programming hybrid algorithm,”European Journal of Operational Research 32 (1987) 47–61.

    Google Scholar 

  31. E. Kranich, “Interior point methods for mathematical programming: a bibliography,” Discussion Paper 171, Institute of Economics and Operations Research, Fern University Hagen, Hagen, Germany (also available via netlib).

  32. I.J. Lustig, R.E. Marsten and D.F. Shanno, “The primal—dual interior point method on the Cray supercomputer,” in: T.F. Coleman and Y. Li, eds.,Large-Scale Numerical Optimization (SIAM, Philadelphia, 1990), pp. 70–80.

    Google Scholar 

  33. 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 

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

    Google Scholar 

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

    Google Scholar 

  36. R.C. Monteiro and I. Adler, “Interior path following primal—dual algorithms. Part II: Convex quadratic programming,”Mathematical Programming 44 (1989) 43–66.

    Google Scholar 

  37. M.J.D. Powell, “Karmarkar's algorithm: a view from nonlinear programming,”Bulletin of the Institute of Mathematics and its Applications 26 (1990) 165–181.

    Google Scholar 

  38. M.J.D. Powell, “The complexity of Karmarkar's algorithm for linear programming,” in: D.F. Griffiths and G.A. Watson, eds.,Numerical Analysis 1991, Pitman Research Notes in Mathematics 260 (Longman, Burnt Hill, UK, 1992), pp. 143–163.

    Google Scholar 

  39. M.J.D. Powell, “On the number of iterations of Karmarkar's algorithm for linear programming,”Mathematical Programming 62 (1993) 153–197.

    Google Scholar 

  40. J. Renegar, “A polynomial-time algorithm based on Newton's method for linear programming,”Mathematical Programming 40 (1988) 59–94.

    Google Scholar 

  41. C. Roos and J.-P. Vial, “A polynomial method of approximate centers for linear programming,”Mathematical Programming 54 (1992) 295–305.

    Google Scholar 

  42. G. Sonnevend, J. Stoer and G. Zhao, “On the complexity of following the central path of linear programs by linear extrapolation,”Methods of Operations Research 63 (1989) 19–31.

    Google Scholar 

  43. A. Steger, “An extension of Karmarkar's algorithm for bounded linear programming problems,” M.S. Thesis, SUNY at Stonybrook, New York (1985).

    Google Scholar 

  44. K. Tanabe, “Centered Newton method for mathematical programming,” in: M. Iri and K. Yajima, eds.,System Modelling and Optimization, Lecture Notes in Control and Information Sciences 113 (Springer-Verlag, Berlin, 1988), pp. 197–206.

    Google Scholar 

  45. M.J. Todd, “Recent developments and new directions in linear programming,” in: M. Iri and K. Tanabe, eds.,Mathematical Programming: Recent Developments and Applications (Kluwer Academic Press, Dordrecht, The Netherlands, 1989), pp. 109–157.

    Google Scholar 

  46. M.J. Todd and B.P. Burrell, “An extension of Karmarkar's algorithm for linear programming using dual variables,”Algorithmica 1 (1986) 409–424.

    Google Scholar 

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

    Google Scholar 

  48. L. Tuncel and M.J. Todd, “Asymptotic behavior of interior-point methods: a view from semi-infinite programming,” Technical Report No. 1031, School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NY, 1992.

    Google Scholar 

  49. P.M. Vaidya, “An algorithm for linear programming which requires O(((m + n)n 2 + (m+n) 1.5n)L) arithmetic operations,”Mathematical Programming 47 (1990) 175–202.

    Google Scholar 

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

    Google Scholar 

  51. H. Yamashita, “A polynomially and quadratically convergent method for linear programming,” Working paper, Mathematical Systems Institute, Inc., Tokyo, Japan, 1986.

    Google Scholar 

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

    Google Scholar 

  53. Y. Ye and M. Kojima, “Recovering optimal dual solutions in Karmarkar's polynomial algorithm for linear programming,”Mathematical Programming 39 (1987) 305–317.

    Google Scholar 

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Authors and Affiliations

  1. School of Operations Research and Industrial Engineering, College of Engineering, Cornell University, 14853-3801, Ithaca, NY, USA

    Michael J. Todd

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  1. Michael J. Todd
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Research supported in part by NSF, AFORS and ONR through NSF grant DMS-8920550.

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Todd, M.J. Interior-point algorithms for semi-infinite programming. Mathematical Programming 65, 217–245 (1994). https://doi.org/10.1007/BF01581697

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  • DOI: https://doi.org/10.1007/BF01581697

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