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Potential Reduction Algorithms

  • Chapter
Interior Point Methods of Mathematical Programming

Part of the book series: Applied Optimization ((APOP,volume 5))

Abstract

Potential reduction algorithms have a distinguished role in the area of interior point methods for mathematical programming. Karmarkar’s [44] algorithm for linear programming, whose announcement in 1984 initiated a torrent of research into interior point methods, used three key ingredients: a non-standard linear programming formulation, projective transformations, and a potential function with which to measure the progress of the algorithm. It was quickly shown that the non-standard formulation could be avoided, and evennally algorithms were developed that eliminated the projective transformations, but retained the use of a potential function. It is then fair to say that the only really essential element of Karmarkar’s analysis was the potential function. Further modifications to Karmarkar’s original potential function gave rise to potential reduction algorithms having the state-of-the-art theoretical complexity of O\(\left( {\sqrt n L} \right)\) iterations, to solve a standard form linear program with n variables, and integer data with total bit size L. In the classical optimization literature, potential reduction algorithms are most closely related to Huard’s [39] “method of centres,” see also Fiacco and McCormick [21, Section 7.2]. However, Karmarkar’s use of a potential function to facilitate acomplexity, as opposed to convergence analysis, was completely novel

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References

  1. F. Alizadeh, “Interior point methods in semidefinite programming with applications to combinatorial optimization,”SIAM J. Opt. 5 (1995) 13–51.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. K.M. Anstreicher, “The worst-case step in Karmarkar’s algorithm,”Math. Oper. Res. 14 (1989) 294–302.

    Article  MathSciNet  MATH  Google Scholar 

  4. K.M. Anstreicher, “A combined phase I-phase II projective algorithm for linear programming,”Math. Prog. 43 (1989) 209–223.

    Article  MathSciNet  MATH  Google Scholar 

  5. K.M. Anstreicher, “A standard form variant, and safeguarded linesearch, for the modified Karmarkar algorithm,”Math. Prog. 47 (1990) 337–351.

    Article  MathSciNet  MATH  Google Scholar 

  6. K.M. Anstreicher, “Dual ellipsoids and degeneracy in the projective algorithm for linear programming,”Contemporary Mathematics 114 (1990) 141–149.

    MathSciNet  Google Scholar 

  7. K.M. Anstreicher, “On the performance of Karmarkar’s algorithm over a sequence of iterations,”SIAM J. Opt. 1 (1991) 22–29.

    Article  MathSciNet  MATH  Google Scholar 

  8. K.M. Anstreicher, “A combined phase I — phase II scaled potential algorithm for linear programming,”Math. Prog. 52 (1991) 429–439.

    Article  MathSciNet  MATH  Google Scholar 

  9. K.M. Anstreicher, “On monotonicity in the scaled potential algorithm for linear programming,”Linear Algebra Appl. 152 (1991) 223–232.

    Article  MathSciNet  MATH  Google Scholar 

  10. K.M. Anstreicher, “Strict monotonicity and improved complexity in the stan¬dard form projective algorithm for linear programming,”Math. Prog. 62 (1993) 517–535.

    Article  MathSciNet  MATH  Google Scholar 

  11. K.M. Anstreicher, “Large step volumetric potential reduction algorithms for linear programming,” to appear inAnnals of O.R. (1996).

    Google Scholar 

  12. K.M. Anstreicher and R.A. Bosch, “Long steps in anO(n 3 L) algorithm for linear programming,”Math. Prog. 54 (1992) 251–265.

    Article  MathSciNet  MATH  Google Scholar 

  13. K.M. Anstreicher and P. Watteyne, “A family of search directions for Karmarkar’s algorithm,”Operations Research 41 (1993), 759–767.

    Article  MathSciNet  MATH  Google Scholar 

  14. M.D. Asic, V.V. Kovacevic-Vujcic, and M.D. Radosavljevcic-Nikolic, “A note on limiting behavior of the projective and the affine rescaling algorithms”Contemporary Mathematics 114 (1990) 151–157.

    Google Scholar 

  15. D. Bayer and J.C. Lagarias, “Karmarkar’s linear programming algorithm and Newton’s method,”Math. Prog. 50 (1991) 291–330.

    Article  MathSciNet  MATH  Google Scholar 

  16. R.A. Bosch, “On Mizuno’s rank one updating algorithm for linear programming,”SIAM J. Opt. 3 (1993) 861–867.

    Article  MathSciNet  MATH  Google Scholar 

  17. R.A. Bosch and K.M. Anstreicher, “On partial updating in a potential reduction linear programming algorithm of Kojima, Mizuno, and Yoshise,”Algorithmica 9 (1993) 184–197.

    Article  MathSciNet  MATH  Google Scholar 

  18. R.A. Bosch and K.M. Anstreicher, “A partial updating algorithm for linear programs with many more variables than constraints,”Optimization Methods and Software 4 (1995) 243–257.

    Article  Google Scholar 

  19. R.W. Cottle, J.-S. Pang, and R.E. Stone,The Linear Complementarity Problem ( Academic Press, Boston, 1992 ).

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  21. A.V. Fiacco and G.P. McCormick,Nonlinear Programming, Sequential Unconstrained Minimization Techniques, (John Wiley, New York, 1968); reprinted asClassics in Applied Mathematics Vol. 4, (SIAM, Philadelphia, 1990).

    MATH  Google Scholar 

  22. C. Fraley, “Linear updates for a single-phase projective method,”O.R. Letters 9 (1990) 169–174.

    Article  MathSciNet  MATH  Google Scholar 

  23. 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,”O.R. Letters 7 (1988) 9–14.

    Article  MathSciNet  MATH  Google Scholar 

  24. R.M. Freund, “Polynomial-time algorithms for linear programming based only on primal scaling and projected gradients of a potential function,”Math. Prog. 51 (1991) 203–222.

    Article  MathSciNet  MATH  Google Scholar 

  25. R.M. Freund, “A potential-function reduction algorithm for solving a linear program directly from an infeasible ‘warm start’,”Math. Prog. 52 (1991) 441– 466.

    Google Scholar 

  26. R.M. Freund,“Projective transformations for interior-point algorithms, and a superlinearly convergent algorithm for the w-center problem,”Math. Prog. 58 (1993) 385–414.

    Article  MathSciNet  MATH  Google Scholar 

  27. R.M. Freund, “A potential reduction algorithm with user-specified phase I- phase II balance for solving a linear program from an infeasible warm start,”SIAM J. Opt. 5 (1995) 247–268.

    Article  MathSciNet  MATH  Google Scholar 

  28. D.M. Gay, “A variant of Karmarkar’s linear programming algorithm for problems in standard form,”Math. Prog. 37 (1987) 81–90.

    Article  MathSciNet  MATH  Google Scholar 

  29. P. Gill, W. Murray, M. Saunders, J. Tomlin, and M. Wright, “On projected Newton barrier methods for linear programming and an equivalence to Karmarkar’s projective method,”Math. Prog. 36 (1986) 183–209.

    Article  MathSciNet  MATH  Google Scholar 

  30. D. Goldfarb and S. Mehrotra, “Relaxed variants of Karmarkar’s algorithm for linear programs with unknown optimal objective value,”Math. Prog. 40 (1988), 183–195.

    Article  MathSciNet  MATH  Google Scholar 

  31. D. Goldfarb and S. Mehrotra, “A relaxed version of Karmarkar’s method,”Math. Prog. 40 (1988), 289–315.

    Article  MathSciNet  MATH  Google Scholar 

  32. C.C. Gonzaga, “Conical projection algorithms for linear programming,”Math. Prog. 43 (1989) 151–173.

    Article  MathSciNet  MATH  Google Scholar 

  33. C.C. Gonzaga, “Polynomial affine algorithms for linear programming,”Math. Prog. 49 (1991) 7–21.

    Article  MathSciNet  Google Scholar 

  34. C.C. Gonzaga, “Large-step path following methods for linear programming, part II: potential reduction method,”SIAM J. Opt. 1 (1991) 280 - 292.

    Article  MathSciNet  MATH  Google Scholar 

  35. C.C. Gonzaga, “Interior point algorithms for linear programs with inequality constraints,”Math. Prog. 52 (1991) 209–225.

    Article  MathSciNet  MATH  Google Scholar 

  36. C.C. Gonzaga, “On lower bound updates in primal potential reduction methods for linear programming,”Math. Prog. 52 (1991) 415–428.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  38. C.C. Gonzaga and M.J. Todd, “An O\(\left( {\sqrt {n} L} \right)\)-iteration large-step primal-dual affine algorithm for linear programming,”SIAM J. Opt. 2 (1992) 349–359.

    Article  MathSciNet  MATH  Google Scholar 

  39. P. Huard, “Resolution of mathematical programming with nonlinear constraints by the method of centres,” inNonlinear Programming, J. Abadie, editor ( North- Holland, Amsterdam, 1967 ).

    Google Scholar 

  40. H. Imai, “On the convexity of the multiplicative version of Karmarkar’s potential function,”Math. Prog. 40 (1988) 29–32.

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  42. B. Jansen, C. Roos, and T. Terlaky, “The theory of linear programming: skew symmetric self-dual problems and the central path,”Optimization 29 (1993) 225–233.

    Article  MathSciNet  Google Scholar 

  43. J. Ji and Y. Ye, “A complexity analysis for interior-point algorithms based on Karmarkar’s potential function,”SIAM J. Opt 4 (1994) 512–520.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  45. M. Kojima, N. Megiddo, T. Noma, and A. Yoshise, “A unified approach to interior point algorithms for linear complementarity problems,”Lecture Notes in Computer Science 538 ( Springer-Verlag, Berlin, 1991 ).

    Google Scholar 

  46. M. Kojima, N. Megiddo, and Y. Ye, “An interior point potential reduction algorithm for the linear complementarity problem,”Math. Prog. 54 (1992) 267– 279.

    Google Scholar 

  47. M. Kojima, S. Mizuno, and A. Yoshise, “An Oxxxiteration potential reduction algorithm for linear complementarity problems,”Math. Prog. 50 (1991) 331–342.

    Article  MathSciNet  MATH  Google Scholar 

  48. C. McDiarmid, “On the improvement per iteration in Karmarkar’s algorithm for linear programming,”Math. Prog. 46 (1990) 299–320.

    Article  MathSciNet  MATH  Google Scholar 

  49. N. Megiddo and M. Shub, “Boundary behavior of interior point algorithms in linear programming,”Math. Oper. Res. 14 (1989), 97–146

    Article  MathSciNet  MATH  Google Scholar 

  50. J.E. Mitchell, “Updating lower bounds when using Karmarkar’s projective algorithm for linear programming,”JOTA 78 (1993) 127 - 142.

    Article  MATH  Google Scholar 

  51. J.E. Mitchell and M.J. Todd, “On the relationship between the search directions in the affine and projective variants of Karmarkar’s linear programming algorithm,” inContributions to Operations Research and Economics:The Twentieth Anniversary of CORE, B. Cornet and H. Tulkens, editors, MIT Press (Cambridge, MA, 1989 ) 237–250.

    Google Scholar 

  52. J.E. Mitchell and M.J. Todd, “A variant of Karmarkar’s linear programming algorithm for problems with some unrestricted variables,”SIAM J. Matrix Anal. Appl. 10 (1989) 30 - 38.

    Article  MathSciNet  MATH  Google Scholar 

  53. S. Mizuno, “A rank one updating algorithm for linear programming,”The Arabian Journal for Science and Engineering 15 (1990) 671–677.

    MathSciNet  Google Scholar 

  54. S. Mizuno, “O(n p L) iteration0(n 3 L) potential reduction algorithms for linear programming,”Linear Algebra Appl. 152 (1991) 155–168.

    Google Scholar 

  55. S. Mizuno, M. Kojima, and M.J. Todd, “Infeasible-interior-point primal-dual potential-reduction algorithms for linear programming,”SIAM J. Opt. 5 (1995) 52–67.

    Article  MathSciNet  MATH  Google Scholar 

  56. S. Mizuno and A. Nagasawa, “A primal-dual affine scaling potential reduction algorithm for linear programming,”Math. Prog. 62 (1993) 119–131.

    Article  MathSciNet  MATH  Google Scholar 

  57. R.D.C. Monteiro, “Convergence and boundary behavior of the projective scaling trajectories for linear programming,”Contemporary Mathematics 114 (1990) 213–229.

    MathSciNet  Google Scholar 

  58. R.D.C. Monteiro, “On the continuous trajectories for a potential reduction algorithm for linear programming,”Math. Oper. Res. 17 (1992) 225–253.

    Article  MathSciNet  MATH  Google Scholar 

  59. M. Muramatsu and T. Tsuchiya, “A convergence analysis of a long-step variant of the projective scaling algorithm,” The Institute of Statistical Mathematics (Tokyo, Japan, 1993); to appear inMath. Prog.

    Google Scholar 

  60. A.S. Nemirovskii, “An algorithm of the Karmarkar type,”Soviet Journal on Computers and Systems Sciences 25 (1987) 61–74.

    MathSciNet  Google Scholar 

  61. Y.E. Nesterov, “Long-step strategies in interior point potential-reduction algorithms,” Dept. SES-COMIN, University of Geneva ( Geneva, Switzerland, 1993 ).

    Google Scholar 

  62. Y.E. Nesterov and M.J. Todd, “Self-scaled barriers and interior-point methods for convex programming,” Technical Report 1091, School of OR/IE, Cornell University (Ithaca, NY, 1994); to appear inMath. Oper. Res..

    Google Scholar 

  63. Y.E. Nesterov and M.J. Todd, “Primal-dual interior point methods for self- scaled cones,” Technical Report 1125, School of OR/IE, Cornell University ( Ithaca, NY, 1995 ).

    Google Scholar 

  64. Y. Nesterov and A. Nemirovskii,Interior-Point Polynomial Algorithms in Convex Programming ( SIAM, Philadelphia, 1994 ).

    MATH  Google Scholar 

  65. C.H. Papadimitriou and K. Steiglitz,Combinatorial Optimization:Algorithms and Complexity (Prentice-Hall, 1982 ).

    MATH  Google Scholar 

  66. M.J.D. Powell, “On the number of iterations of Karmarkar’s algorithm for linear programming,”Math. Prog. 62 (1993) 153–197

    Google Scholar 

  67. A.E. Steger, “An extension of Karmarkar’s algorithm for bounded linear programming problems,” M.S. Thesis, State University of New York (Stonybrook, NY, 1985 )

    Google Scholar 

  68. D.F. Shanno, “Computing Karmarkar projections quickly,”Math. Prog. 41 (1988) 61–71.

    Article  MathSciNet  MATH  Google Scholar 

  69. D. Shaw and D. Goldfarb, “A path-following projective interior point method for linear programming,”SIAM J. Opt. 4 (1994) 65–85.

    Article  MathSciNet  MATH  Google Scholar 

  70. K. Tanabe, “Centered Newton method for mathematical programming,”Lecture Notes in Control and Information Sciences 113 ( Springer-Verlag, Berlin, 1988 ) 197–206.

    Google Scholar 

  71. M.J. Todd, “Exploiting special structure in Karmarkar’s linear programming algorithm,”Math. Prog. 41 (1988) 97–113.

    Article  MathSciNet  MATH  Google Scholar 

  72. M.J. Todd, “Improved bounds and containing ellipsoids in Karmarkar’s linear programming algorithm,”Mathematics of Operations Research 13 (1988) 650–659.

    Article  MathSciNet  MATH  Google Scholar 

  73. M.J. Todd, “The effects of degeneracy and null and unbounded variables on variants of Karmarkar’s linear programming algorithm,” inLarge Scale Numerical Optimization, T.F. Coleman and Y. Li, editors ( SIAM, Philadelphia, 1990 ).

    Google Scholar 

  74. M.J. Todd, “A Dantzig-Wolfe-like variant of Karmarkar’s interior-point linear programming algorithm,”Operations Research 38 (1990) 1006–1018.

    Article  MathSciNet  MATH  Google Scholar 

  75. M.J. Todd, “On Anstreicher’s combined phase I-phase II projective algorithm for linear programming,”Math. Prog. 55 (1992) 1–15.

    Article  MathSciNet  MATH  Google Scholar 

  76. M.J. Todd, “Combining phase I and phase II in a potential reduction algorithm for linear programming,”Math. Prog. 59 (1993) 133–150.

    Article  MathSciNet  MATH  Google Scholar 

  77. M.J. Todd, “Interior-point algorithms for semi-infinite programming,”Math. Prog. 65 (1994) 217–245.

    Article  MathSciNet  MATH  Google Scholar 

  78. M.J. Todd, “Potential-reduction methods in mathematical programming,” School of IE/OR, Cornell University (Ithaca, NY, 1995); to appear inMath. Prog.

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  80. M.J. Todd and Y. Ye, “A centered projective algorithm for linear programming,”Math. Oper. Res. 15 (1990) 508–529.

    Article  MathSciNet  MATH  Google Scholar 

  81. L. Tungel, “Constant potential primal-dual algorithms: a framework,”Math. Prog 66 (1994) 145 - 159.

    Article  Google Scholar 

  82. L. Vandenberghe and S. Boyd, “Positive definite programming,” Dept. of Electrical Engineering, Stanford University (Stanford, CA, 1994); to appear inSIAM Review

    Google Scholar 

  83. Y. Ye, “A class of projective transformations for linear programming,”SIAM J. Comp. 19 (1990) 457–466.

    Article  Google Scholar 

  84. Y. Ye, “A ‘build down’ scheme for linear programming,”Mathematical Programming 46 (1990) 61–72.

    Article  MathSciNet  MATH  Google Scholar 

  85. Y. Ye, “AnO(n 3 L) potential reduction algorithm for linear programming,”Math. Prog. 50 (1991) 239–258.

    Article  MATH  Google Scholar 

  86. Y. Ye, “A potential reduction algorithm allowing column generation,”SIAM J. Opt. 2 (1992), 7–20.

    Article  MATH  Google Scholar 

  87. Y. Ye, “A fully polynomial-time approximation algorithm for computing a stationary point of the general LCP,”Math. Oper. Res. 18 (1993) 334–345.

    Article  MathSciNet  MATH  Google Scholar 

  88. Y. Ye, “On the complexity of approximating a KKT point of quadratic programming,” Dept. of Management Sciences, University of Iowa ( Iowa City, IA, 1995 ).

    Google Scholar 

  89. Y. Ye and M. Kojima, “Recovering optimal dual solutions in Karmarkar’s polynomial algorithm for linear programming,”Math. Prog. 39 (1987) 305–317.

    Article  MathSciNet  MATH  Google Scholar 

  90. Y. Ye, K.O. Kortanek, J.A. Kaliski, and S. Huang, “Near-boundary behavior of primal-dual potential reduction algorithms for linear programming,”Math. Prog. 58 (1993) 243–255.

    Article  MathSciNet  MATH  Google Scholar 

  91. Y.Ye,M.J.Todd,and S.Mizuno,“An O\(\left( {\sqrt {n} L} \right)\)-iteration homogeneous and self-dual linear programming algorithm,”Math. Oper. Res. 19 (1994) 53–67.

    Article  MathSciNet  MATH  Google Scholar 

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  1. Department of Management Sciences, University of Iowa, IA 52242, Iowa City, USA

    Kurt M. Anstreicher

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  1. Delft University of Technology, The Netherlands

    Tamás Terlaky

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© 1996 Kluwer Academic Publishers

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Anstreicher, K.M. (1996). Potential Reduction Algorithms. In: Terlaky, T. (eds) Interior Point Methods of Mathematical Programming. Applied Optimization, vol 5. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-3449-1_4

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  • DOI: https://doi.org/10.1007/978-1-4613-3449-1_4

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