Mathematical Programming

, Volume 49, Issue 1–3, pp 7–21

Polynomial affine algorithms for linear programming

  • Clovis C. Gonzaga
Article

Abstract

The method of steepest descent with scaling (affine scaling) applied to the potential functionq logc′x — ∑i=1n logxi solves the linear programming problem in polynomial time forq ⩾ n. Ifq = n, then the algorithm terminates in no more than O(n2L) iterations; if q ⩾ n +\(\sqrt n \) withq = O(n) then it takes no more than O(nL) iterations. A modified algorithm using rank-1 updates for matrix inversions achieves respectively O(n4L) and O(n3.5L) arithmetic computions.

Key words

Linear programming affine algorithms Karmarkar's algorithm interior methods 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    D. Bayer and J.C. Lagarias, “The non-linear geometry of linear programming, I. Affine and projective scaling trajectories, II. Legendre transform coordinates, III. Central trajectories,” Preprints, AT&T Bell Laboratories (Murray Hill, NJ, 1986).Google Scholar
  2. [2]
    C. Gonzaga, “An algorithm for solving linear programming problems in O(n 3 L) operations,” in: N. Megiddo, ed.,Progress in Mathematical Programming: Interior-Point and Related Methods (Springer, New York, 1988) pp. 1–28.Google Scholar
  3. [3]
    C. Gonzaga, “Conical projection algorithms for linear programming,”Mathematical Programming 43 (1989) 151–173.Google Scholar
  4. [4]
    N. Karmarkar, “A new polynomial time algorithm for linear programming,”Combinatorica 4 (1984) 373–395.Google Scholar
  5. [5]
    M. Kojima, S. Mizuno and A. Yoshise, “A primal—dual interior point method for linear programming,” in: N. Megiddo, ed.,Progress in Mathematical Programming: Interior-Point and Related Methods (Springer, New York, 1988), pp. 29–48.Google Scholar
  6. [6]
    N. Megiddo, “On the complexity of linear programming,” in: T. Bewley, ed.,Advances in Economic Theory (Cambridge University Press, Cambridge, 1987) pp. 225–268.Google Scholar
  7. [7]
    N. Megiddo and M. Shub, “Boundary behaviour of interior point algorithms in linear programming,” Research Report RJ 5319, IBM Thomas J. Watson Research Center (Yorktown Heights, NY, 1986).Google Scholar
  8. [8]
    R.C. Monteiro and I. Adler, “An O(n 3 L) primal—dual interior point algorithm for linear programming,” Manuscript, Department of Industrial Engineering and Operations Research, University of California (Berkeley, CA, 1987).Google Scholar
  9. [9]
    J. Renegar, “A polynomial-time algorithm based on Newton's method for linear programming,”Mathematical Programming 40 (1988) 59–94.Google Scholar
  10. [10]
    M. Todd and B. Burrell, “An extension of Karmarkar's algorithm for linear programming using dual variables,”Algorithmica 1 (1986) 409–424.Google Scholar
  11. [11]
    M.J. Todd and Y. Ye, “A centered projective algorithm for linear programming,” Technical Report 763, School of Operations Research and Industrial Engineering, Cornell University (Ithaca, NY, 1987).Google Scholar
  12. [12]
    Pravin M. Vaidya, “An algorithm for linear programming which requires O(((m+n)n 2+(m+n) 1.5 n)L) arithmetic operations,” Preprint, AT&T Bell Laboratories (Murray Hill, NJ, 1987).Google Scholar

Copyright information

© The Mathematical Programming Society, Inc. 1990

Authors and Affiliations

  • Clovis C. Gonzaga
    • 1
  1. 1.Department of Systems Engineering and Computer SciencesCOPPE-Federal University of Rio de JaneiroRio de JaneiroBrasil

Personalised recommendations