Linear Programming Part II: Interior-Point Methods


A paper by Karmarkar in 1984 [1] and substantial progress made since that time have led to the field of modern interior-point methods for linear programming (LP). Unlike the family of simplex methods considered in Chap. 11, which approach the solution through a sequence of iterates that move from vertex to vertex along the edges on the boundary of the feasible polyhedron, the iterates generated by interior-point algorithms approach the solution from the interior of a polyhedron. Although the claims about the efficiency of the algorithm in [1] have not been substantiated in general, extensive computational testing has shown that a number of interior-point algorithms are much more efficient than simplex methods for large-scale LP problems [2].


Linear Program Problem Central Path Apply Algorithm Barrier Parameter Logarithmic Barrier Function 
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  1. 1.
    N. K. Karmarkar, “A new polynomial time algorithm for linear programming,” Combinatorica, vol. 4, pp. 373–395, 1984.MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    M. H. Wright, “Interior methods for constrained optimization,” Acta Numerica, pp. 341–407, Cambridge Univ. Press, Cambridge, UK, 1992.Google Scholar
  3. 3.
    S. J. Wright, Primal-Dual Interior-Point Methods, SIAM, Philadelphia, 1997.MATHGoogle Scholar
  4. 4.
    E. R. Barnes, “A variation on Karmarkar’s algorithm for solving linear programming problems,” Math. Programming, vol. 36, pp. 174–182, 1986.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    R. J. Vanderbei, M. S. Meketon, and B. A. Freedman, “A modification of Karmarkar’s linear programming algorithm,” Algorithmica, vol. 1, pp. 395–407, 1986.MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    S. G. Nash and A. Sofer, Linear and Nonlinear Programming, McGraw-Hill, New York, 1996.Google Scholar
  7. 7.
    P. E. Gill, W. Murray, M. A. Saunders, J. A. Tomlin, and M. H. Wright, “On projected Newton barrier methods for linear programming and an equivalence to Karmarkar’s projective method,” Math. Programming, vol. 36, pp. 183–209, 1986.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    A. V. Fiacco and G. P. McCormick, Nonlinear Programming: Sequential Unconstrained Minimization Techniques, Wiley, New York, 1968 (Republished by SIAM, 1990).MATHGoogle Scholar
  9. 9.
    K. Jittorntrum, Sequential Algorithms in Nonlinear Programming, Ph.D. thesis, Australian National University, 1978.Google Scholar
  10. 10.
    K. Jittorntrum and M. R. Osborne, “Trajectory analysis and extrapolation in barrier function methods,” J. Australian Math. Soc., Series B, vol. 20, pp. 352–369, 1978.MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    M. Kojima, S. Mizuno, and A. Yoshise, “A primal-dual interior point algorithm for linear programming,” Progress in Mathematical Programming: Interior Point and Related Methods, N. Megiddo ed., pp. 29–47, Springer Verlag, New York, 1989.Google Scholar
  12. 12.
    N. Megiddo, “Pathways to the optimal set in linear programming,” Progress in Mathematical Programming: Interior Point and Related Methods, N. Megiddo ed., pp. 131–158, Springer Verlag, New York, 1989.Google Scholar
  13. 13.
    R. D. C. Monteiro and I. Adler, “Interior path following primal-dual algorithms, Part I: Linear programming,” Math. Programming, vol. 44, pp. 27–41, 1989.MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Y. Ye, Interior-Point Algorithm: Theory and Analysis, Wiley, New York, 1997.MATHGoogle Scholar
  15. 15.
    I. J. Lustig, R. E. Marsten, and D. F. Shanno, “Computational experience with a primal-dual interior point method for linear programming,” Linear Algebra and Its Applications, vol. 152, pp. 191–222, 1991.MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    S. Mehrotra, “On the implementation of a primal-dual interior point method,” SIAM J. Optimization, vol. 2, pp. 575–601, 1992.MATHCrossRefMathSciNetGoogle Scholar

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