Abstract
We describe a primal-dual interior point algorithm for linear programming problems which requires a total of\(O\left( {\sqrt n L} \right)\) number of iterations, whereL is the input size. Each iteration updates a penalty parameter and finds the Newton direction associated with the Karush-Kuhn-Tucker system of equations which characterizes a solution of the logarithmic barrier function problem. The algorithm is based on the path following idea.
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References
D.A. Bayer and J.C. Lagarias, “The nonlinear geometry of linear programming,” manuscripts, AT&T Bell Laboratories (Murray Hill, NJ, 1986), to appear inTransactions of the American Mathematical Society.
A. Fiacco and G. McCormick,Nonlinear Programming: Sequential Unconstrained Minimization Techniques (John Wiley and Sons, New York, 1968).
K.R. Frisch, “The logarithmic potential method of convex programming,” unpublished manuscript, University Institute of Economics (Oslo, Norway, 1955).
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,”Mathematical Programming 36 (1986) 183–209.
C.C. Gonzaga, “An algorithm for solving linear programming problems in O(n 3 L) operations,” Memorandum Number UCB/ERL M87/10, Electronics Research Laboratory, Universtiy of California (Berkeley, CA, March, 1987).
N. Karmarkar, “A new polynomial time algorithm for linear programming,”Combinatorica 4 (1984) 373–395.
N. Karmarkar, Talk at the University of California at Berkeley (Berkeley, CA, 1984).
M. Kojima, S. Mizuno and A. Yoshise, “A primal-dual interior point algorithm for linear programming,” Report No. B-188, Department of Information Sciences, Tokyo Institute of Technology (Tokyo, Japan, February, 1987).
N. Megiddo, “Pathways to the optimal set in linear programming,” Research Report, IBM Almaden Research Center (San Jose, CA, 1986).
C.R. Papadimitriou and K. Steiglitz,Combinatorial Optimization: Algorithms and Complexity (Prentice-Hall, Englewood Cliffs, New Jersey, 1982).
J. Renegar, “A polynomial-time algorithm based on Newton's method for linear programming,”Mathematical Programming 40 (1988) 59–93.
P.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).
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Monteiro, R.D.C., Adler, I. Interior path following primal-dual algorithms. part I: Linear programming. Mathematical Programming 44, 27–41 (1989). https://doi.org/10.1007/BF01587075
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DOI: https://doi.org/10.1007/BF01587075