×

zbMATH — the first resource for mathematics

Interior path following primal-dual algorithms. I: Linear programming. (English) Zbl 0676.90038
Authors’ abstract: “We describe a primal-dual interior point algorithm for linear programming problems which requires a total of O(\(\sqrt{n}L)\) number of iterations, where L 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.”
[For part II see the authors, ibid. A 44, No.1, 43-66 (1989; Zbl 0676.90039).]
Reviewer: N.Deng

MSC:
90C05 Linear programming
65K05 Numerical mathematical programming methods
68Q25 Analysis of algorithms and problem complexity
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] 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. · Zbl 0671.90045
[2] A. Fiacco and G. McCormick,Nonlinear Programming: Sequential Unconstrained Minimization Techniques (John Wiley and Sons, New York, 1968). · Zbl 0193.18805
[3] K.R. Frisch, ”The logarithmic potential method of convex programming,” unpublished manuscript, University Institute of Economics (Oslo, Norway, 1955).
[4] 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. · Zbl 0624.90062 · doi:10.1007/BF02592025
[5] 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).
[6] N. Karmarkar, ”A new polynomial time algorithm for linear programming,”Combinatorica 4 (1984) 373–395. · Zbl 0557.90065 · doi:10.1007/BF02579150
[7] N. Karmarkar, Talk at the University of California at Berkeley (Berkeley, CA, 1984).
[8] 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). · Zbl 0708.90049
[9] N. Megiddo, ”Pathways to the optimal set in linear programming,” Research Report, IBM Almaden Research Center (San Jose, CA, 1986). · Zbl 0612.90082
[10] C.R. Papadimitriou and K. Steiglitz,Combinatorial Optimization: Algorithms and Complexity (Prentice-Hall, Englewood Cliffs, New Jersey, 1982). · Zbl 0503.90060
[11] J. Renegar, ”A polynomial-time algorithm based on Newton’s method for linear programming,”Mathematical Programming 40 (1988) 59–93. · Zbl 0654.90050 · doi:10.1007/BF01580724
[12] 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). · Zbl 0708.90047
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.