Abstract
We propose a polynomial time primal—dual potential reduction algorithm for linear programming. The algorithm generates sequencesd k andv k rather than a primal—dual interior point (x k,s k), where\(d_i^k = \sqrt {{{x_i^k } \mathord{\left/ {\vphantom {{x_i^k } {s_i^k }}} \right. \kern-\nulldelimiterspace} {s_i^k }}} \) and\(v_i^k = \sqrt {x_i^k s_i^k }\) fori = 1, 2,⋯,n. Only one element ofd k is changed in each iteration, so that the work per iteration is bounded by O(mn) using rank-1 updating techniques. The usual primal—dual iteratesx k ands k are not needed explicitly in the algorithm, whereasd k andv k are iterated so that the interior primal—dual solutions can always be recovered by aforementioned relations between (x k, sk) and (d k, vk) with improving primal—dual potential function values. Moreover, no approximation ofd k is needed in the computation of projection directions. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.
Similar content being viewed by others
References
N.K. Karmarkar, A new polynomial-time algorithm for linear programming, Combinatorica 4 (1984) 373–395.
P.M. Vaidya, An algorithm for linear programming which requires O(((m + n)n 2 + (m + n)1.5n)L) arithmetic operations, Proceedings of the 19th ACM Symposium on the Theory of Computing, 1987, pp. 29–38.
C.C. Gonzaga, An algorithm for solving linear programming problems in O(n 3L) operations, in: N. Megiddo (Ed.), Progress in Mathematical Programming, Springer, Berlin, 1989, pp. 1–28.
M. Kojima, S. Mizuno, A. Yoshise, A polynomial-time algorithm for a class of linear complementarity problems, Mathematical Programming 44 (1989) 1–26.
R.D.C. Monteiro, I. Adler, Interior path following primal—dual algorithms, Part II: convex quadratic programming, Mathematical Programming 44 (1989) 43–66.
K.M. Anstreicher, A standard form variant, and safeguarded linesearch, for the modified Karmarkar algorithm, Mathematical Programming 47 (1990) 337–351.
K.M. Anstreicher, R.A. Bosch, Long steps in a O(n 3L) algorithm for linear programming, Mathematical Programming 54 (1992) 251–265.
D. Den Hertog, Interior Point Approach to Linear, Quadratic and Convex Programming, Algorithms and Complexity, Kluwer Academic Publishers, Dordrecht, 1994.
D. Den Hertog, C. Roos, J.-Ph. Vial, A complexity reduction for the long-step path-following algorithm for linear programming, SIAM Journal on Optimization 2 (1992) 71–87.
Y. Ye, Further developments in potential reduction algorithm, Technical Report, Department of Management Sciences, University of Iowa, 1989.
S. Mizuno, “O(n ρL) iteration O(n 3L) potential reduction algorithms for linear programming, Linear Algebra and its Applications 152 (1991) 155–168.
S. Mizuno, A rank one updating algorithm for linear programming, The Arabian Journal for Science and Engineering 15 (1990) 671–677.
R.A. Bosch, On Mizuno's rank one updating algorithm for linear programming, SIAM Journal on Optimization 3 (1993) 861–867.
M.J. Todd, Y. Ye, A centered projective algorithm for linear programming, Mathematics of Operations Research 15 (1990) 508–529.
Y. Ye, An O(n 3L) potential reduction algorithm for linear programming, Mathematical Programming 50 (1991) 239–258.
R.A. Bosch, K.M. Anstreicher, On partial updating in a potential reduction linear programming algorithm of Kojima, Mizuno, and Yoshise, Algorithmica 9 (1993) 184–197.
S. Mehrotra, Deferred rank one updates in O(n 3L) interior point algorithms, Technical Report 90-11, Department of Industrial Engineering and Management Sciences, Northwestern University, Evanston, IL, 1990.
J. Renegar, A polynomial time algorithm, based on Newton's method, for linear programming, Mathematical Programming 40 (1988) 59–93.
D.F. Shanno, Computing Karmarkar projections quickly, Mathematical Programming 41 (1988) 61–71.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Sturm, J.F., Zhang, S. An interior point method, based on rank-1 updates, for linear programming. Mathematical Programming 81, 77–87 (1998). https://doi.org/10.1007/BF01584845
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01584845