Abstract
This chapter describes a short-step penalty function algorithm that solves linear programming problems in no more than O(n 0.5 L) iterations. The total number of arithmetic operations is bounded by O(n 3 L), carried on with the same precision as that in Karmarkar’s algorithm. Each iteration updates a penalty multiplier and solves a Newton-Raphson iteration on the traditional logarithmic barrier function using approximated Hessian matrices. The resulting sequence follows the path of optimal solutions for the penalized functions as in a predictor-corrector homotopy algorithm.
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References
K. 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, Trans. Amer. Math. Soc. (to appear).
A. Fiacco and G. McCormick. Nonlinear Programming: Sequential Unconstrained Minimization Techniques, Wiley, New York, 1955.
K. R. Frisch. The logarithmic potential method of convex programming, Memorandum, University Institute of Economics, Oslo, Norway (May 1955).
C. B. Garcia and W. I. Zangwill, Pathways to Solutions, Fixed Points, and Equilibria, Prentice-Hall, Englewood cliffs, N.J., 1981.
P. Gill, W. Murray, M. Saunders, J. Tomlin, and M. Wright, On projected Newton barrier methods for linear programming and an equivalence to Karmarkar’s projective method, Report SOL 85–11, Systems Optimization Laboratory, Department of Operations Research, Stanford University, Stanford, Calif. (1985).
C. Gonzaga, A conical projection algorithm for linear programming, Memorandum UCB/ERL M85/61, Electronics Research Laboratory, University of California, Berkeley (July 1985).
C. Gonzaga, Search directions for interior linear programming methods, Memorandum UCB/ERL M87/44, Electronics Research Laboratory, University of California, Berkeley (March 1987).
N. Karmarkar, A new polynomial time algorithm for linear programming, Combinatorica 4 (1984), 373–395.
F. A. Lootsma, Numerical Methods for Nonlinear Optimization, Academic Press, New York, 1972.
N. Megiddo, Pathways to the optimal set in linear programming, Chapter 8, this volume.
E. Polak, Computational Method in Optimization, Academic Press, New York, 1971.
James Renegar, A polynomial-time algorithm based on Newton’s method for linear programming. Mathematical Programmings 40 (1988), 50–94.
M. Todd and B. Burrell, An extension of Karmarkar’s algorithm for linear programming using dual variables, Algorithmica 1 (1986), 409–424.
Pravin M. Vaidya, An algorithm for linear programming which requires O(((m + n)n 2 + (m + n) 1.5 n)L) arithmetic operations, Proc. 19th Annual ACM Symposium on Theory of Computing (1987), pp. 29–38.
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© 1989 Springer-Verlag New York Inc.
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Gonzaga, C.C. (1989). An Algorithm for Solving Linear Programming Problems in O(n 3 L) Operations. In: Megiddo, N. (eds) Progress in Mathematical Programming. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-9617-8_1
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DOI: https://doi.org/10.1007/978-1-4613-9617-8_1
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