Abstract
We consider the generalization of a variant of Karmarkar's algorithm to semi-infinite programming. The extension of interior point methods to infinite-dimensional linear programming is discussed and an algorithm is derived. An implementation of the algorithm for a class of semi-infinite linear programs is described and the results of a number of test problems are given. We pay particular attention to the problem of Chebyshev approximation. Some further results are given for an implementation of the algorithm applied to a discretization of the semi-infinite linear program, and a convergence proof is given in this case.
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References
I. Adler, N.K. Karmarkar, M.G.C. Resende and G. Veiga. “An implementation of Karmarkar's algorithm for linear programming,” Department of Industrial Engineering and Operations Research, University of California, Berkeley, CA 94720.
E.R. Barnes. “A variation on Karmarkar's algorithm for solving linear programming problems,”Mathematical Programming 36 (1986) 174–182.
T.M. Cavalier and A.L. Soyster, “Some Computational Experience and a Modification of the Karmarkar Algorithm,” ISME Working Paper 85–105, Pennsylvania State University, Pennsylvania (1985).
I.I. Dikin, “Iterative solution of problems of linear and quadratic programming,”Soviet Mathematics Doklady 8(3) (1967) 674–675.
K. Glashoff and S.-Å. Gustafson,Linear Optimization and Approximation (Springer-Verlag, New York, 1983).
G.H. Golub and C.F. Van Loan,Matrix Computations (The John Hopkins University Press, Baltimore, Maryland, 1983).
S.-Å. Gustafson. “On numerical analysis in semi-infinite programming,” in: R. Hettich, ed.,Semi-Infinite Programming (Springer-Verlag, Berlin, 1979).
N. Karmarkar, “A new polynomial time algorithm for linear programming,”Combinatorica 4 (1984) 373–395.
K.O. Kortanek. “Vector-Supercomputer Experiments with the Linear Programming Scaling Algorithm,” Working Paper 87-2, College of Business Adminstration, University of Iowa, Iowa City (1987).
K.O. Kortanek and M. Shi. “Convergence results and numerical experiments on a linear programming hybrid algorithm,”European Journal of Operational Research 32 (1987) 47–61.
R.J. Vanderbei, M.S. Meketon and B.A. Freedman, “A modification of Karmarkar's algorithm,”Algorithmica 1 (1986) 395–407.
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Ferris, M.C., Philpott, A.B. An interior point algorithm for semi-infinite linear programming. Mathematical Programming 43, 257–276 (1989). https://doi.org/10.1007/BF01582293
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DOI: https://doi.org/10.1007/BF01582293