Abstract
We propose a sufficient condition that allows an optimal basis to be identified from a central path point in a linear programming problem. This condition can be applied when there is a “gap” in the sorted list of slack values. Unlike previously known conditions, this condition is valid for real-number data and does not involve the number of bits in the data.
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D.A. Bayer and J.C. Lagarias, The nonlinear geometry of linear programming, I. Affine and projective scaling trajectories, Transactions of the AMS 314(1989)499–526.
D.A. Bayer and J.C. Lagarias, The nonlinear geometry of linear programming. II. Legendre transform coordinates and central trajectories, Transactions of the AMS 314(1989)527–581.
D.A. Bayer and J.C. Lagarias, The nonlinear geometry of linear programming. III. Projective Legendre transform coordinates and Hilbert geometry, Transactions of the AMS 320(1990) 193–225.
O. Güler and Y. Ye, Convergence behavior of interior point algorithms, Mathematical Programming 60(1993)215–228.
J.A. Kaliski and Y. Ye, A short-cut potential reduction algorithm for linear programming, Management Science 39(1993)757–773.
N. Karmarkar, A new polynomial-time algorithm for linear programming, Combinatorica 4(1984) 373–395.
L.G. Khachiyan, A polynomial algorithm in linear programming, Dokl. Akad. Nauk SSSR 244(1979)1093–1086, translated in: Soviet Math. Dokl. 20(1979)191–194.
L. McLinden, An analogue of Moreau's proximation theorem, with applications to the nonlinear complementarity problem, Pacific Journal of Mathematics 88(1980)101–161.
N. Megiddo, Pathways to the optimal set in linear programming, in:Progress in Mathematical Programming: Interior Point and Related Method, ed. N. Megiddo (Springer, New York, 1989) pp. 131–158.
D.P. O'Leary, On bounds for scaled projections and pseudoinverses, Linear Algebra and its Applications 132(1990)115–117.
J. Renegar, A polynomial-time algorithm based on Newton's method for linear programming, Mathematical Programming 40(1988)59–94.
G. Sonnevend, An analytical center for polyhedrons and new classes of global algorithms for linear (smooth, convex) programming, in:Lecture Notes in Control and Information Sciences 84 (Springer, New York, 1985) pp. 866–876.
G.W. Stewart, On scaled projections and pseudoinverses, Linear Algebra and its Applications 112(1989)189–193.
M.J. Todd, A Dantzig-Wolfe-like variant of Karmarkar's interior-point linear programming algorithm, Operations Research 38(1990)1006–1018.
K. Tone, An active-set strategy in an interior point method for linear programming, Mathematical Programming 59(1993)345–360.
S.A. Vavasis and Y. Ye, A primal-dual interior point method whose running time depends only on the constraint matrix, Mathematical Programming (1996), to appear.
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This work is supported in part by the National Science Foundation, the Air Force Office of Scientific Research, and the Office of Naval Research, through NSF Grant DMS-8920550. Also supported in part by an NSF Presidential Young Investigator Award with matching funds received from AT&T and the Xerox Corporation. Part of this work was carried out while the author was visiting the Sandia National Laboratories, supported by the U.S. Department of Energy under Contract DE-AC04-76DP00789.
The author is supported in part by NSF Grant DDM-9207347. Part of this work was carried out while the author was on a sabbatical leave from the University of Iowa and visiting the Cornell Theory Center, Cornell University, Ithaca, NY 14853, supported in part by the Cornell Center for Applied Mathematics and by the Advanced Computing Research Institute, a unit of the Cornell Theory Center, which receives major funding from the National Science Foundation and the IBM Corporation, with additional support from New York State and members of its Corporate Research Institute.
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Vavasis, S.A., Ye, Y. Identifying an optimal basis in linear programming. Ann Oper Res 62, 565–572 (1996). https://doi.org/10.1007/BF02206830
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DOI: https://doi.org/10.1007/BF02206830