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A primal-dual interior point method whose running time depends only on the constraint matrix

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Abstract

We propose a primal-dual “layered-step” interior point (LIP) algorithm for linear programming with data given by real numbers. This algorithm follows the central path, either with short steps or with a new type of step called a “layered least squares” (LLS) step. The algorithm returns an exact optimum after a finite number of steps—in particular, after O(n 3.5 c(A)) iterations, wherec(A) is a function of the coefficient matrix. The LLS steps can be thought of as accelerating a classical path-following interior point method. One consequence of the new method is a new characterization of the central path: we show that it composed of at mostn 2 alternating straight and curved segments. If the LIP algorithm is applied to integer data, we get as another corollary a new proof of a well-known theorem by Tardos that linear programming can be solved in strongly polynomial time provided thatA contains small-integer entries.

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Correspondence to Stephen A. Vavasis.

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This paper represents a simplification of an earlier manuscript “An accelerated interior point method whose running time depends only onA” by the same authors.

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 Xerox Corp. Part of this work was done while the author was visiting Sandia National Laboratories, supported by the U.S. Department of Energy under contract DE-AC04-76DP00789.

This author is supported in part by NSF Grant DDM-9207347. Part of this work was done 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 IBM Corporation, with additional support from New York State and members of its Corporate Research Institute. E-mail: yyye@col.biz.uiowa.edu.

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Vavasis, S.A., Ye, Y. A primal-dual interior point method whose running time depends only on the constraint matrix. Mathematical Programming 74, 79–120 (1996). https://doi.org/10.1007/BF02592148

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  • DOI: https://doi.org/10.1007/BF02592148

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