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Characterizations, bounds, and probabilistic analysis of two complexity measures for linear programming problems

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Abstract.

This note studies A , a condition number used in the linear programming algorithm of Vavasis and Ye [14] whose running time depends only on the constraint matrix A∈ℝm×n, and (A), a variant of another condition number due to Ye [17] that also arises in complexity analyses of linear programming problems. We provide a new characterization of A and relate A and (A). Furthermore, we show that if A is a standard Gaussian matrix, then E(ln A )=O(min{mlnn,n}). Thus, the expected running time of the Vavasis-Ye algorithm for linear programming problems is bounded by a polynomial in m and n for any right-hand side and objective coefficient vectors when A is randomly generated in this way. As a corollary of the close relation between A and (A), we show that the same bound holds for E(ln(A)).

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Received: September 1998 / Accepted: September 2000¶Published online January 17, 2001

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Todd, M., Tunçel, L. & Ye, Y. Characterizations, bounds, and probabilistic analysis of two complexity measures for linear programming problems. Math. Program. 90, 59–69 (2001). https://doi.org/10.1007/PL00011420

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  • DOI: https://doi.org/10.1007/PL00011420

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