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

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∈ℝ^m×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)).