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

This note studies <(<chi>)over bar>(A), a condition number used in the line ar programming algorithm of Vavasis and Ye [14] whose running time depends only on the constraint matrix A is an element of R-mxn, and <(<sigma>)under bar>(A), a variant of another condition number due to Ye [17] that also ar ises in complexity analyses of linear programming problems. We provide a ne w characterization of <(<chi>)over bar>(A) and relate <(<chi>)over bar>(A) and <(<sigma>)under bar>(A). Furthermore, we show that if A is a standard G aussian matrix, then E(ln <(<chi>)over bar>(A)) = O(min{m ln n, n}). Thus, the expected running time of the Vavasis-Ye algorithm for linear programmin g problems is bounded by a polynomial in m and n for any right-hand side an d objective coefficient vectors when A is randomly generated in this way. A s a corollary of the close relation between <(<chi>)over bar>(A) and <(<sig ma>)under bar>(A), we show that the same bound holds for E(ln<(<sigma>)unde r bar>(A)).