Probabilistic analysis of two complexity measures for linear programming problems

This note provides a probabilistic analysis of χ̄A, a condition number used in the linear programming algorithm of Vavasis and Ye [14] whose running time depends only on the constraint matrix A ∈ IRm×n. We show that if A is a standard Gaussian matrix, then E(ln χ̄A) = O(min{m ln n, n}). Thus, the expected running time of linear programming 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. We show that the same bound holds for E(ln σ(A)), where σ(A) is another condition number of A arising in complexity analyses of linear programming problems.