Smoothed Analysis of Interior-Point Algorithms: Condition Number

A linear program is typically specified by a matrix A together with two vectors b and c, where A is an n-by-d matrix, b is an n-vector and c is a d-vector. There are several canonical forms for defining a linear program using (A, b, c). One commonly used canonical form is: max cx s.t. Ax ≤ b and its dual min by s.t A y = c, y ≥ 0. In [Ren95b, Ren95a, Ren94], Renegar defined the condition number C(A, b, c) of a linear program and proved that an interior point algorithmwhose complexity wasO(n log(C(A, b , c)/ǫ)) could solve a linear program in this canonical form to relative accuracy ǫ, or determine that the program was infeasible or unbounded. In this paper, we prove that for any (Ā, b̄, c̄) such that ∥