Condition measures and properties of the central trajectory of a linear program

The central trajectory of a linear program consists of the set of optimal solutions x(Mu) and (y(t), s(j)) to the logarithmic barrier problems: (P,(d)): min{cT x + --p(x) : Ax = b, x > 0), (D,(d)): maxb T y /p(s) : ATy + s = c,s > 0), where p(ul) -= ln(u), is the logarithmic barrier function, d = (A, b, c) is a data instance in the space of all data D = {(A, b, c) : A C Wmn, b E Rm, c E fn), and the parameter u is a positive scalar considered independent of the data instance d E D. This study shows that certain properties of solutions along the central trajectory of a linear program are inherently related to the condition number C(d) of the data instance d = (A, b, c), where the condition number C(d) and a closely-related measure p(d) called the "distance to ill-posedness" were introduced by Renegar in a recent series of papers [17, 15, 16]. In the context of the central trajectory problem, p(d) essentially measures how close the data instance d = (A, b, c) is to be being infeasible for (P(d)), and C(d) A Ildll/p(d) is a scale-invariant reciprocal of the distance to illposedness p(d), and so C(d) goes to o as the data instance d = (A, b, c) approaches infeasibility. We present lower and upper bounds on sizes of optimal solutions along the central trajectory, and on rates of change of solutions along the central trajectory as either ,u changes or the data d changes, where these bounds are all polynomial functions of ,l and are linear or polynomial functions of the condition number C(d) and the related distance to ill-posedness p(d) of the data instance d = (A, b, c).