CONDITION MEASURES AND PROPERTIES OF THE CENTRAL TRAJECTORY OF A LINEAR PROGRAM

Given a data instance d = (A, b, c) of a linear program, we show that certain properties of solutions along the central trajectory of the li near program are inherently related to the condition number C(d) of th e data instance d = (A, b, c), where C(d) is a scale-invariant recipro cal of a closely-related measure p(d) called the ''distance to ill-pos edness'', (The distance to ill-posedness essentially measures how clos e the data instance d = (A, b, c) is to being primal or dual infeasibl e.) We present lower and upper bounds on sizes of optimal solutions al ong the central trajectory, and on rates of change of solutions along the central trajectory, as either the barrier parameter mu or the data d = (A, b, c) of the linear program is changed. These bounds are all linear or polynomial functions of certain natural parameters associate d with the linear program, namely the condition number C(d), the dista nce to ill-posedness p(d), the norm of the data parallel to d parallel to, and the dimensions m and n. (C) 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.