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

Given a data instanced=(A, b, c) of a linear program, we show that certain properties of solutions along the central trajectory of the linear program are inherently related to the condition number C(d) of the data instanced=(A, b, c), where C(d) is a scale-invariant reciprocal of a closely-related measure ρ(d) called the “distance to ill-posedness”. (The distance to ill-posedness essentially measures how close the data instanced=(A,b,c) is to being primal or dual infeasible.) 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 the barrier parameter μ or the datad=(A, b, c) of the linear program is changed. These bounds are all linear or polynomial functions of certain natural parameters associated with the linear program, namely the condition number C(d), the distance to ill-posedness ρ(d), the norm of the data ‖d‖, and the dimensionsm andn.