Dennis Cheung

A new condition number for linear programming

Abstract.In this paper we define a new condition number ?(A) for the following problem: given a m by n matrix A, find x∈ℝn, s.t. Ax<0. We characterize this condition number in terms of distance to ill-posedness and we compare it with existing condition numbers for the same problem.

Unifying Condition Numbers for Linear Programming

In recent years, several condition numbers were defined for a variety of linear programming problems based upon relative distances to ill-posedness. In this paper, we provide a unifying view of some of these condition numbers. To do so, we introduce yet another linear programming problem and show that its distance to ill-posedness naturally captures the most commonly used distances to ill-posedness.

Probabilistic analysis of condition numbers for linear programming

In this paper, we provide bounds for the expected value of the log of the condition number C(A) of a linear feasibility problem given by a n × m matrix A (Ref. 1). We show that this expected value is O(min{n, m log n}) if n > m and is O(log n) otherwise. A similar bound applies for the log of the condition number CR(A) introduced by Renegar (Ref. 2).

Solving linear programs with finite precision: I. condition numbers and random programs

Abstract.We define a condition number (A,b,c) for a linear program min x s.t. Ax=b,x≥0 and give two characterizations via distances to degeneracy and singularity. We also give bounds for the expected value, as well as for higher moments, of log (A,b,c) when the entries of A,b and c are i.i.d. random variables with normal distribution.

A note on level-2 condition numbers

We prove that for a large class of problems, including those with a discrete set of inputs, their level-2 condition number (essentially) coincides with the original one.

Tail Decay and Moment Estimates of a Condition Number for Random Linear Conic Systems

In this paper we study the distribution of

Linear Programming and Condition Numbers under the Real Number Computation Model

\mathscr C(A)

A Condition Number for Multifold Conic Systems

and

On strata of degenerate polyhedral cones I: Condition and distance to strata

\log\mathscr C(A)

On the average condition of random linear programs

, where