On the condition numbers for polyhedra in Kasmarkar's form

We present formulations of two condition measures (one for linear programmi ng (LP) due to Ye, and the other for a matrix) as optimization problems ove r sign constraints. We construct, based on LP duality, a dual characterizat ion of Ye's condition measure in the setting of Karmarkar's form. The eleme ntary formulations (utilizing the dual characterization) lead to trivial pr oofs of some results relating these two condition measures for polyhedra in Karmarkar's form. Such interpretations, using the sign constraints, allow for the definition of families of condition measures that are "between" the two condition measures. Our viewpoint provides further understanding of th e relationship of the two condition measures. As a result of this new under standing, we point to a connection with oriented matroids and prove that a conjecture of Vavasis and Ye on the relationship of these two condition mea sures is false. (C) 1999 Elsevier Science B.V. All rights reserved.