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

Systems Ay ≥ 0 with a degenerate cone of solutions are considered ill-posed since finite-precision algorithms are not expected to find points in the cone of solutions. Consequently, common condition numbers for these systems, such as C(A) [15] and C (A) [2], which are based on the notion of distance to the nearest ill-posed problem, become infinite on such ill-posed instances. In this paper we extend these two condition numbers to versions C(A) and C (A) which are always finite. Both condition numbers can be expressed in terms of a distance to a change in the geometry of the cone of solutions. The Partially supported by CERG grant CityU 100707