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) [J. Renegar. Some perturbation theory for linear programming, Mathematical Programming 65 (1994) 73-91] and C (A) [D. Cheung, F. Cucker, A new condition number for linear programming, Mathematical Programming 91 (2001) 163-174], 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 over(C, -) (A) and over(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 main result shows that for both of them, the distance corresponds to a notion of best conditioned solution for a canonical complementarity problem associated to the system Ay ≥ 0. © 2008 Elsevier B.V. All rights reserved.