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

Systems Ay⩾0Ay⩾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)C(A) [J. Renegar. Some perturbation theory for linear programming, Mathematical Programming 65 (1994) 73–91] and C(A)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 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 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⩾0Ay⩾0.