Perturbation analysis of a condition number for convex inequality systems and global error bounds for analytic systems

Abstract

In this paper several types of perturbations on a convex inequality system are considered, and conditions are obtained for the system to be well-conditioned under these types of perturbations, where the well-conditionedness of a convex inequality system is defined in terms of the uniform boundedness of condition numbers under a set of perturbations. It is shown that certain types of perturbations can be used to characterize the well-conditionedness of a convex inequality system, in which either the system has a bounded solution set and satisfies the Slater condition or an associated convex inequality system, which defines the recession cone of the solution set for the system, satisfies the Slater condition. Finally, sufficient conditions are given for the existence of a global error bound for an analytic system. It is shown that such a global error bound always holds for any inequality system defined by finitely many convex analytic functions when the zero vector is in the relative interior of the domain of an associated convex conjugate function.