On Condition Number Theorems in Mathematical Programming

Abstract

A condition number of nonconvex mathematical programming problems is defined as a measure of the sensitivity of their global optimal solutions under canonical perturbations. A (pseudo-)distance among problems is defined via the corresponding augmented Kojima functions. A characterization of well-conditioning is obtained. In the nonconvex case, we prove that the distance from ill-conditioning is bounded from above by a multiple of the reciprocal of the condition number. Moreover, a lower bound of the distance from a special class of ill-conditioned problems is obtained in terms of the condition number. The proof is based on a new theorem about the permanence of the Lipschitz character of set-valued inverse mappings. A uniform version of the condition number theorem is proved for classes of convex problems defined through bounds of some constants available from problem’s data.