A Condition Number Theorem in Convex Programming without Uniqueness

A condition number of mathematical programming problems with convex data is defined as a suitable measure of the sensitivity of their optimal solutions under canonical perturbations. A pseudo-distance among mathematical programming problems is introduced via the corresponding Kojima functions. Characterizations of well-conditioning are obtained. We prove that the distance to ill-conditioning is bounded from above by a multiple of the reciprocal of the condition number, thereby generalizing previous results dealing with problems with a unique optimal solution.