A Partial Condition Number Theorem in Mathematical Programming

A condition number of mathematical programming problems is defined as a measure of the sensitivity of their global optimal solutions under general perturbations described by parameters acting on their data. A (pseudo-) distance among problems fulfilling prescribed bounds is defined via the corresponding augmented Kojima functions. A characterisation of well-conditioning is obtained. It is shown that the distance to ill-conditioning is bounded from above by a multiple of the reciprocal of the condition number. This upper bound extends to general perturbed problems known results dealing with canonical perturbations.