Typical convex program is very well posed

In this paper we consider the collection of convex programming problems with inequality and equality constraints, in which every problem of the collection is obtained by linear perturbations of the cost function and right-hand side perturbation of the constraints, while the ``core' cost function and the left-hand side constraint functions are kept fixed. The main result shows that the set of the problems which are not well-posed is σ-porous in a certain strong sense. Our results concern both the infinite and finite dimensional case. In the last case the conclusions are significantly sharper. Research of A. Ioffe was supported in part by the US-Israel Binational Fund under the grant 2000157. research of R. E. Lucchetti was partially supported by Ministero dell'Istruzione, dell'Università e della Ricerca (COFIN 2001).