Almost Every Convex or Quadratic Programming Problem Is Well Posed

We provide an abstract principle aimed at proving that classes of optimization problems are typically well posed in the sense that the collection of ill-posed problems within each class is σ-porous. As a consequence, we establish typical well-posedness in the above sense for unconstrained minimization of certain classes of functions (e.g., convex and quasi-convex continuous), as well as of convex programming with inequality constraints. We conclude the paper by showing that the collection of consistent ill-posed problems of quadratic programming of any fixed size has Lebesgue measure zero in the corresponding Euclidean space.