Facial reduction in partially finite convex programming

The paper considers the problem of minimizing an extended-value convex function on a locally convex space subject to a finite number of linear (in)equalities. When the standard constraint qualification fails a reduction technique is needed to derive necessary optimality conditions. Facial reduction is usually applied in the range of the constraints. In this paper it is applied in the domain space, thus maintaining any structure (and in particular lattice properties) of the underlying domain. Applications include constrained approximation and best entropy estimation.