  Integral functionals based on convex normal integrands are minimized subject
to finitely many moment constraints. The integrands are finite on the positive
and infinite on the negative numbers, strictly convex but not necessarily
differentiable. The minimization is viewed as a primal problem and studied
together with a dual one in the framework of convex duality. The effective
domain of the value function is described by a conic core, a modification of
the earlier concept of convex core. Minimizers and generalized minimizers are
explicitly constructed from solutions of modified dual problems, not assuming
the primal constraint qualification. A generalized Pythagorean identity is
presented using Bregman distance and a correction term for lack of essential
smoothness in integrands. Results are applied to minimization of Bregman
distances. Existence of a generalized dual solution is established whenever the
dual value is finite, assuming the dual constraint qualification. Examples of
`irregular' situations are included, pointing to the limitations of generality
of certain key results.
