  We extend in two ways the standard Karush-Kuhn-Tucker optimality conditions
to problems with a convex objective, convex functional constraints, and the
extra requirement that some of the variables must be integral. While the
standard Karush-Kuhn-Tucker conditions involve separating hyperplanes, our
extension is based on lattice-free polyhedra. Our optimality conditions allow
us to define an exact dual of our original mixed-integer convex problem.
