Mathematical programming formulation of the convex lexicographic multi-criteria problems typically lacks a constraint qualification. Therefore the classical Kuhn-tucker theory fails to characterize their optimal solutions. Furthermore, numerical methods for solving the lexicographic problems are virtually nonexistent. This paper shows that using a recent theory of convex programming, which is free of a constraint qualification assumption, it is possible both to characterize and to calculate the optimal solutions of the convex lexicographic problem.