Partially finite convex programming, part II: explicit lattice models

In Part I of this work we derived a duality theorem for partially finite convex programs, problems for which the standard Slater condition fails almost invariably. Our result depended on a constraint qualification involving the notion of quasi relative interior. The derivation of the primal solution from a dual solution depended on the differentiability of the dual objective function: the differentiability of various convex functions in lattices was considered at the end of Part I. In Part II we shall apply our results to a number of more concrete problems, including variants of semi-infinite linear programming, L¹ approximation, constrained approximation and interpolation, spectral estimation, semi-infinite transportation problems and the generalized market area problem of Lowe and Hurter (1976). As in Part I, we shall use lattice notation extensively, but, as we illustrated there, in concrete examples lattice-theoretic ideas can be avoided, if preferred, by direct calculation