Perturbational duality (continued) and Applications

A subset S of R is said to be polyhedral if it is the intersection of a finite number of closed halfspaces, i.e., if there exist J ∈ N and collections {y1, . . . , yJ} ⊂ R, {α1, . . . , αJ} ⊂ R such that S = ∩j=1{x ∈ R : y jx ≤ αj}. A function f : R → [−∞,+∞] is polyhedral if its epigraph epi f ⊂ R is a polyhedral set. Clearly, any polyhedral set is automatically convex and closed. Consequently, any polyhedral function is convex and lower semicontinuous (exercise). Polyhedral functions have a special form, which can also be used to obtain refinements of the previous duality results (for instance, refinements that include linear programming duality). We shall not go into the details of this, but the following example illustrates the main point, which is to be confirmed by Proposition 1.2 below: the conditions under which polyhedral functions are subdifferentiable are less stringent than those of arbitrary convex functions.