Cut-Generating Functions

In optimization problems such as integer programs or their relaxations, one encounters feasible regions of the form {x ∈ R+ : Rx ∈ S} where R is a general real matrix and S ⊂ R is a specific closed set with 0 / ∈ S. For example, in a relaxation of integer programs introduced in [ALWW2007], S is of the form Z − b where b 6∈ Z . One would like to generate valid inequalities that cut off the infeasible solution x = 0. Formulas for such inequalities can be obtained through cut-generating functions. This paper presents a formal theory of minimal cut-generating functions and maximal S-free sets which is valid independently of the particular S. This theory relies on tools of convex analysis.