Cut-Generating Functions and S-Free Sets

Abstract

We consider the separation problem for sets X that are pre-images of a given set S by a linear mapping. Classical examples occur in integer programming, as well as in other optimization problems such as complementarity. One would like to generate valid inequalities that cut off some point not lying in X, without reference to the linear mapping. To this aim, we introduce a concept: cut-generating functions (cgf) and we develop a formal theory for them, largely based on convex analysis. They are intimately related to S-free sets and we study this relation, disclosing several definitions for minimal cgf's and maximal S-free sets. Our work unifies and puts in perspective a number of existing works on S-free sets; in particular, we show how cgf's recover the celebrated Gomory cuts.

On sublinear inequalities for mixed integer conic programs

“…This setup naturally arises in the context of separating a fractional solution from the feasible region of an MILP [13] and has attracted some attention in the recent literature. In particular, some of the previous literature, including [13,14], has specifically focused on the separation of the origin from conv(S(A, R n + , B)) under the assumption that B is closed and 0 / ∈ B. In this case, 0 / ∈ conv(S(A, R n + , B)), which ensures the existence of valid inequalities separating the origin from conv(S(A, R n + , B)).…”

“…Recently, Conforti et al [13] studied a variant of the set S(A, R n + , B) where B ∈ R m is a fixed nonempty and closed set such that 0 / ∈ B, yet n and A ∈ R m×n are varying. It is easy to see [13, Lemma 2.1] that in such a setup, 0 / ∈ conv(S(A, R n + , B)).…”

“…In this setup, various properties of cut-generating functions and their relations to Bfree sets were studied in [13]. One question left open in [13] was whether or not CGFs are sufficient to give all of the cuts separating the origin from conv(S(A, R n + , B)).…”

“…One question left open in [13] was whether or not CGFs are sufficient to give all of the cuts separating the origin from conv(S(A, R n + , B)). In the follow up work [14], by using the value functions of linear programs and arguments from convex analysis, a positive answer to this question was provided under an additional structural assumption.…”

“…From Remark 1, Proposition 4, and Corollary 2, we immediately deduce a related result. That is, without any further technical assumptions, the relaxed CGFs are always sufficient to generate all cuts separating the origin from conv(S(A, R n + , B)) and A and n can be taken as varying exactly as in the framework of [13]. In contrast to this, Example 6.1 of [13] shows that there are sets of the form S(A, R n + , B) such that CGFs are not sufficient to generate all cuts separating the origin.…”