Cut-Generating Functions

Michele Conforti, Gérard Cornuéjols, Aris Daniilidis, Claude Lemaréchal, Jérôme Malick

Abstract: In optimization problems such as integer programs or their relaxations, one encounters feasible regions of the form $\{x\in\mathbb{R}_+^n:\: Rx\in S\}$ where R is a general real matrix and S ⊂ ℝ q 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 q  − b where $b \not\in \mathbb{Z}^q$ . 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.