Light on the infinite group relaxation I: foundations and taxonomy

Abstract

This is a survey on the infinite group problem, an infinite-dimensional relaxation of integer linear optimization problems introduced by Ralph Gomory and Ellis Johnson in their groundbreaking papers titled Some continuous functions related to corner polyhedra I, II [Math. Programming 3 (1972), 23-85, 359-389]. The survey presents the infinite group problem in the modern context of cut generating functions. It focuses on the recent developments, such as algorithms for testing extremality and breakthroughs for the k-row problem for general k ≥ 1 that extend previous work on the single-row and two-row problems. The survey also includes some previously unpublished results; among other things, it unveils piecewise linear extreme functions with more than four different slopes. An interactive companion program, implemented in the open-source computer algebra package Sage, provides an updated compendium of known extreme functions.1 This notation for functions of finite support is used, for example, in [2]. 2 This model is called the mixed-integer infinite relaxation, for example in the survey [22], or sometimes the mixed-integer group problem, but we shall not use either of these terms in the remainder of our survey.3 This model is called the continuous infinite relaxation, for example in the survey [22], or sometimes the continuous group problem.

Theoretical challenges towards cutting-plane selection

“…Many beautiful and non-trivial structural results have been proved regarding valid inequalities for these relaxations [47,44,48,122,209]. See [190,39,45,46] for reviews on the group relaxation and its variants.…”

Minimal cut-generating functions are nearly extreme

“…We say that a CGF π is extreme if there do not exist distinct CGFs π 1 and π 2 such that π = π 1 +π 2 2 . This is a subset of strongly minimal functions [18] that corresponds to a notion of "facets" in the context of CGFs (see also [5,6] for other notions of "facet" for CGFs). Because of the importance of facet-defining cuts in Integer Programming, there has been substantial interest in obtaining and understanding extreme functions (see [5,6] for a survey).…”

“…This is a subset of strongly minimal functions [18] that corresponds to a notion of "facets" in the context of CGFs (see also [5,6] for other notions of "facet" for CGFs). Because of the importance of facet-defining cuts in Integer Programming, there has been substantial interest in obtaining and understanding extreme functions (see [5,6] for a survey). For example, a celebrated result is Gomory and Johnson's 2-Slope Theorem (Theorem 2.4 below) that gives a sufficient condition for a CGF to be extreme (in the affine lattice setting with n = 1; see [7,10,18] for generalizations).…”

“…For example, even verifying the extremality of a function is not completely understood (see [5,6] for preliminary steps in this direction); a simple characterization like Theorem 1.1 seems all the more unlikely.…”

“…Cut-generating functions have received significant attention in the literature (see the surveys [3,5,6] and the references therein). One important feature is that cut-generating functions capture known general purpose cuts, for example the prominent GMI cuts and more generally split cuts (when S is an affine lattice) and the lopsided cuts of Balas and Qualizza [1] (when S is a truncated affine lattice).…”