Cut-generating functions and S-free sets

International audienceWe 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

Software for cut-generating functions in the Gomory--Johnson model and beyond

We present software for investigations with cut generating functions in the Gomory-Johnson model and extensions, implemented in the computer algebra system SageMath.Comment: 8 pages, 3 figures; to appear in Proc. International Congress on Mathematical Software 201

Operations that preserve the covering property of the lifting region

We contribute to the theory for minimal liftings of cut-generating functions. In particular, we give three operations that preserve the so-called covering property of certain structured cut-generating functions. This has the consequence of vastly expanding the set of undominated cut generating functions which can be used computationally, compared to known examples from the literature. The results of this paper are significant generalizations of previous results from the literature on such operations, and also use completely different proof techniques which we feel are more suitable for attacking future research questions in this area.Comment: 23 page

On the development of cut-generating functions

Cut-generating functions are tools for producing cutting planes for generic mixed-integer sets. Historically, cutting planes have advanced the progress of algorithms for solving mixed- integer programs. When used alone, cutting-planes provide a finite time algorithm for solving a large family of integer programs [12, 70]. Used in tandem with other algorithmic techniques, cutting planes play a large role in popular commercial solvers for mixed-integer programs [9, 34, 35]. Considering the benefit that cutting planes bring, it becomes important to understand how to construct good cutting planes. Sometimes information about the motivating prob- lem can be used to construct problem-specific cutting planes. One prominent example is the history of the Traveling Salesman Problem [43]. However, it is unclear how much insight into the particular problem is required for these types of cutting-planes. In contrast, cut- generating functions (a term coined by Cornu ́ejols et al. [40]) provide a way to construct cutting planes without using inherent structure that a problem may have. Some of the earliest examples of cut-generating functions are due to Gomory [70] and these have been very successful in practice [34]. Moreover, cut-generating functions produce the strongest cutting planes for some commonly used mixed-integer sets such as Gomory’s corner poly- hedron [66, 95]. In this thesis, we examine the theory of cut-generating functions. Due to the success of the cut-generating function created by Gomory, there has been a proliferation of research in this direction with one end goal being the further advancement of algorithms for mixed- integer programs [78, 40, 28]. We contribute to the theory by assessing the usefulness of certain cut-generating functions and developing methods for constructing new ones. Primary Reader: Amitabh Basu Secondary Reader: Daniel Robinso

On Minimal Valid Inequalities for Mixed Integer Conic Programs

We study disjunctive conic sets involving a general regular (closed, convex, full dimensional, and pointed) cone K such as the nonnegative orthant, the Lorentz cone or the positive semidefinite cone. In a unified framework, we introduce K-minimal inequalities and show that under mild assumptions, these inequalities together with the trivial cone-implied inequalities are sufficient to describe the convex hull. We study the properties of K-minimal inequalities by establishing algebraic necessary conditions for an inequality to be K-minimal. This characterization leads to a broader algebraically defined class of K- sublinear inequalities. We establish a close connection between K-sublinear inequalities and the support functions of sets with a particular structure. This connection results in practical ways of showing that a given inequality is K-sublinear and K-minimal. Our framework generalizes some of the results from the mixed integer linear case. It is well known that the minimal inequalities for mixed integer linear programs are generated by sublinear (positively homogeneous, subadditive and convex) functions that are also piecewise linear. This result is easily recovered by our analysis. Whenever possible we highlight the connections to the existing literature. However, our study unveils that such a cut generating function view treating the data associated with each individual variable independently is not possible in the case of general cones other than nonnegative orthant, even when the cone involved is the Lorentz cone

Light on the Infinite Group Relaxation

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.Comment: 45 page