Unique Minimal Liftings for Simplicial Polytopes

For a minimal inequality derived from a maximal lattice-free simplicial polytope in $\R^n$, we investigate the region where minimal liftings are uniquely defined, and we characterize when this region covers $\R^n$. We then use this characterization to show that a minimal inequality derived from a maximal lattice-free simplex in $\R^n$ with exactly one lattice point in the relative interior of each facet has a unique minimal lifting if and only if all the vertices of the simplex are lattice points.Comment: 15 page

Relaxations of mixed integer sets from lattice-free polyhedra

This paper gives an introduction to a recently established link between the geometry of numbers and mixed integer optimization. The main focus is to provide a review of families of lattice-free polyhedra and their use in a disjunctive programming approach. The use of lattice-free polyhedra in the context of deriving and explaining cutting planes for mixed integer programs is not only mathematically interesting, but it leads to some fundamental new discoveries, such as an understanding under which conditions cutting planes algorithms converge finitel

A (k+1)-Slope Theorem for the k-Dimensional Infinite Group Relaxation

We prove that any minimal valid function for the k-dimensional infinite group relaxation that is piecewise linear with at most k+1 slopes and does not factor through a linear map with non-trivial kernel is extreme. This generalizes a theorem of Gomory and Johnson for k=1, and Cornuejols and Molinaro for k=2.Comment: 25 pages, 2 figure

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

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

Combinatorial Optimization

Combinatorial Optimization is a very active field that benefits from bringing together ideas from different areas, e.g., graph theory and combinatorics, matroids and submodularity, connectivity and network flows, approximation algorithms and mathematical programming, discrete and computational geometry, discrete and continuous problems, algebraic and geometric methods, and applications. We continued the long tradition of triannual Oberwolfach workshops, bringing together the best researchers from the above areas, discovering new connections, and establishing new and deepening existing international collaborations

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