Approximation of corner polyhedra with families of intersection cuts

We study the problem of approximating the corner polyhedron using intersection cuts derived from families of lattice-free sets in $\mathbb{R}^n$. In particular, we look at the problem of characterizing families that approximate the corner polyhedron up to a constant factor, which depends only on $n$ and not the data or dimension of the corner polyhedron. The literature already contains several results in this direction. In this paper, we use the maximum number of facets of lattice-free sets in a family as a measure of its complexity and precisely characterize the level of complexity of a family required for constant factor approximations. As one of the main results, we show that, for each natural number $n$, a corner polyhedron with $n$ basic integer variables and an arbitrary number of continuous non-basic variables is approximated up to a constant factor by intersection cuts from lattice-free sets with at most $i$ facets if $i> 2^{n-1}$ and that no such approximation is possible if $i \leq 2^{n-1}$. When the approximation factor is allowed to depend on the denominator of the fractional vertex of the linear relaxation of the corner polyhedron, we show that the threshold is $i > n$ versus $i \leq n$. The tools introduced for proving such results are of independent interest for studying intersection cuts

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

Comparing Intersection Cut Closures using Simple Families of Lattice-Free Convex Sets

Mixed integer programs are a powerful mathematical tool, providing a general model for expressing both theoretically difficult and practically useful problems. One important subroutine of algorithms solving mixed integer programs is a cut generation procedure. The job of a cut generation procedure is to produce a linear inequality that separates a given infeasible point x* (usually a basic feasible solution of the linear programming relaxation) from the set of feasible solutions for the problem at hand. Early and well-known cut generation procedures rely on analyzing a single row of the simplex tableau for x*. Andersen et al. renewed interest in d-row cuts (i.e. cuts derived from d rows of the simplex tableau) by showing that these cuts afford some theoretical benefit. One lens from which to study d-row cuts is in the context of the intersection cuts of Balas and, in particular, intersection cuts obtained from lattice-free convex sets. The strongest d-row intersection cuts are obtained from maximal lattice-free convex sets in $R^d$ - all of which are polyhedra with at most $2^d$ facets. This thesis is concerned with theoretical comparison of the d-row cuts generated by different families of maximal lattice-free convex sets. We use the gauge measure to appraise the quality of the approximation. The main area of focus is 2-row cuts. The problem of generating 2-row cuts can be re-posed as generating valid inequalities for a mixed integer linear set F with two free integer variables and any number of non-negative continuous variables, where there are two defining equations. Every minimal valid inequality for the convex hull of F is an intersection cut generated by a maximal lattice-free split, triangle or quadrilateral. The family of maximal lattice-free triangles can be subdivided into the families of type 1, type 2, and type 3 triangles. Previous results of Basu et al. and Awate et al. compare how well the inequalities from one of these families approximates the convex hull of F (a.k.a. the corner polyhedron). In particular, the closure of all type 2 triangle inequalities is shown to be within a factor of 3/2 of the corner polyhedron. The authors also provide an instance where all type 2 triangles inequalities cannot approximate the corner polyhedron better than a factor of 9/8. The same bounds are shown for type 3 triangles and quadrilaterals. These results are obtained not by directly comparing the given closures to the convex hull of F, but rather to each other. In this thesis, we tighten one of the underlying bounds, showing that the closure of all type 2 triangle inequalities are within a factor of 5/4 of the closure of all quadrilateral inequalities. We also consider the sub-family of quadrilaterals where opposite edges have equal slope. We show that these parallelogram cuts can be approximated by all type 2 triangle inequalities within a factor of 9/8 and there exist instances where no better approximation is possible. In proving both these bounds, we use a subset of the family of type 2 triangles; we call the members of this sub-family ray-sliding triangles. A secondary area of focus in this thesis is d-row cuts for d >= 3. For d-row cuts in general, the underlying maximal lattice-free convex sets in $R^d$ are not easily classified. Absent a classification, Averkov et al, show that all inequalities generated by lattice-free convex sets with at most $i$ facets approximate the corner polyhedron within a finite factor only when $i > 2^{d-1}$. Here we take a different tact and try to prove analogues of 2-row cut results. We extend the proof techniques to obtain a constant factor approximation between two structured families of maximal lattice-free convex sets in $R^d$ for d >= 3

The computational complexity of convex bodies

We discuss how well a given convex body B in a real d-dimensional vector space V can be approximated by a set X for which the membership question: ``given an x in V, does x belong to X?'' can be answered efficiently (in time polynomial in d). We discuss approximations of a convex body by an ellipsoid, by an algebraic hypersurface, by a projection of a polytope with a controlled number of facets, and by a section of the cone of positive semidefinite quadratic forms. We illustrate some of the results on the Traveling Salesman Polytope, an example of a complicated convex body studied in combinatorial optimization.Comment: 24 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