The maximum number of faces of the Minkowski sum of three convex polytopes

We derive tight expressions for the maximum number of $k$-faces, $0\le{}k\le{}d-1$, of the Minkowski sum, $P_1+P_2+P_3$, of three $d$-dimensional convex polytopes $P_1$, $P_2$ and $P_3$ in $\reals^d$, as a function of the number of vertices of the polytopes, for any $d\ge{}2$. Expressing the Minkowski sum as a section of the Cayley polytope $\mathcal{C}$ of its summands, counting the $k$-faces of $P_1+P_2+P_3$ reduces to counting the $(k+2)$-faces of $\mathcal{C}$ which meet the vertex sets of the three polytopes. In two dimensions our expressions reduce to known results, while in three dimensions, the tightness of our bounds follows by exploiting known tight bounds for the number of faces of $r$ $d$-polytopes in $\reals^d$, where $r\ge d$. For $d\ge{}4$, the maximum values are attained when $P_1$, $P_2$ and $P_3$ are $d$-polytopes, whose vertex sets are chosen appropriately from three distinct $d$-dimensional moment-like curves

Embedding Formulations and Complexity for Unions of Polyhedra

It is well known that selecting a good Mixed Integer Programming (MIP) formulation is crucial for an effective solution with state-of-the art solvers. While best practices and guidelines for constructing good formulations abound, there is rarely a systematic construction leading to the best possible formulation. We introduce embedding formulations and complexity as a new MIP formulation paradigm for systematically constructing formulations for disjunctive constraints that are optimal with respect to size. More specifically, they yield the smallest possible ideal formulation (i.e. one whose LP relaxation has integral extreme points) among all formulations that only use 0-1 auxiliary variables. We use the paradigm to characterize optimal formulations for SOS2 constraints and certain piecewise linear functions of two variables. We also show that the resulting formulations can provide a significant computational advantage over all known formulations for piecewise linear functions.United States. National Science Foundation. (Grant CMMI-13516

Small and strong formulations for unions of convex sets from the Cayley embedding

There is often a significant trade-off between formulation strength and size in mixed integer programming (MIP). When modelling convex disjunctive constraints (e.g. unions of convex sets) this trade-off can be resolved by adding auxiliary continuous variables. However, adding these variables can result in a deterioration of the computational effectiveness of the formulation. For this reason, there has been considerable interest in constructing strong formulations that do not use continuous auxiliary variables. We introduce a technique to construct formulations without these detrimental continuous auxiliary variables. To develop this technique we introduce a natural nonpolyhedral generalization of the Cayley embedding of a family of polytopes and show it inherits many geometric properties of the original embedding. We then show how the associated formulation technique can be used to construct small and strong formulation for a wide range of disjunctive constraints. In particular, we show it can recover and generalize all known strong formulations without continuous auxiliary variables.National Science Foundation (U.S.) (grant CMMI-1351619

A geometric approach for the upper bound theorem for Minkowski sums of convex polytopes

We derive tight expressions for the maximum number of $k$-faces, $0\le{}k\le{}d-1$, of the Minkowski sum, $P_1+...+P_r$, of $r$ convex $d$-polytopes $P_1,...,P_r$ in $\mathbb{R}^d$, where $d\ge{}2$ and $r<d$, as a (recursively defined) function on the number of vertices of the polytopes. Our results coincide with those recently proved by Adiprasito and Sanyal [2]. In contrast to Adiprasito and Sanyal's approach, which uses tools from Combinatorial Commutative Algebra, our approach is purely geometric and uses basic notions such as $f$- and $h$-vector calculus and shellings, and generalizes the methodology used in [15] and [14] for proving upper bounds on the $f$-vector of the Minkowski sum of two and three convex polytopes, respectively. The key idea behind our approach is to express the Minkowski sum $P_1+...+P_r$ as a section of the Cayley polytope $\mathcal{C}$ of the summands; bounding the $k$-faces of $P_1+...+P_r$ reduces to bounding the subset of the $(k+r-1)$-faces of $\mathcal{C}$ that contain vertices from each of the $r$ polytopes. We end our paper with a sketch of an explicit construction that establishes the tightness of the upper bounds.Comment: 43 pages; minor changes (mostly typos

The maximum number of faces of the Minkowski sum of three convex polytopes

We derive tight expressions for the maximum number of $k$-faces, $0\le{}k\le{}d-1$, of the Minkowski sum, $P_1+P_2+P_3$, of three $d$-dimensional convex polytopes $P_1$, $P_2$ and $P_3$ in $\mathbb{R}^d$, as a function of the number of vertices of the polytopes, for any $d\ge{}2$. Expressing the Minkowski sum as a section of the Cayley polytope $\mathcal{C}$ of its summands, counting the $k$-faces of $P_1+P_2+P_3$ reduces to counting the $(k+2)$-faces of $\mathcal{C}$ that contain vertices from each of the three polytopes. In two dimensions our expressions reduce to known results, while in three dimensions, the tightness of our bounds follows by exploiting known tight bounds for the number of faces of $r$ $d$-polytopes in $\mathbb{R}^d$, where $r\ge d$. For $d\ge{}4$, the maximum values are attained when $P_1$, $P_2$ and $P_3$ are $d$-polytopes, whose vertex sets are chosen appropriately from three distinct $d$-dimensional moment-like curves