Sebastian Pokutta Augmentation methods for mixed-integer (linear) programs are a class of primal solution approaches in which a current iterate is augmented to a better solution or proved optimal. It is well known that the performance of these methods, i.e., number of iterations needed, can theoretically be improved by scaling methods. We extend these results by an … Read more
Robust optimization is an important technique to immunize optimization problems against data uncertainty. In the case of a linear program and an ellipsoidal uncertainty set, the robust counterpart turns into a second-order cone program. In this work, we investigate the efficiency of linearizing the second-order cone constraints of the latter. This is done using the … Read more
We introduce a natural abstraction of propositional proof systems that are based on cut- ting planes. This leads to a new class of proof systems that includes many well-known meth- ods, such as Gomory-Chvátal cuts, lift-and-project cuts, Sherali-Adams cuts, or split cuts. The rank of a proof system corresponds to the number of rounds that … Read more
We show that there are 0-1 and unbounded knapsack polytopes with super-polynomial extension complexity. More specifically, for each n in N we exhibit 0-1 and unbounded knapsack polyhedra in dimension n with extension complexity \Omega(2^\sqrt{n}). ArticleDownload View PDF
Valid inequalities or cutting planes for (mixed-) integer programming problems are an essential theoretical tool for studying combinatorial properties of polyhedra. They are also of utmost importance for solving optimization problems in practice; in fact any modern solver relies on several families of cutting planes. The Chvátal-Gomory procedure, one such approach, has a peculiarity that … Read more
A cutting-plane procedure for integer programming (IP) problems usually involves invoking a black-box procedure (such as the Gomory-Chvatal (GC) procedure) to compute a cutting-plane. In this paper, we describe an alternative paradigm of using the same cutting-plane black-box. This involves two steps. In the first step, we design an inequality cx = d + 1\} … Read more
We provide a complete characterization of all polytopes P ⊆ [0,1]^n with empty integer hull whose Gomory-Chvátal rank is n (and, therefore, maximal). In particular, we show that the first Gomory- Chvátal closure of all these polytopes is identical. ArticleDownload View PDF
We show that half-integral polytopes obtained as the convex hull of a random set of half-integral points of the 0/1 cube have rank as high as Ω(logn/loglogn) with positive probability — even if the size of the set relative to the total number of half-integral points of the cube tends to 0. The high rank … Read more
Although well studied, important questions on the rank of the Chvátal-Gomory operator when restricting to polytopes contained in the n-dimensional 0/1 cube have not been answered yet. In particular, the question on the maximal rank of the Chvátal-Gomory procedure for this class of polytopes is still open. So far, the best-known upper bound is O(n^2 … Read more
We will analyze mixed 0/1 second order cone programs where the fractional and binary variables are solely coupled via the conic constraints. For this special type of mixed-integer second order cone programs we devise a cutting-plane framework based on the generalized Benders cut and an implicit Sherali-Adams reformulation. We show that the resulting cuts are … Read more