Multi-cover Inequalities for Totally-Ordered Multiple Knapsack Sets: Theory and Computation

We propose a method to generate cutting-planes from multiple covers of knapsack constraints. The covers may come from different knapsack inequalities if the weights in the inequalities form a totally-ordered set. Thus we introduce and study the structure of a totally-ordered multiple knapsack set. The valid multi-cover inequalities we derive for its convex hull have … Read more

Linear Programming and Community Detection

The problem of community detection with two equal-sized communities is closely related to the minimum graph bisection problem over certain random graph models. In the stochastic block model distribution over networks with community structure, a well-known semidefinite programming (SDP) relaxation of the minimum bisection problem recovers the underlying communities whenever possible. Motivated by their superior … Read more

Proximity in Concave Integer Quadratic Programming

A classic result by Cook, Gerards, Schrijver, and Tardos provides an upper bound of n∆ on the proximity of optimal solutions of an Integer Linear Programming problem and its standard linear relaxation. In this bound, n is the number of variables and ∆ denotes the maximum of the absolute values of the subdeterminants of the … Read more

Integer packing sets form a well-quasi-ordering

An integer packing set is a set of non-negative integer vectors with the property that, if a vector x is in the set, then every non-negative integer vector y with y ≤ x is in the set as well. Integer packing sets appear naturally in Integer Optimization. In fact, the set of integer points in … Read more

Short simplex paths in lattice polytopes

We consider the problem of optimizing a linear function over a lattice polytope P contained in [0,k]^n and defined via m linear inequalities. We design a simplex algorithm that, given an initial vertex, reaches an optimal vertex by tracing a path along the edges of P of length at most O(n^6 k log k). The … Read more

On the complexity of binary polynomial optimization over acyclic hypergraphs

In this work we advance the understanding of the fundamental limits of computation for Binary Polynomial Optimization (BPO), which is the problem of maximizing a given polynomial function over all binary points. In our main result we provide a novel class of BPO that can be solved efficiently both from a theoretical and computational perspective. … Read more

Sparse PCA on fixed-rank matrices

Sparse PCA is the optimization problem obtained from PCA by adding a sparsity constraint on the principal components. Sparse PCA is NP-hard and hard to approximate even in the single-component case. In this paper we settle the computational complexity of sparse PCA with respect to the rank of the covariance matrix. We show that, if … Read more

Chvatal rank in binary polynomial optimization

Recently, several classes of cutting planes have been introduced for binary polynomial optimization. In this paper, we present the first results connecting the combinatorial structure of these inequalities with their Chvatal rank. We show that almost all known cutting planes have Chvatal rank 1. All these inequalities have an associated hypergraph that is beta-acyclic, thus, … Read more

Subset selection in sparse matrices

In subset selection we search for the best linear predictor that involves a small subset of variables. From a computational complexity viewpoint, subset selection is NP-hard and few classes are known to be solvable in polynomial time. Using mainly tools from discrete geometry, we show that some sparsity conditions on the original data matrix allow … Read more

On the impact of running intersection inequalities for globally solving polynomial optimization problems

We consider global optimization of nonconvex problems whose factorable reformulations contain a collection of multilinear equations. Important special cases include multilinear and polynomial optimization problems. The multilinear polytope is the convex hull of a set of binary points satisfying a number of multilinear equations. Running intersection inequalities are a family of facet-defining inequalities for the … Read more