Alberto Del Pia 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 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
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
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
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
In this article, we provide an overview of some of our recent results on the facial structure of the multilinear polytope with a special focus on its decomposability properties. Namely, we demonstrate that, in the context of mixed-integer nonlinear optimization, the decomposability of the multilinear polytope plays a key role from both theoretical and algorithmic … Read more
The multilinear polytope MP_G of a hypergraph G is the convex hull of a set of binary points satisfying a collection of multilinear equations. We introduce the running-intersection inequalities, a new class of facet-defining inequalities for the multilinear polytope. Accordingly, we define a new polyhedral relaxation of MP_G referred to as the running-intersection relaxation and … Read more
We consider the NP-hard problem of minimizing a separable concave quadratic function over the integral points in a polyhedron, and we denote by D the largest absolute value of the subdeterminants of the constraint matrix. In this paper we give an algorithm that finds an epsilon-approximate solution for this problem by solving a number of … Read more
In this paper, we present lower bounds on the rank of the split closure, the multi-branch closure and the lattice-free closure for packing sets as a function of the integrality gap. We also provide a similar lower bound on the split rank of covering polyhedra. These results indicate that whenever the integrality gap is high, … Read more
Representability results play a fundamental role in optimization since they provide characterizations of the feasible sets that arise from optimization problems. In this paper we study the sets that appear in the feasibility version of mixed binary convex quadratic optimization problems. We provide a complete characterization of the sets that can be obtained as the … Read more