Annie Raymond

0–1 Multiband Robust Optimization

We provide an overview of new theoretical results that we obtained while further investigating multiband robust optimization, a new model for robust optimization that we recently proposed to tackle uncertainty in mixed-integer linear programming. This new model extends and refines the classical \(\varGamma \)-robustness model of Bertsimas and Sim and is particularly useful in the common case of arbitrary asymmetric distributions of the uncertainty. Here, we focus on uncertain 0–1 programs and we analyze their robust counterparts when the uncertainty is represented through a multiband set. Our investigations were inspired by the needs of our industrial partners in the research project ROBUKOM [2].

Symmetric sums of squares over k -subset hypercubes

We consider the problem of finding sum of squares (sos) expressions to establish the non-negativity of a symmetric polynomial over a discrete hypercube whose coordinates are indexed by the k-element subsets of [n]. For simplicity, we focus on the case

The bullet problem with discrete speeds

k=2

Multiband Robust Optimization and its Adoption in Harvest Scheduling

k=2, but our results extend naturally to all values of

New Conjectures for Union-Closed Families

k \ge 2

The Tur\'an Polytope

k≥2. We develop a variant of the Gatermann–Parrilo symmetry-reduction method tailored to our setting that allows for several simplifications and a connection to flag algebras. We show that every symmetric polynomial that has a sos expression of a fixed degree also has a succinct sos expression whose size depends only on the degree and not on the number of variables. Our method bypasses much of the technical difficulties needed to apply the Gatermann–Parrilo method, and offers flexibility in obtaining succinct sos expressions that are combinatorially meaningful. As a byproduct of our results, we arrive at a natural representation-theoretic justification for the concept of flags as introduced by Razborov in his flag algebra calculus. Furthermore, this connection exposes a family of non-negative polynomials that cannot be certified with any fixed set of flags, answering a question of Razborov in the context of our finite setting.