Polynomial optimization: matrix factorization ranks, portfolio selection, and queueing theory

Inspired by Leonhard Euler’s belief that every event in the world can be understood in terms of maximizing or minimizing a specific quantity, this thesis delves into the realm of mathematical optimization. The thesis is divided into four parts, with optimization acting as the unifying thread. Part 1 introduces a particular class of optimization problems called generalized moment problems (GMPs) and explores the moment method, a powerful tool used to solve GMPs. We introduce the new concept of ideal sparsity, a technique that aids in solving GMPs by improving the bounds of their associated hierarchy of semidefinite programs. Part 2 focuses on matrix factorization ranks, in particular, the nonnegative rank, the completely positive rank, and the separable rank. These ranks are extensively studied using the moment method, and ideal sparsity is applied (whenever possible) to enhance the bounds on these ranks and speed-up their computation. Part 3 centers around portfolio optimization and the mean-variance-skewness kurtosis (MVSK) problem. Multi-objective optimization techniques are employed to uncover Pareto optimal solutions to the MVSK problem. We show that most linear scalarizations of the MVSK problem result in specific convex polynomial optimization problems which can be solved efficiently. Part 4 explores hypergraph-based polynomials emerging from queueing theory in the setting of parallel-server systems with job redundancy policies. By exploiting the symmetry inherent in the polynomials and some classical results on matrix algebras, the convexity of these polynomials is demonstrated, thereby allowing us to prove that the polynomials attain their optima at the barycenter of the simplex.<br/

Matrix factorization ranks via polynomial optimization

In light of recent data science trends, new interest has fallen in alternative matrix factorizations. By this, we mean various ways of factorizing particular data matrices so that the factors have special properties and reveal insights into the original data. We are interested in the specialized ranks associated with these factorizations, but they are usually difficult to compute. In particular, we consider the nonnegative-, completely positive-, and separable ranks. We focus on a general tool for approximating factorization ranks, the moment hierarchy, a classical technique from polynomial optimization, further augmented by exploiting ideal-sparsity. Contrary to other examples of sparsity, the resulting sparse hierarchy yields equally strong, if not superior, bounds while potentially delivering a speed-up in computation

Optimizing hypergraph-based polynomials modeling job-occupancy in queuing with redundancy scheduling

We investigate two classes of multivariate polynomials with variables indexed by the edges of a uniform hypergraph and coefficients depending on certain patterns of unions of edges. These polynomials arise naturally to model job-occupancy in some queuing problems with redundancy scheduling policies. The question, posed by Cardinaels, Borst, and van Leeuwaarden in [Redundancy Scheduling with Locally Stable Compatibility Graphs, arXiv preprint, 2020], is to decide whether their global minimum over the standard simplex is attained at the uniform probability distribution. By exploiting symmetry properties of these polynomials we can give a positive answer for the first class and partial results for the second one, where we in fact show a stronger convexity property of these polynomials over the simplex

Bounding the separable rank via polynomial optimization

We investigate questions related to the set SEPd consisting of the linear maps ρ acting on Cd⊗Cd that can be written as a convex combination of rank one matrices of the form xx∗⊗yy∗. Such maps are known in quantum information theory as the separable bipartite states, while nonseparable states are called entangled. In particular we introduce bounds for the separable rank ranksep(ρ), defined as the smallest number of rank one states xx∗⊗yy∗ entering the decomposition of a separable state ρ. Our approach relies on the moment method and yields a hierarchy of semidefinite-based lower bounds, that converges to a parameter τsep(ρ), a natural convexification of the combinatorial parameter ranksep(ρ). A distinguishing feature is exploiting the positivity constraint ρ −xx∗⊗yy∗ 0 to impose positivity of a polynomial matrix localizing map, the dual notion of the notion of sum-of-squares polynomial matrices. Our approach extends naturally to the multipartite setting and to the real separable rank, and it permits strengthening some known bounds for the completely positive rank. In addition, we indicate how the moment approach also applies to define hierarchies of semidefinite relaxations for the set SEPd and permits to give new proofs, using only tools from moment theory, for convergence results on the DPS hierarchy from Doherty et al. (2002) [16]

JAndriesJ /ju-cp-rank

This code models the problem of bounding the completely positive rank of a symmetric nonnegative matrix via moment hierarchies