Optimizing hypergraph-based polynomials modeling job-occupancy in queueing 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 union of edges. These polynomials arise naturally to model job-occupancy in some queuing problems with redundancy scheduling policy. The question, posed by Cardinaels, Borst and van Leeuwaarden (arXiv:2005.14566, 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.Comment: 39 pages, including 2 figures and 10 table

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

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/

Exploiting ideal-sparsity in the generalized moment problem with application to matrix factorization ranks

We explore a new type of sparsity for the generalized moment problem (GMP) that we call ideal-sparsity. This sparsity exploits the presence of equality constraints requiring the measure to be supported on the variety of an ideal generated by bilinear monomials modeled by an associated graph. We show that this enables an equivalent sparse reformulation of the GMP, where the single (high dimensional) measure variable is replaced by several (lower-dimensional) measure variables supported on the maximal cliques of the graph. We explore the resulting hierarchies of moment-based relaxations for the original dense formulation of GMP and this new, equivalent ideal-sparse reformulation, when applied to the problem of bounding nonnegative- and completely positive matrix factorization ranks. We show that the ideal-sparse hierarchies provide bounds that are at least as good (and often tighter) as those obtained from the dense hierarchy. This is in sharp contrast to the situation when exploiting correlative sparsity, as is most common in the literature, where the resulting bounds are weaker than the dense bounds. Moreover, while correlative sparsity requires the underlying graph to be chordal, no such assumption is needed for ideal-sparsity. Numerical results show that the ideal-sparse bounds are often tighter and much faster to compute than their dense analogs.Comment: 36 pages, 3 figure

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]

Erythrocyte, Platelet, Serum Ferritin, and P-Selectin Pathophysiology Implicated in Severe Hypercoagulation and Vascular Complications in COVID-19

CITATION: Venter, C. et al. 2020. Erythrocyte, Platelet, Serum Ferritin, and P-Selectin Pathophysiology Implicated in Severe Hypercoagulation and Vascular Complications in COVID-19. International Journal of Molecular Sciences, 21(21). doi:10.3390/ijms21218234The original publication is available at https://www.mdpi.com/journal/ijmsProgressive respiratory failure is seen as a major cause of death in severe acute respiratory syndrome coronavirus 2 (SARS-Cov-2)-induced infection. Relatively little is known about the associated morphologic and molecular changes in the circulation of these patients. In particular, platelet and erythrocyte pathology might result in severe vascular issues, and the manifestations may include thrombotic complications. These thrombotic pathologies may be both extrapulmonary and intrapulmonary and may be central to respiratory failure. Previously, we reported the presence of amyloid microclots in the circulation of patients with coronavirus disease 2019 (COVID-19). Here, we investigate the presence of related circulating biomarkers, including C-reactive protein (CRP), serum ferritin, and P-selectin. These biomarkers are well-known to interact with, and cause pathology to, platelets and erythrocytes. We also study the structure of platelets and erythrocytes using fluorescence microscopy (using the markers PAC-1 and CD62PE) and scanning electron microscopy. Thromboelastography and viscometry were also used to study coagulation parameters and plasma viscosity. We conclude that structural pathologies found in platelets and erythrocytes, together with spontaneously formed amyloid microclots, may be central to vascular changes observed during COVID-19 progression, including thrombotic microangiopathy, diffuse intravascular coagulation, and large-vessel thrombosis, as well as ground-glass opacities in the lungs. Consequently, this clinical snapshot of COVID-19 strongly suggests that it is also a true vascular disease and considering it as such should form an essential part of a clinical treatment regime.https://www.mdpi.com/1422-0067/21/21/8234Publishers versio

Conceptualizing and measuring strategy implementation – a multi-dimensional view

Through quantitative methodological approaches for studying the strategic management and planning process, analysis of data from 208 senior managers involved in strategy processes within ten UK industrial sectors provides evidence on the measurement properties of a multi-dimensional instrument that assesses ten dimensions of strategy implementation. Using exploratory factor analysis, results indicate the sub-constructs (the ten dimensions) are uni-dimensional factors with acceptable reliability and validity; whilst using three additional measures, and correlation and hierarchical regression analysis, the nomological validity for the multi-dimensional strategy implementation construct was established. Relative importance of ten strategy implementation dimensions (activities) for practicing managers is highlighted, with the mutually and combinative effects drawing conclusion that senior management involvement leads the way among the ten key identified activities vital for successful strategy implementation