Symmetry reduction in convex optimization with applications in combinatorics

This dissertation explores different approaches to and applications of symmetry reduction in convex optimization. Using tools from semidefinite programming, representation theory and algebraic combinatorics, hard combinatorial problems are solved or bounded. The first chapters consider the Jordan reduction method, extend the method to optimization over the doubly nonnegative cone, and apply it to quadratic assignment problems and energy minimization on a discrete torus. The following chapter uses symmetry reduction as a proving tool, to approach a problem from queuing theory with redundancy scheduling. The final chapters propose generalizations and reductions of flag algebras, a powerful tool for problems coming from extremal combinatorics

New lower bounds on crossing numbers of Km,nK_{m,n} from semidefinite programming

In this paper, we use semidefinite programming and representation theory to compute new lower bounds on the crossing number of the complete bipartite graph $K_{m,n}$, extending a method from de Klerk et al. [SIAM J. Discrete Math. 20 (2006), 189--202] and the subsequent reduction by De Klerk, Pasechnik and Schrijver [Math. Prog. Ser. A and B, 109 (2007) 613--624]. We exploit the full symmetry of the problem using a novel decomposition technique. This results in a full block-diagonalization of the underlying matrix algebra, which we use to improve bounds on several concrete instances. Our results imply that $\text{cr}(K_{10,n}) \geq 4.87057 n^2 - 10n$, $\text{cr}(K_{11,n}) \geq 5.99939 n^2-12.5n$, $\text{cr}(K_{12,n}) \geq 7.25579 n^2 - 15n$, $\text{cr}(K_{13,n}) \geq 8.65675 n^2-18n$ for all $n$. The latter three bounds are computed using a new and well-performing relaxation of the original semidefinite programming bound. This new relaxation is obtained by only requiring one small matrix block to be positive semidefinite.Comment: 17 pages, 3 figures, 3 tables. Revisions have been made based on comments of the referees. Accepted for publication in Mathematical Programmin

Jordan symmetry reduction for conic optimization over the doubly nonnegative cone: theory and software

A common computational approach for polynomial optimization problems (POPs) is to use (hierarchies of) semidefinite programming (SDP) relaxations. When the variables in the POP are required to be nonnegative, these SDP problems typically involve nonnegative matrices, i.e. they are conic optimization problems over the doubly nonnegative cone. The Jordan reduction, a symmetry reduction method for conic optimization, was recently introduced for symmetric cones by Parrilo and Permenter [Mathematical Programming 181(1), 2020]. We extend this method to the doubly nonnegative cone, and investigate its application to known relaxations of the quadratic assignment and maximum stable set problems. We also introduce new Julia software where the symmetry reduction is implemented.Comment: 19 pages, titled change from earlier version. arXiv admin note: text overlap with arXiv:1908.0087

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

The MtZ Strain: Molecular Characteristics and Outbreak Investigation of the Most Successful Mycobacterium tuberculosis Strain in Aragon Using Whole-Genome Sequencing

Since 2004, a tuberculosis surveillance protocol has been carried out in Aragon, thereby managing to detect all tuberculosis outbreaks that take place in the community. The largest outbreak was caused by a strain named Mycobacterium tuberculosis Zaragoza (MtZ), causing 242 cases as of 2020. The main objective of this work was to analyze this outbreak and the molecular characteristics of this successful strain that could be related to its greater transmission. To do this, we first applied whole-genome sequencing to 57 of the isolates. This revealed two principal transmission clusters and six subclusters arising from them. The MtZ strain belongs to L4.8 and had eight specific single nucleotide polymorphisms (SNPs) in genes considered to be virulence factors [ptpA, mc3D, mc3F, VapB41, pks15 (two SNPs), virS, and VapC50]. Second, a transcriptomic study was carried out to better understand the multiple IS6110 copies present in its genome. This allowed us to observe three effects of IS6110: the disruption of the gene in which the IS6110 is inserted (desA3), the overexpression of a gene (ppe38), and the absence of transcription of genes (cut1:Rv1765c) due to the recombination of two IS6110 copies. Finally, because of the disruption of ppe38 and ppe71 genes by an IS6110, a study of PE_PGRS secretion was carried out, showing that MtZ secretes these factors in higher amounts than the reference strain, thereby differing from the hypervirulent phenotype described for the Beijing strains. In conclusion, MtZ consists of several SNPs in genes related to virulence, pathogenesis, and survival, as well as other genomic polymorphisms, which may be implicated in its success among our population

A large margin algorithm for automated segmentation of white matter hyperintensity

Precise detection and quantification of white matter hyperintensity (WMH) is of great interest in studies of neurological and vascular disorders. In this work, we propose a novel method for automatic WMH segmentation with both supervised and semi-supervised large margin algorithms provided by the framework. The proposed algorithms optimize a kernel based max-margin objective function which aims to maximize the margin between inliers and outliers. We show that the semi-supervised learning problem can be formulated to learn a classifier and label assignment simultaneously, which can be solved efficiently by an iterative algorithm. The model is learned first via the supervised approach and then fine-tuned on a target image by using the semi-supervised algorithm. We evaluate our method on 88 brain fluid-attenuated inversion recovery (FLAIR) magnetic resonance (MR) images from subjects with vascular disease. Quantitative evaluation of the proposed approach shows that it outperforms other well known methods for WMH segmentation

Resistance to First-Line Anti-TB Drugs Is Associated with Reduced Nitric Oxide Susceptibility in Mycobacterium tuberculosis

Background and objective: The relative contribution of nitric oxide (NO) to the killing of Mycobacterium tuberculosis in human tuberculosis (TB) is controversial, although this has been firmly established in rodents. Studies have demonstrated that clinical strains of M. tuberculosis differ in susceptibility to NO, but how this correlates to drug resistance and clinical outcome is not known. Methods: In this study, 50 sputum smear- and culture-positive patients with pulmonary TB in Gondar, Ethiopia were included. Clinical parameters were recorded and drug susceptibility profile and spoligotyping patterns were investigated. NO susceptibility was studied by exposing the strains to the NO donor DETA/NO. Results: Clinical isolates of M. tuberculosis showed a dose- and time-dependent response when exposed to NO. The most frequent spoligotypes found were CAS1-Delhi and T3_ETH in a total of nine known spoligotypes and four orphan patterns. There was a significant association between reduced susceptibility to NO (>10% survival after exposure to 1mM DETA/NO) and resistance against first-line anti-TB drugs, in particular isoniazid (INH). Patients infected with strains of M. tuberculosis with reduced susceptibility to NO showed no difference in cure rate or other clinical parameters, but a tendency towards lower rate of weight gain after two months of treatment. Conclusion: There is a correlation between resistance to first-line anti-TB drugs and reduced NO susceptibility in clinical strains of M. tuberculosis. Further studies including the mechanisms of reduced NO susceptibility are warranted and could identify targets for new therapeutic interventions

Detection and Molecular Characterization of 9000-Year-Old Mycobacterium tuberculosis from a Neolithic Settlement in the Eastern Mediterranean

Background: Mycobacterium tuberculosis is the principal etiologic agent of human tuberculosis. It has no environmental reservoir and is believed to have co-evolved with its host over millennia. This is supported by skeletal evidence of the disease in early humans, and inferred from M. tuberculosis genomic analysis. Direct examination of ancient human remains for M. tuberculosis biomarkers should aid our understanding of the nature of prehistoric tuberculosis and the host/pathogen relationship.Methodology/Principal Findings: We used conventional PCR to examine bone samples with typical tuberculosis lesions from a woman and infant, who were buried together in the now submerged site of Atlit-Yam in the Eastern Mediterranean, dating from 9250-8160 years ago. Rigorous precautions were taken to prevent contamination, and independent centers were used to confirm authenticity of findings. DNA from five M. tuberculosis genetic loci was detected and had characteristics consistent with extant genetic lineages. High performance liquid chromatography was used as an independent method of verification and it directly detected mycolic acid lipid biomarkers, specific for the M. tuberculosis complex.Conclusions/Significance: Human tuberculosis was confirmed by morphological and molecular methods in a population living in one of the first villages with evidence of agriculture and animal domestication. The widespread use of animals was not a source of infection but may have supported a denser human population that facilitated transmission of the tubercle bacillus. The similarity of the M. tuberculosis genetic signature with those of today gives support to the theory of a long-term co-existence of host and pathogen

TBVAC2020: Advancing tuberculosis vaccines from discovery to clinical development

TBVAC2020 is a research project supported by the Horizon 2020 program of the European Commission (EC). It aims at the discovery and development of novel tuberculosis (TB) vaccines from preclinical research projects to early clinical assessment. The project builds on previous collaborations from 1998 onwards funded through the EC framework programs FP5, FP6, and FP7. It has succeeded in attracting new partners from outstanding laboratories from all over the world, now totaling 40 institutions. Next to the development of novel vaccines, TB biomarker development is also considered an important asset to facilitate rational vaccine selection and development. In addition, TBVAC2020 offers portfolio management that provides selection criteria for entry, gating, and priority settings of novel vaccines at an early developmental stage. The TBVAC2020 consortium coordinated by TBVI facilitates collaboration and early data sharing between partners with the common aim of working toward the development of an effective TB vaccine. Close links with funders and other consortia with shared interests further contribute to this goal