A unified approach to mixed-integer optimization with logical constraints

We propose a united framework to address a family of classical mixed-component analysis, and sparse learning problems. These problems exhibit logical relationships between continuous and discrete variables, which are usually reformulated linearly using a big-M formulation. In this work, we challenge this longstanding modeling practice and express the logical constraints in a non-linear way. By imposing a regularization condition, we reformulate these problems as convex binary optimization problems, which are solvable using an outer-approximation procedure. In numerical experiments, we establish that a general-purpose numerical strategy, which combines cutting-plane, first-order, and local search methods, solves these problems faster and at a larger scale than state-of-the-art mixed-integer linear or second-order cone methods. Our approach successfully solves network design problems with 100s of nodes and provides solutions up to 40% better than the state-of-the-art; sparse portfolio selection problems with up to 3,200 securities compared with 400 securities for previous attempts; and sparse regression problems with up to 100,000 covariates

A new perspective on low-rank optimization

A key question in many low-rank problems throughout optimization, machine learning, and statistics is to characterize the convex hulls of simple low-rank sets and judiciously apply these convex hulls to obtain strong yet computationally tractable relaxations. We invoke the matrix perspective function the matrix analog of the perspective function to characterize explicitly the convex hull of epigraphs of simple matrix convex functions under low-rank constraints. Further, we combine the matrix perspective function with orthogonal projection matrices{the matrix analog of binary variables which capture the row-space of a matrix{to develop a matrix perspective reformulation technique that reliably obtains strong relaxations for a variety of low-rank problems, including reduced rank regression, non-negative matrix factorization, and factor analysis. Moreover, we establish that these relaxations can be modeled via semidenite constraints and thus optimized over tractably. The proposed approach parallels and generalizes the perspective reformulation technique in mixed-integer optimization and leads to new relaxations for a broad class of problems

Mixed-Projection Conic Optimization: A New Paradigm for Modeling Rank Constraints

We propose a framework for modeling and solving low-rank optimization problems to certifiable optimality. We introduce symmetric projection matrices that satisfy Y 2 =Y , the matrix analog of binary variables that satisfy z2 = z, to model rank constraints. By leveraging regularization and strong duality, we prove that this modeling paradigm yields tractable convex optimization problems over the non-convex set of orthogonal projection matrices. Furthermore, we design outer-approximation algorithms to solve low-rank problems to certifiable optimality, compute lower bounds via their semidenite relaxations, and provide near optimal solutions through rounding and local search techniques. We implement these numerical ingredients and, for the first time, solve low-rank optimization problems to certifiable optimality. Our algorithms also supply certifiably near-optimal solutions for larger problem sizes and outperform existing heuristics, by deriving an alternative to the popular nuclear norm relaxation which generalizes the perspective relaxation from vectors to matrices. Using currently available spatial branch-and-bound codes, not tailored to projection matrices, we can scale our exact (resp. near-exact) algorithms to matrices with up to 30 (resp. 600) rows/columns. All in all, our framework, which we name Mixed-Projection Conic Optimization, solves low-rank problems to certifiable optimality in a tractable and unified fashion

Solving Large-Scale Sparse PCA to Certifiable (Near) Optimality

Sparse principal component analysis (PCA) is a popular dimensionality reduction technique for obtaining principal components which are linear combinations of a small subset of the original features. Existing approaches cannot supply certifiably optimal principal components with more than *p* = 100s of variables. By reformulating sparse PCA as a convex mixed-integer semidefinite optimization problem, we design a cutting-plane method which solves the problem to certifiable optimality at the scale of selecting *k* = 5 covariates from *p*= 300 variables, and provides small bound gaps at a larger scale. We also propose a convex relaxation and greedy rounding scheme that provides bound gaps of 1 - 2% in practice within minutes for *p* = 100s or hours for *p*= 1; 000s and is therefore a viable alternative to the exact method at scale. Using real-world financial and medical data sets, we illustrate our approach's ability to derive interpretable principal components tractably at scale

High-Speed Operation of Interband Cascade Lasers

Optical sources operating in the atmospheric window of 3-5 microns are of particular interest for the development of free-space optical communication link. It is more advantageous to operate the free-space optical communication link in 3-5-microns atmospheric transmission window than at the telecom wavelength of 1.5 m due to lower optical scattering, scintillation, and background radiation. However, the realization of optical communications at the longer wavelength has encountered significant difficulties due to lack of adequate optical sources and detectors operating in the desirable wavelength regions. Interband Cascade (IC) lasers are novel semiconductor lasers that have a great potential for the realization of high-power, room-temperature optical sources in the 3-5-microns wavelength region, yet no experimental work, until this one, was done on high-speed direct modulation of IC lasers. Here, highspeed interband cascade laser, operating at wavelength 3.0 m, has been developed and the first direct measurement of the laser modulation bandwidth has been performed using a unique, highspeed quantum well infrared photodetector (QWIP). The developed laser has modulation bandwidth exceeding 3 GHz. This constitutes a significant increase of the IC laser modulation bandwidth over currently existing devices. This result has demonstrated suitability of IC lasers as a mid-IR light source for multi-GHz free-space optical communications link

Enhanced stability and local structure in biologically relevant amorphous materials containing pyrophosphate

There is increasing evidence that amorphous inorganic materials play a key role in biomineralisation in many organisms, however the inherent instability of synthetic analogues in the absence of the complex in vivo matrix limits their study and clinical exploitation. To address this, we report here an approach that enhances long-term stability to >1 year of biologically relevant amorphous metal phosphates, in the absence of any complex stabilisers, by utilising pyrophosphates (P2O7 4-); species themselves ubiquitous in vivo. Ambient temperature precipitation reactions were employed to synthesise amorphous Ca2P2O7.nH2O and Sr2P2O7.nH2O (3.8 < n < 4.2) and their stability and structure were investigated. Pair distribution functions (PDF) derived from synchrotron X-ray data indicated a lack of structural order beyond ~8 A° in both phases, with this local order found to resemble crystalline analogues. Further studies, including 1H and 31P solid state NMR, suggest the unusually high stability of these purely inorganic amorphous phases is partly due to disorder in the P–O–P bond angles within the P2O7 units, which impede crystallization, and to water molecules, which are involved in H-bonds of various strengths within the structures and hamper the formation of an ordered network. In situ high temperature powder X-ray diffraction data indicated that the amorphous nature of both phases surprisingly persisted to ~450° C. Further NMR and TGA studies found that above ambient temperature some water molecules reacted with P2O7 anions, leading to the hydrolysis of some P–O–P linkages and the formation of HPO4 2- anions within the amorphous matrix. The latter anions then recombined into P2O7 ions at higher temperatures prior to crystallization. Together, these findings provide important new materials with unexplored potential for enzyme-assisted resorption and establish factors crucial to isolate further stable amorphous inorganic materials

Correlations of Gene Expression with Blood Lead Levels in Children with Autism Compared to Typically Developing Controls

The objective of this study was to examine the correlation between gene expression and lead (Pb) levels in blood in children with autism (AU, n = 37) compared to typically developing controls (TD, n = 15). We postulated that, though lead levels did not differ between the groups, AU children might metabolize lead differently compared to TD children. RNA was isolated from blood and processed on Affymetrix microarrays. Separate analyses of covariance (ANCOVA) corrected for age and gender were performed for TD, AU, and all subjects (AU + TD). To reduce false positives, only genes that overlapped these three ANCOVAs were considered. Thus, 48 probe sets correlated with lead levels in both AU and TD subjects and were significantly different between the groups (p(Diagnosis × log2 Pb) < 0.05). These genes were related mainly to immune and inflammatory processes, including MHC Class II family members and CD74. A large number (n = 791) of probe sets correlated (P ≤ 0.05) with lead levels in TD but not in AU subjects; and many probe sets (n = 162) correlated (P ≤ 0.05) with lead levels in AU but not in TD subjects. Only 30 probe sets correlated (P ≤ 0.05) with lead levels in a similar manner in the AU and TD groups. These data show that AU and TD children display different associations between transcript levels and low levels of lead. We postulate that this may relate to the underlying genetic differences between the two groups, though other explanations cannot be excluded

Genetic predisposition to mosaic Y chromosome loss in blood.

Mosaic loss of chromosome Y (LOY) in circulating white blood cells is the most common form of clonal mosaicism1-5, yet our knowledge of the causes and consequences of this is limited. Here, using a computational approach, we estimate that 20% of the male population represented in the UK Biobank study (n = 205,011) has detectable LOY. We identify 156 autosomal genetic determinants of LOY, which we replicate in 757,114 men of European and Japanese ancestry. These loci highlight genes that are involved in cell-cycle regulation and cancer susceptibility, as well as somatic drivers of tumour growth and targets of cancer therapy. We demonstrate that genetic susceptibility to LOY is associated with non-haematological effects on health in both men and women, which supports the hypothesis that clonal haematopoiesis is a biomarker of genomic instability in other tissues. Single-cell RNA sequencing identifies dysregulated expression of autosomal genes in leukocytes with LOY and provides insights into why clonal expansion of these cells may occur. Collectively, these data highlight the value of studying clonal mosaicism to uncover fundamental mechanisms that underlie cancer and other ageing-related diseases.This research has been conducted using the UK Biobank Resource under application 9905 and 19808. This work was supported by the Medical Research Council [Unit Programme number MC_UU_12015/2]. Full study-specific and individual acknowledgements can be found in the supplementary information

Rare coding variants in PLCG2, ABI3, and TREM2 implicate microglial-mediated innate immunity in Alzheimer's disease

We identified rare coding variants associated with Alzheimer’s disease (AD) in a 3-stage case-control study of 85,133 subjects. In stage 1, 34,174 samples were genotyped using a whole-exome microarray. In stage 2, we tested associated variants (P<1×10-4) in 35,962 independent samples using de novo genotyping and imputed genotypes. In stage 3, an additional 14,997 samples were used to test the most significant stage 2 associations (P<5×10-8) using imputed genotypes. We observed 3 novel genome-wide significant (GWS) AD associated non-synonymous variants; a protective variant in PLCG2 (rs72824905/p.P522R, P=5.38×10-10, OR=0.68, MAFcases=0.0059, MAFcontrols=0.0093), a risk variant in ABI3 (rs616338/p.S209F, P=4.56×10-10, OR=1.43, MAFcases=0.011, MAFcontrols=0.008), and a novel GWS variant in TREM2 (rs143332484/p.R62H, P=1.55×10-14, OR=1.67, MAFcases=0.0143, MAFcontrols=0.0089), a known AD susceptibility gene. These protein-coding changes are in genes highly expressed in microglia and highlight an immune-related protein-protein interaction network enriched for previously identified AD risk genes. These genetic findings provide additional evidence that the microglia-mediated innate immune response contributes directly to AD development