Geometry of the copositive and completely positive cones

The copositive cone, and its dual the completely positive cone, have useful applications in optimisation, however telling if a general matrix is in the copositive cone is a co-NP-complete problem. In this paper we analyse some of the geometry of these cones. We discuss a way of representing all the maximal faces of the copositive cone along with a simple equation for the dimension of each one. In doing this we show that the copositive cone has faces which are isomorphic to positive semidefinite cones. We also look at some maximal faces of the completely positive cone and find their dimensions. Additionally we consider extreme rays of the copositive and completely positive cones and show that every extreme ray of the completely positive cone is also an exposed ray, but the copositive cone has extreme rays which are not exposed rays

Considering copositivity locally

Let $A$ be an element of the copositive cone \copos{n}. A zero \vu of $A$ is a nonnegative vector whose elements sum up to one and such that \vu^TA\vu = 0. The support of \vu is the index set \Supp{\vu} \subset \{1,\dots,n\} corresponding to the nonzero entries of \vu. A zero \vu of $A$ is called minimal if there does not exist another zero \vv of $A$ such that its support \Supp{\vv} is a strict subset of \Supp{\vu}. Our main result is a characterization of the cone of feasible directions at $A$, i.e., the convex cone \VarK{A} of real symmetric $n \times n$ matrices $B$ such that there exists $\delta > 0$ satisfying A + \delta B \in \copos{n}. This cone is described by a set of linear inequalities on the elements of $B$ constructed from the set of zeros of $A$ and their supports. This characterization furnishes descriptions of the minimal face of $A$ in \copos{n}, and of the minimal exposed face of $A$ in \copos{n}, by sets of linear equalities and inequalities constructed from the set of minimal zeros of $A$ and their supports. In particular, we can check whether $A$ lies on an extreme ray of \copos{n} by examining the solution set of a system of linear equations. In addition, we deduce a simple necessary and sufficient condition on the irreducibility of $A$ with respect to a copositive matrix $C$. Here $A$ is called irreducible with respect to $C$ if for all $\delta > 0$ we have A - \delta C \not\in \copos{n}

A new approximation hierarchy for polynomial conic optimization

In this paper we consider polynomial conic optimization problems, where the feasible set is defined by constraints in the form of given polynomial vectors belonging to given nonempty closed convex cones, and we assume that all the feasible solutions are non-negative. This family of problems captures in particular polynomial optimization problems (POPs), polynomial semi-definite polynomial optimization problems (PSDPs) and polynomial second-order cone-optimization problems (PSOCPs). We propose a new general hierarchy of linear conic optimization relaxations inspired by an extension of Pólyaʼs Positivstellensatz for homogeneous polynomials being positive over a basic semi-algebraic cone contained in the non-negative orthant, introduced in Dickinson and Povh (J Glob Optim 61(4):615-625, 2015). We prove that based on some classic assumptions, these relaxations converge monotonically to the optimal value of the original problem. Adding a redundant polynomial positive semi-definite constraint to the original problem drastically improves the bounds produced by our method. We provide an extensive list of numerical examples that clearly indicate the advantages and disadvantages of our hierarchy. In particular, in comparison to the classic approach of sum-of-squares, our new method provides reasonable bounds on the optimal value for POPs, and strong bounds for PSDPs and PSOCPs, even outperforming the sum-of-squares approach in these latter two cases.V članku obravnavamo polinomske konične optimizacijske probleme, kjer je dopustna množica definirana z omejitvami, da morajo biti dani polinomski vektorji v danih nepraznih zaprtih konveksnih stožcih. Dodatno morajo dopustne rešitve zadoščati pogoju nenegativnosti. Ta družina problemov zajema zlasti klasične probleme polinomske optimizacije (POP), probleme polinomske semidefinitne optimizacije (PSDP) in probleme polinomske optimizacije nad stožci drugega reda (PSOCP). Predlagamo novo splošno hierarhijo linearnih koničnih optimizacijskih poenostavitev, ki naravno sledijo iz razširitve Pólya-jevega izreka o pozitivnosti iz Dickinson in Povh (J Glob Optim 61 (4): 615-625, 2015). Ob nekaterih klasičnih predpostavkah te poenostavitve monotono konvergirajo k optimalni vrednosti izvirnega problema. Kot zanimivost pokažemo, da dodajanje posebne redundantne omejitve k osnovnemu problemu ne spremeni optimalne rešitve tega problema, a bistveno izboljša kvaliteto poenostavitev. V članku tudi predstavimo obsežen seznam številčnih primerov, ki jasno kažejo na prednosti in slabosti naše hierarhije

On the Exhaustivity of Simplicial Partitioning

Abstract During the last 40 years, simplicial partitioning has shown itself to be highly useful, including in the field of Nonlinear Optimisation. In this article, we consider results on the exhaustivity of simplicial partitioning schemes. We consider conjectures on this exhaustivity which seem at first glance to be true (two of which have been stated as true in published articles). However, we will provide counter examples to these conjectures. We also provide a new simplicial partitioning scheme, which provides a lot of freedom, whilst guaranteeing exhaustivity. Mathematics Subject Classification: 65K99; 90C2

Planck 2015 results: XXV. Diffuse low-frequency Galactic foregrounds

We discuss the Galactic foreground emission between 20 and 100 GHz based on observations by Planck and WMAP. The total intensity in this part of the spectrum is dominated by free-free and spinning dust emission, whereas the polarized intensity is dominated by synchrotron emission. The Commander component-separation tool has been used to separate the various astrophysical processes in total intensity. Comparison with radio recombination line templates verifies the recovery of the free-free emission along the Galactic plane. Comparison of the high-latitude Hα emission with our free-free map shows residuals that correlate with dust optical depth, consistent with a fraction (≈30%) of Hα having been scattered by high-latitude dust. We highlight a number of diffuse spinning dust morphological features at high latitude. There is substantial spatial variation in the spinning dust spectrum, with the emission peak (in Iν) ranging from below 20 GHz to more than 50 GHz. There is a strong tendency for the spinning dust component near many prominent H ii regions to have a higher peak frequency, suggesting that this increase in peak frequency is associated with dust in the photo-dissociation regions around the nebulae. The emissivity of spinning dust in these diffuse regions is of the same order as previous detections in the literature. Over the entire sky, the Commander solution finds more anomalous microwave emission (AME) than the WMAP component maps, at the expense of synchrotron and free-free emission. This can be explained by the difficulty in separating multiple broadband components with a limited number of frequency maps. Future surveys, particularly at 5–20 GHz, will greatly improve the separation by constraining the synchrotron spectrum. We combine Planck and WMAP data to make the highest signal-to-noise ratio maps yet of the intensity of the all-sky polarized synchrotron emission at frequencies above a few GHz. Most of the high-latitude polarized emission is associated with distinct large-scale loops and spurs, and we re-discuss their structure. We argue that nearly all the emission at 40deg > l > −90deg is part of the Loop I structure, and show that the emission extends much further in to the southern Galactic hemisphere than previously recognised, giving Loop I an ovoid rather than circular outline. However, it does not continue as far as the “Fermi bubble/microwave haze”, making it less probable that these are part of the same structure. We identify a number of new faint features in the polarized sky, including a dearth of polarized synchrotron emission directly correlated with a narrow, roughly 20deg long filament seen in Hα at high Galactic latitude. Finally, we look for evidence of polarized AME, however many AME regions are significantly contaminated by polarized synchrotron emission, and we find a 2σ upper limit of 1.6% in the Perseus region

Fractionation effects in phase equilibria of polydisperse hard sphere colloids

The equilibrium phase behaviour of hard spheres with size polydispersity is studied theoretically. We solve numerically the exact phase equilibrium equations that result from accurate free energy expressions for the fluid and solid phases, while accounting fully for size fractionation between coexisting phases. Fluids up to the largest polydispersities that we can study (around 14%) can phase separate by splitting off a solid with a much narrower size distribution. This shows that experimentally observed terminal polydispersities above which phase separation no longer occurs must be due to non-equilibrium effects. We find no evidence of re-entrant melting; instead, sufficiently compressed solids phase separate into two or more solid phases. Under appropriate conditions, coexistence of multiple solids with a fluid phase is also predicted. The solids have smaller polydispersities than the parent phase as expected, while the reverse is true for the fluid phase, which contains predominantly smaller particles but also residual amounts of the larger ones. The properties of the coexisting phases are studied in detail; mean diameter, polydispersity and volume fraction of the phases all reveal marked fractionation. We also propose a method for constructing quantities that optimally distinguish between the coexisting phases, using Principal Component Analysis in the space of density distributions. We conclude by comparing our predictions to perturbative theories for near-monodisperse systems and to Monte Carlo simulations at imposed chemical potential distribution, and find excellent agreement.Comment: 21 pages, 23 figures, 2 table

The Infrared Luminosity Function of Galaxies in the Coma Cluster

An infrared survey of the central 650 arcmin$^2$ of the Coma cluster is used to determine the $H$ band luminosity function for the cluster. Redshifts are available for all galaxies in the survey with $H < 14.5$, and for this sample we obtain a good fit to a Schechter function with $H^*=11.13$ and $\alpha=-0.78$. These luminosity function parameters are similar to those measured for field galaxies in the infrared, which is surprising considering the very different environmental densities and, presumably, merger histories for field galaxies. For fainter galaxies, we use two independent techniques to correct for field galaxy contamination in the cluster population: the $B-R$ color-magnitude relation and an estimate for the level of background and foreground contamination from the literature. Using either method we find a steep upturn for galaxies with $14 < H < 16$, with slope $\alpha \simeq 1.7$.Comment: 15 pages, 2 figures Accepted by ApJ Letter

FGF4 retrogene on CFA12 is responsible for chondrodystrophy and intervertebral disc disease in dogs.

Chondrodystrophy in dogs is defined by dysplastic, shortened long bones and premature degeneration and calcification of intervertebral discs. Independent genome-wide association analyses for skeletal dysplasia (short limbs) within a single breed (PBonferroni = 0.01) and intervertebral disc disease (IVDD) across breeds (PBonferroni = 4.0 × 10-10) both identified a significant association to the same region on CFA12. Whole genome sequencing identified a highly expressed FGF4 retrogene within this shared region. The FGF4 retrogene segregated with limb length and had an odds ratio of 51.23 (95% CI = 46.69, 56.20) for IVDD. Long bone length in dogs is a unique example of multiple disease-causing retrocopies of the same parental gene in a mammalian species. FGF signaling abnormalities have been associated with skeletal dysplasia in humans, and our findings present opportunities for both selective elimination of a medically and financially devastating disease in dogs and further understanding of the ever-growing complexity of retrogene biology

Planck 2015 results: XXV. Diffuse low-frequency Galactic foregrounds

We discuss the Galactic foreground emission between 20 and 100 GHz based on observations by Planck and WMAP. The total intensity in this part of the spectrum is dominated by free-free and spinning dust emission, whereas the polarized intensity is dominated by synchrotron emission. The Commander component-separation tool has been used to separate the various astrophysical processes in total intensity. Comparison with radio recombination line templates verifies the recovery of the free-free emission along the Galactic plane. Comparison of the high-latitude Hα emission with our free-free map shows residuals that correlate with dust optical depth, consistent with a fraction (≈30%) of Hα having been scattered by high-latitude dust. We highlight a number of diffuse spinning dust morphological features at high latitude. There is substantial spatial variation in the spinning dust spectrum, with the emission peak (in Iν) ranging from below 20 GHz to more than 50 GHz. There is a strong tendency for the spinning dust component near many prominent H ii regions to have a higher peak frequency, suggesting that this increase in peak frequency is associated with dust in the photo-dissociation regions around the nebulae. The emissivity of spinning dust in these diffuse regions is of the same order as previous detections in the literature. Over the entire sky, the Commander solution finds more anomalous microwave emission (AME) than the WMAP component maps, at the expense of synchrotron and free-free emission. This can be explained by the difficulty in separating multiple broadband components with a limited number of frequency maps. Future surveys, particularly at 5–20 GHz, will greatly improve the separation by constraining the synchrotron spectrum. We combine Planck and WMAP data to make the highest signal-to-noise ratio maps yet of the intensity of the all-sky polarized synchrotron emission at frequencies above a few GHz. Most of the high-latitude polarized emission is associated with distinct large-scale loops and spurs, and we re-discuss their structure. We argue that nearly all the emission at 40deg > l > −90deg is part of the Loop I structure, and show that the emission extends much further in to the southern Galactic hemisphere than previously recognised, giving Loop I an ovoid rather than circular outline. However, it does not continue as far as the “Fermi bubble/microwave haze”, making it less probable that these are part of the same structure. We identify a number of new faint features in the polarized sky, including a dearth of polarized synchrotron emission directly correlated with a narrow, roughly 20deg long filament seen in Hα at high Galactic latitude. Finally, we look for evidence of polarized AME, however many AME regions are significantly contaminated by polarized synchrotron emission, and we find a 2σ upper limit of 1.6% in the Perseus region

Multiaperture UBVRIzJHKUBVRIzJHK Photometry of Galaxies in the Coma Cluster

We present a set of $UBVRIzJHK_s$ photometry for 745 $J+H$ band selected objects in a $22.5' \times 29.2'$ region centered on the core of the Coma cluster. This includes 516 galaxies and is at least 80% complete to H=16, with a spectroscopically complete sample of 111 cluster members (nearly all with morphological classification) for $H < 14.5$. For each object we present total \cite{kron80} magnitudes and aperture photometry. As an example, we use these data to derive color-magnitude relations for Coma early-type galaxies, measure the intrinsic scatter of these relations and its dependence on galaxy mass, and address the issue of color gradients. We find that the color gradients are mild and that the intrinsic scatter about the color-magnitude relation is small ($\sim 0.05$ mag in $U-V$ and less than $\sim 0.03$ in $B-R$, $V-I$ or $J-K$). There is no evidence that the intrinsic scatter varies with galaxy luminosity, suggesting that the cluster red sequence is established at early epochs over a range of $\sim 100$ in stellar mass.Comment: 41 pages, 5 figures, 18 data tables attached to source files or available on request from R. De propris. Accepted for publication in Astrophysical Journal Supplement Serie