Convergence of the Lasserre Hierarchy of SDP Relaxations for Convex Polynomial Programs without Compactness

The Lasserre hierarchy of semidefinite programming (SDP) relaxations is an effective scheme for finding computationally feasible SDP approximations of polynomial optimization over compact semi-algebraic sets. In this paper, we show that, for convex polynomial optimization, the Lasserre hierarchy with a slightly extended quadratic module always converges asymptotically even in the face of non-compact semi-algebraic feasible sets. We do this by exploiting a coercivity property of convex polynomials that are bounded below. We further establish that the positive definiteness of the Hessian of the associated Lagrangian at a saddle-point (rather than the objective function at each minimizer) guarantees finite convergence of the hierarchy. We obtain finite convergence by first establishing a new sum-of-squares polynomial representation of convex polynomials over convex semi-algebraic sets under a saddle-point condition. We finally prove that the existence of a saddle-point of the Lagrangian for a convex polynomial program is also necessary for the hierarchy to have finite convergence.Comment: 17 page

Magnetic Field Strengths in Photodissociation Regions

We measure carbon radio recombination line (RRL) emission at 5.3 GHz toward four H ii regions with the Green Bank Telescope to determine the magnetic field strength in the photodissociation region (PDR) that surrounds the ionized gas. Roshi suggests that the non-thermal line widths of carbon RRLs from PDRs are predominantly due to magneto-hydrodynamic waves, thus allowing the magnetic field strength to be derived. We model the PDR with a simple geometry and perform the non-LTE radiative transfer of the carbon RRL emission to solve for the PDR physical properties. Using the PDR mass density from these models and the carbon RRL non-thermal line width we estimate total magnetic field strengths of B ~ 100-300 µG in W3 and NGC 6334A. Our results for W49 and NGC 6334D are less well constrained with total magnetic field strengths between B ~ 200-1000 µG. H i and OH Zeeman measurements of the line of sight magnetic field strength (B_(los)), taken from the literature, are between a factor of ~ 0.5-1 of the lower bound of our carbon RRL magnetic field strength estimates. Since |B_(los)| ⩽ B, our results are consistent with the magnetic origin of the non-thermal component of carbon RRL widths

Iminosugar-Based Inhibitors of Glucosylceramide Synthase Increase Brain Glycosphingolipids and Survival in a Mouse Model of Sandhoff Disease

The neuropathic glycosphingolipidoses are a subgroup of lysosomal storage disorders for which there are no effective therapies. A potential approach is substrate reduction therapy using inhibitors of glucosylceramide synthase (GCS) to decrease the synthesis of glucosylceramide and related glycosphingolipids that accumulate in the lysosomes. Genz-529468, a blood-brain barrier-permeant iminosugar-based GCS inhibitor, was used to evaluate this concept in a mouse model of Sandhoff disease, which accumulates the glycosphingolipid GM2 in the visceral organs and CNS. As expected, oral administration of the drug inhibited hepatic GM2 accumulation. Paradoxically, in the brain, treatment resulted in a slight increase in GM2 levels and a 20-fold increase in glucosylceramide levels. The increase in brain glucosylceramide levels might be due to concurrent inhibition of the non-lysosomal glucosylceramidase, Gba2. Similar results were observed with NB-DNJ, another iminosugar-based GCS inhibitor. Despite these unanticipated increases in glycosphingolipids in the CNS, treatment nevertheless delayed the loss of motor function and coordination and extended the lifespan of the Sandhoff mice. These results suggest that the CNS benefits observed in the Sandhoff mice might not necessarily be due to substrate reduction therapy but rather to off-target effects

Optimality criteria without constraint qualications for linear semidenite problems

We consider two closely related optimization problems: a problem of convex Semi- Infinite Programming with multidimensional index set and a linear problem of Semidefinite Programming. In study of these problems we apply the approach suggested in our recent paper [14] and based on the notions of immobile indices and their immobility orders. For the linear semidefinite problem, we define the subspace of immobile indices and formulate the first order optimality conditions in terms of a basic matrix of this subspace. These conditions are explicit, do not use constraint qualifications, and have the form of criterion. An algorithm determining a basis of the subspace of immobile indices in a finite number of steps is suggested. The optimality conditions obtained are compared with other known optimality conditions