On the local stability of semidefinite relaxations

We consider a parametric family of quadratically constrained quadratic programs (QCQP) and their associated semidefinite programming (SDP) relaxations. Given a nominal value of the parameter at which the SDP relaxation is exact, we study conditions (and quantitative bounds) under which the relaxation will continue to be exact as the parameter moves in a neighborhood around the nominal value. Our framework captures a wide array of statistical estimation problems including tensor principal component analysis, rotation synchronization, orthogonal Procrustes, camera triangulation and resectioning, essential matrix estimation, system identification, and approximate GCD. Our results can also be used to analyze the stability of SOS relaxations of general polynomial optimization problems.Comment: 23 pages, 3 figure

Cool dwarfs in wide multiple systems. Paper 6: A curious quintuple system of a compact Sun-like triple and a close pair of an M dwarf and a very cool white dwarf at a wide separation

The system WDS 16329+0315 is an old, nearby quintuple physical system in the thick Galactic disc formed by a close-resolved, triple primary of solar metallicity, namely HD 149162, and a very wide, common proper motion, secondary pair, formed by the mid-M dwarf G-17-23 and the white dwarf LSPM J1633+0311S. We present an exhaustive astrometric and photometric data compilation of the system, including Gaia DR2 parallaxes and proper motions, and the first analysis of the nature of the faintest component. LSPM J1633+0311S (HD 149162 C) is a very cool white dwarf with an effective temperature of only about 5500 K, near the coolest end of the grid of theoretical models.Comment: The Observatory, in press, to appear in December 201