An Alternative Perspective on Copositive and Convex Relaxations of Nonconvex Quadratic Programs

We study convex relaxations of nonconvex quadratic programs. We identify a family of so-called feasibility preserving convex relaxations, which includes the well-known copositive and doubly nonnegative relaxations, with the property that the convex relaxation is feasible if and only if the nonconvex quadratic program is feasible. We observe that each convex relaxation in this family implicitly induces a convex underestimator of the objective function on the feasible region of the quadratic program. This alternative perspective on convex relaxations enables us to establish several useful properties of the corresponding convex underestimators. In particular, if the recession cone of the feasible region of the quadratic program does not contain any directions of negative curvature, we show that the convex underestimator arising from the copositive relaxation is precisely the convex envelope of the objective function of the quadratic program, providing another proof of Burer's well-known result on the exactness of the copositive relaxation. We also present an algorithmic recipe for constructing instances of quadratic programs with a finite optimal value but an unbounded doubly nonnegative relaxation.Comment: 26 page

Income inequality and opinion expression gap in the American public: an analysis of policy priorities

Past scholarship has documented that the poor are more likely to withhold their policy preferences in public opinion surveys, suggesting income gaps in political engagement. Despite the wealth of scholarly interest in opinion formation, however, previous studies focused almost exclusively on opinion gaps in preferences, leaving income-related gaps in policy prioritisation virtually unexamined. Drawing on 596 public opinion surveys conducted with nearly 700,000 Americans over 55 years, we make a comprehensive attempt to examine income-level differences in “don’t know” responses to the most important problem (MIP) question. Our results show that the less affluent are more likely to say “don’t know” when asked about the MIP facing their country, even after controlling for various factors including educational attainment and political attention. Importantly, we also show that income-related differences in opinionation cross cut other socio-economic differences in policy prioritisation. These results have important implications for the study of public opinion.publishedVersio

A GEOMETRIC VARIATIONAL APPROACH TO SHAPE INVERSION FOR RADAR

In this thesis, we develop a novel method for dense shape reconstruction of scenes using radar. For a given scene and antennas taking measurements from the scene, our method iteratively estimates the scene shape using the measurements. To this end, we use a deformable shape evolution approach which seeks to match the received signal to a computed forward model based on the evolving shape. Adopting such an approach comes with important advantages such as the ability to naturally embed the shape priors into the estimation and being able to model self-occlusions which cannot be easily incorporated into classical radar imaging techniques. Iterations start with an initial shape model which is gradually deformed until its image under the forward model gets sufficiently close to the actual measurements. Since we use a gradient-based scheme to minimize our error and radar signals are highly oscillatory, a special attention is required to prevent these oscillations to manifest in the cost functional as local minima. For this purpose, we develop a novel technique by which we can extract the geometric information embedded in the radar signals that is used to formulate a well behaving cost functional. We test our approach with synthetic simulations performed in 2D which shows the promise of our approach on some challenging scenarios.Ph.D