![]() ![]() PUBLICATIONS PRODUCED AS A RESULT OF THIS RESEARCH ![]() This research considers both improvements in our understanding of the theoretical aspects of polynomial optimization, and the application of these methods to problems in computer vision. Such problems have a wide array of applications and admit methods from both the algebraic and analytic sides of mathematics. The work of the PI with her collaborators improvies our understanding of the algebraic and geometric structures that underlie optimization problems that involve polynomials. Here the application is primarily to object reconstruction from images taken by multiple cameras. The methods from polynomial optimization and convex algebraic geometry can be applied to problems from computer vision. Various open questions about these bodies are posed. The investigator and collaborators have recently constructed a new hierarchy of convex relaxations for algebraic sets called theta bodies. This study provides a uniform view of all lift-and-project methods via new notions of cone factorizations of certain operators associated to the convex body. This phenomena is central to all lift-and-project methods for discrete and polynomial optimization. The first set of questions studies the general phenomenon of when a given convex body is the linear projection of a slice (by an affine plane) of a closed convex cone. ![]() The key tool is the use of efficient algorithms in semidefinite programming, a branch of convex optimization that is used in polynomial optimization. The latter is a new research area that concerns convex sets and convex hulls of sets that are described algebraically and arise in optimization. This study focuses on problems from polynomial optimization and convex algebraic geometry. Primary Place of Performance Congressional District: Rekha Thomas (Principal Investigator) Sponsored Research Office:.Junping Wang (703)292-4488 DMS Division Of Mathematical Sciences MPS Direct For Mathematical & Physical Scien Polynomial Optimization and Convex Algebraic Geometry NSF Org: ![]()
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