Asymptotic Cones of Quadratically Defined Sets and Their Applications to QCQPs

Quadratically constrained quadratic programs (QCQPs) are a set of optimization problems defined by a quadratic objective function and quadratic constraints. QCQPs cover a diverse set of problems, but the nonconvexity and unboundedness of quadratic constraints lead to difficulties in globally solving a QCQP. This thesis covers properties of unbounded quadratic constraints via a description of the asymptotic cone of a set defined by a single quadratic constraint. A description of the asymptotic cone is provided, including properties such as retractiveness and horizon directions. Using the characterization of the asymptotic cone, we generalize existing results for bounded quadratically defined regions with non-intersecting constraints. The newer result provides a sufficient condition for when the intersection of the lifted convex hulls of quadratically defined sets equals the lifted convex hull of the intersection. This document goes further by expanding the non-intersecting property to cover affine linear constraints. The Frank-Wolfe theorem provides conditions for when a problem defined by a quadratic objective function over affine linear constraints has an optimal solution. Over time, this theorem has been extended to cover cases involving convex quadratic constraints. We discuss more current results through the lens of the asymptotic cone of a quadratically defined set. This discussion expands current results and provides a sufficient condition for when a QCQP with one quadratic constraint with an indefinite Hessian has an optimal solution