We study the trust region subproblem (TRS) of minimizing a nonconvex quadratic function over the unit ball with additional conic constraints. Despite having a nonconvex objective, it is known that the TRS and a number of its variants are polynomial-time solvable. In this paper, we follow a second-order cone based approach to derive an exact convex formulation of the TRS, and under slightly stronger conditions, give a low-complexity characterization of the convex hull of its epigraph. As a result, we make the connection between the nonconvex TRS and smooth convex quadratic minimization, which allows for the application of cheap iterative methods to the TRS. We also explore the inclusion of additional hollow constraints to the domain of the TRS, and convexify the associated epigraph.
Technical report, Tepper School of Business, Carnegie Mellon University, March 2016