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We study the trust-region subproblem (TRS) of minimizing a nonconvexquadratic function over the unit ball with additional conic constraints.Despite having a nonconvex objective, it is known that the classical TRS and anumber of its variants are polynomial-time solvable. In this paper, we follow asecond-order cone (SOC) based approach to derive an exact convex reformulationof the TRS under a structural condition on the conic constraint. Our structuralcondition is immediately satisfied when there is no additional conicconstraints, and it generalizes several such conditions studied in theliterature. As a result, our study highlights an explicit connection betweenthe classical nonconvex TRS and smooth convex quadratic minimization, whichallows for the application of cheap iterative methods such as Nesterov'saccelerated gradient descent, to the TRS. Furthermore, under slightly strongerconditions, we give a low-complexity characterization of the convex hull of theepigraph of the nonconvex quadratic function intersected with the constraintsdefining the domain without any additional variables. We also explore theinclusion of additional hollow constraints to the domain of the TRS, andconvexification of the associated epigraph.

A Second-Order Cone Based Approach for Solving the Trust Region Subproblem and Its Variants