A conic optimization approach to variants of the trust region subproblem

Abstract

The Trust Region Subproblem (TRS), which minimizes a nonconvex quadratic function over the unit ball, is an important subproblem in trust region methods for nonlinear optimization. Even though TRS is a nonconvex problem, it can be solved in polynomial time using, for example, a semidefinite programming (SDP) relaxation. Different variants of TRS have been considered from both theoretical and practical perspectives. In this thesis, we study three variants of TRS and their SDP/conic relaxations. We first study an extended trust region subproblem (eTRS) in which the trust region equals the intersection of the unit ball with m linear cuts. When m = 0, when m = 1, or when m = 2 and the linear cuts are parallel, it is known that the eTRS optimal value equals the optimal value of a particular conic relaxation, which is solvable in polynomial time. However, it is also known that, when m ≥ 2 and at least two of the linear cuts intersect within the ball, i.e., some feasible point of the eTRS satisfies both linear constraints at equality, then the same conic relaxation may admit a gap with eTRS. We show that the conic relaxation admits no gap for arbitrary m as long as the linear cuts are non-intersecting. We then extend our result to a more general setting. We study an eTRS in which a quadratic function is minimized over a structured nonconvex feasible region: the unit ball with m linear cuts and r hollows. In the special case when m = 0 and r = 1, it is known that the eTRS has a tight polynomial-time solvable conic