  We study large scale extended trust region subproblems (eTRS) i.e., the
minimization of a general quadratic function subject to a norm constraint,
known as the trust region subproblem (TRS) but with an additional linear
inequality constraint. It is well known that strong duality holds for the TRS
and that there are efficient algorithms for solving large scale TRS problems.
It is also known that there can exist at most one local non-global minimizer
(LNGM) for TRS. We combine this with known characterizations for strong duality
for eTRS and, in particular, connect this with the so-called hard case for TRS.
  We begin with a recent characterization of the minimum for the TRS via a
generalized eigenvalue problem and extend this result to the LNGM. We then use
this to derive an efficient algorithm that finds the global minimum for eTRS by
solving at most three generalized eigenvalue problems.
