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We study large scale extended trust region subproblems (eTRS) i.e., theminimization of a general quadratic function subject to a norm constraint,known as the trust region subproblem (TRS) but with an additional linearinequality constraint. It is well known that strong duality holds for the TRSand 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 dualityfor 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 ageneralized eigenvalue problem and extend this result to the LNGM. We then usethis to derive an efficient algorithm that finds the global minimum for eTRS bysolving at most three generalized eigenvalue problems.

Local Nonglobal Minima for Solving Large Scale Extended Trust Region Subproblems