  Generalized trust-region subproblem (GT) is a nonconvex quadratic
optimization with a single quadratic constraint. It reduces to the classical
trust-region subproblem (T) if the constraint set is a Euclidean ball. (GT) is
polynomially solvable based on its inherent hidden convexity. In this paper, we
study local minimizers of (GT). Unlike (T) with at most one local nonglobal
minimizer, we can prove that two-dimensional (GT) has at most two local
nonglobal minimizers, which are shown by example to be attainable. The main
contribution of this paper is to prove that, at any local nonglobal minimizer
of (GT), not only the strict complementarity condition holds, but also the
standard second-order sufficient optimality condition remains necessary. As a
corollary, finding all local nonglobal minimizers of (GT) or proving the
nonexistence can be done in polynomial time. Finally, for (GT) in complex
domain, we prove that there is no local nonglobal minimizer, which demonstrates
that real-valued optimization problem may be more difficult to solve than its
complex version.
