  We study nonconvex homogeneous quadratically constrained quadratic
optimization with one or two constraints, denoted by (QQ1) and (QQ2),
respectively. (QQ2) contains (QQ1), trust region subproblem (TRS) and ellipsoid
regularized total least squares problem as special cases. It is known that
there is a necessary and sufficient optimality condition for the global
minimizer of (QQ2). In this paper, we first show that any local minimizer of
(QQ1) is globally optimal. Unlike its special case (TRS) with at most one local
non-global minimizer, (QQ2) may have infinitely many local non-global
minimizers. At any local non-global minimizer of (QQ2), both linearly
independent constraint qualification and strict complementary condition hold,
and the Hessian of the Lagrangian has exactly one negative eigenvalue. As a
main contribution, we prove that the standard second-order sufficient
optimality condition for any strict local non-global minimizer of (QQ2) remains
necessary. Applications and the impossibility of further extension are
discussed.
