  The quadratic programming over one inequality quadratic constraint (QP1QC) is
a very special case of quadratically constrained quadratic programming (QCQP)
and attracted much attention since early 1990's. It is now understood that,
under the primal Slater condition, (QP1QC) has a tight SDP relaxation (PSDP).
The optimal solution to (QP1QC), if exists, can be obtained by a matrix rank
one decomposition of the optimal matrix X? to (PSDP). In this paper, we pay a
revisit to (QP1QC) by analyzing the associated matrix pencil of two symmetric
real matrices A and B, the former matrix of which defines the quadratic term of
the objective function whereas the latter for the constraint. We focus on the
\undesired" (QP1QC) problems which are often ignored in typical literature:
either there exists no Slater point, or (QP1QC) is unbounded below, or (QP1QC)
is bounded below but unattainable. Our analysis is conducted with the help of
the matrix pencil, not only for checking whether the undesired cases do happen,
but also for an alternative way to compute the optimal solution in comparison
with the usual SDP/rank-one-decomposition procedure.
