  In the Celis-Dennis-Tapia (CDT) problem a quadratic function is minimized
over a region defined by two strictly convex quadratic constraints. In this
paper we re-derive a necessary and optimality condition for the exactness of
the dual Lagrangian bound (equivalent to the Shor relaxation bound in this
case). Starting from such condition we propose to strengthen the dual
Lagrangian bound by adding one or two linear cuts to the Lagrangian relaxation.
Such cuts are obtained from supporting hyperplanes of one of the two
constraints. Thus, they are redundant for the original problem but they are not
for the Lagrangian relaxation. The computational experiments show that the new
bounds are effective and require limited computing times. In particular, one of
the proposed bounds is able to solve all but one of the 212 hard instances of
the CDT problem presented in [Burer, Anstreicher, 2013].
