  For the Lagrangian-DNN relaxation of quadratic optimization problems (QOPs),
we propose a Newton-bracketing method to improve the performance of the
bisection-projection method implemented in BBCPOP [to appear in ACM Tran.
Softw., 2019]. The relaxation problem is converted into the problem of finding
the largest zero $y^*$ of a continuously differentiable (except at $y^*$)
convex function $g : \mathbb{R} \rightarrow \mathbb{R}$ such that $g(y) = 0$ if
$y \leq y^*$ and $g(y) > 0$ otherwise. In theory, the method generates lower
and upper bounds of $y^*$ both converging to $y^*$. Their convergence is
quadratic if the right derivative of $g$ at $y^*$ is positive. Accurate
computation of $g'(y)$ is necessary for the robustness of the method, but it is
difficult to achieve in practice. As an alternative, we present a
secant-bracketing method. We demonstrate that the method improves the quality
of the lower bounds obtained by BBCPOP and SDPNAL+ for binary QOP instances
from BIQMAC. Moreover, new lower bounds for the unknown optimal values of large
scale QAP instances from QAPLIB are reported.
