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 [ACM Tran. Softw., 45(3):34 (2019)]. The relaxation problem is converted into the problem of finding the largest zero $y^{\ast }$ of a continuously differentiable (except at $y^{\ast }$) convex function $g:\mathbb{R}\to \mathbb{R}$ such that $g\left(y\right)=0$ if $y\le y^{\ast }$ and $g\left(y\right)>0$ otherwise. In theory, the method generates lower and upper bounds of $y^{\ast }$ both converging to $y^{\ast }$. Their convergence is quadratic if the right derivative of g at $y^{\ast }$ is positive. Accurate computation of $g^{\prime }\left(y\right)$ 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.

对于二次优化问题（QOP）的Lagrangian-DNN松弛，我们提出了一种牛顿括号法，以改善BBCPOP [ACM Tran。Softw。，45（3）：34（2019）]。松弛问题转化为找到最大零的问题${ÿ}^{\ast }$ 的连续可微数（除 ${ÿ}^{\ast }$）凸函数 $G：\mathbb{[R}\to \mathbb{[R}$ 这样 $G（ÿ）=0$ 如果 $ÿ\le {ÿ}^{\ast }$ 和 $G（ÿ）>0$除此以外。从理论上讲，该方法生成${ÿ}^{\ast }$ 都收敛到 ${ÿ}^{\ast }$。如果g的右导数在${ÿ}^{\ast }$是积极的。精确计算$G^{\prime }（ÿ）$该方法的鲁棒性是必需的，但是在实践中很难实现。作为替代方案，我们提出了一种割线包围方法。我们证明了该方法提高了BBCPOP和SDPNAL +对于BIQMAC二进制QOP实例获得的下限的质量。此外，报告了来自QAPLIB的大规模QAP实例的未知最优值的新下界。