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 yβ of a continuously differentiable (except at yβ)
convex function g:RβR such that g(y)=0 if
yβ€yβ and g(y)>0 otherwise. In theory, the method generates lower
and upper bounds of yβ both converging to yβ. Their convergence is
quadratic if the right derivative of g at yβ is positive. Accurate
computation of gβ²(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.Comment: 19 pages, 2 figure