A Newton-bracketing method for a simple conic optimization problem

Abstract

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* of a continuously differentiable (except at y*) convex function g : R -> R such that g(y) = 0 if y <= 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.