A Newton-bracketing method for a simple conic optimization problem. (4th May 2021) - Kim, Sunyoung; Kojima, Masakazu; Toh, Kim-Chuan
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.