- A Newton-bracketing method for a simple conic optimization problem Sunyoung Kim(skimewha.ac.kr) Masakazu Kojima(kojimais.titech.ac.jp) Kim-Chuan Toh(mattohkcnus.edu.sg) 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 [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 $g : R \rightarrow R$ 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. Keywords: Nonconvex quadratic optimization problems, conic relaxations, robust numerical algorithms, Newton-bracketing method, secant-bracketing method for generating valid bounds. Category 1: Linear, Cone and Semidefinite Programming Citation: Download: [PDF]Entry Submitted: 05/29/2019Entry Accepted: 05/30/2019Entry Last Modified: 05/29/2019Modify/Update this entry Visitors Authors More about us Links Subscribe, Unsubscribe Digest Archive Search, Browse the Repository Submit Update Policies Coordinator's Board Classification Scheme Credits Give us feedback Optimization Journals, Sites, Societies Optimization Online is supported by the Mathematical Optmization Society.