A Newton-bracketing method for a simple conic optimization problem

TL;DR

This paper introduces a Newton-bracketing method to efficiently find the largest zero of a convex function, improving bounds in quadratic optimization problems and demonstrating effectiveness on large-scale instances.

Contribution

It proposes a novel Newton-bracketing approach for conic optimization, enhancing existing methods like BBCPOP and SDPNAL+ for quadratic and quadratic assignment problems.

Findings

Improves lower bounds for binary quadratic optimization problems.

Achieves quadratic convergence under certain conditions.

Provides new bounds for large-scale quadratic assignment problems.

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 $g : \mathbb{R} \rightarrow \mathbb{R}$ 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.