Sharp and fast bounds for the Celis-Dennis-Tapia problem

TL;DR

This paper develops sharp, fast bounds for the Celis-Dennis-Tapia quadratic optimization problem by strengthening the dual Lagrangian bound with linear cuts, leading to significant computational improvements.

Contribution

It introduces a novel method to tighten dual bounds for the CDT problem using supporting hyperplanes, improving solution efficiency and effectiveness.

Findings

New bounds solve nearly all hard instances of CDT problem

Strengthened bounds are computationally efficient

Proposed bounds outperform existing methods in experiments

Abstract

In the Celis-Dennis-Tapia (CDT) problem a quadratic function is minimized over a region defined by two strictly convex quadratic constraints. In this paper we re-derive a necessary and optimality condition for the exactness of the dual Lagrangian bound (equivalent to the Shor relaxation bound in this case). Starting from such condition we propose to strengthen the dual Lagrangian bound by adding one or two linear cuts to the Lagrangian relaxation. Such cuts are obtained from supporting hyperplanes of one of the two constraints. Thus, they are redundant for the original problem but they are not for the Lagrangian relaxation. The computational experiments show that the new bounds are effective and require limited computing times. In particular, one of the proposed bounds is able to solve all but one of the 212 hard instances of the CDT problem presented in [Burer, Anstreicher, 2013].