Mathematical Programming Computation

, Volume 11, Issue 1, pp 119–171 | Cite as

New global algorithms for quadratic programming with a few negative eigenvalues based on alternative direction method and convex relaxation

  • Hezhi Luo
  • Xiaodi Bai
  • Gino Lim
  • Jiming PengEmail author
Full Length Paper


We consider a quadratic program with a few negative eigenvalues (QP-r-NE) subject to linear and convex quadratic constraints that covers many applications and is known to be NP-hard even with one negative eigenvalue (QP1NE). In this paper, we first introduce a new global algorithm (ADMBB), which integrates several simple optimization techniques such as alternative direction method, and branch-and-bound, to find a globally optimal solution to the underlying QP within a pre-specified \(\epsilon \)-tolerance. We establish the convergence of the ADMBB algorithm and estimate its complexity. Second, we develop a global search algorithm (GSA) for QP1NE that can locate an optimal solution to QP1NE within \(\epsilon \)-tolerance and estimate the worst-case complexity bound of the GSA. Preliminary numerical results demonstrate that the ADMBB algorithm can effectively find a global optimal solution to large-scale QP-r-NE instances when \(r\le 10\), and the GSA outperforms the ADMBB for most of the tested QP1NE instances. The software reviewed as part of this submission was given the DOI (digital object identifier)


Quadratic programming Alternative direction method Convex relaxation Branch-and-bound Line search Computational complexity 

Mathematics Subject Classification

90C20 90C22 90C26 



We would like to thank all the anonymous reviewers and the associate editor for their useful suggestions that has helped to substantially improve the presentation of this work.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature and The Mathematical Programming Society 2018

Authors and Affiliations

  1. 1.Department of Management Science & Engineering, College of Economics and ManagementZhejiang University of TechnologyHangzhouP. R. China
  2. 2.Department of Applied Mathematics, College of ScienceZhejiang University of TechnologyHangzhouP. R. China
  3. 3.Department of Industrial EngineeringUniversity of HoustonHoustonUSA

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