Mathematical Programming

, Volume 113, Issue 2, pp 259–282

A finite branch-and-bound algorithm for nonconvex quadratic programming via semidefinite relaxations


DOI: 10.1007/s10107-006-0080-6

Cite this article as:
Burer, S. & Vandenbussche, D. Math. Program. (2008) 113: 259. doi:10.1007/s10107-006-0080-6


Existing global optimization techniques for nonconvex quadratic programming (QP) branch by recursively partitioning the convex feasible set and thus generate an infinite number of branch-and-bound nodes. An open question of theoretical interest is how to develop a finite branch-and-bound algorithm for nonconvex QP. One idea, which guarantees a finite number of branching decisions, is to enforce the first-order Karush-Kuhn-Tucker (KKT) conditions through branching. In addition, such an approach naturally yields linear programming (LP) relaxations at each node. However, the LP relaxations are unbounded, a fact that precludes their use. In this paper, we propose and study semidefinite programming relaxations, which are bounded and hence suitable for use with finite KKT-branching. Computational results demonstrate the practical effectiveness of the method, with a particular highlight being that only a small number of nodes are required.


Nonconcave quadratic maximization Nonconvex quadratic programming Branch-and-bound Lift-and-project relaxations Semidefinite programming 

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Department of Management SciencesUniversity of IowaIowa CityUSA
  2. 2.Axioma, Inc.MariettaUSA

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