Mathematical Programming

, Volume 113, Issue 2, pp 259-282

First online:

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

  • Samuel BurerAffiliated withDepartment of Management Sciences, University of Iowa Email author 
  • , Dieter VandenbusscheAffiliated withAxioma, Inc.

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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