Mathematical Programming

, 129:129 | Cite as

Semidefinite relaxations for quadratically constrained quadratic programming: A review and comparisons

  • Xiaowei Bao
  • Nikolaos V. Sahinidis
  • Mohit Tawarmalani
Full Length Paper Series B


At the intersection of nonlinear and combinatorial optimization, quadratic programming has attracted significant interest over the past several decades. A variety of relaxations for quadratically constrained quadratic programming (QCQP) can be formulated as semidefinite programs (SDPs). The primary purpose of this paper is to present a systematic comparison of SDP relaxations for QCQP. Using theoretical analysis, it is shown that the recently developed doubly nonnegative relaxation is equivalent to the Shor relaxation, when the latter is enhanced with a partial first-order relaxation-linearization technique. These two relaxations are shown to theoretically dominate six other SDP relaxations. A computational comparison reveals that the two dominant relaxations require three orders of magnitude more computational time than the weaker relaxations, while providing relaxation gaps averaging 3% as opposed to gaps of up to 19% for weaker relaxations, on 700 randomly generated problems with up to 60 variables. An SDP relaxation derived from Lagrangian relaxation, after the addition of redundant nonlinear constraints to the primal, achieves gaps averaging 13% in a few CPU seconds.


Quadratic programming Semidefinite programming Lagrangian relaxation Nonconvex optimization 

Mathematics Subject Classification (2000)

49M29 65K05 90C22 90C26 90C30 


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Copyright information

© Springer and Mathematical Optimization Society 2011

Authors and Affiliations

  • Xiaowei Bao
    • 1
  • Nikolaos V. Sahinidis
    • 2
  • Mohit Tawarmalani
    • 3
  1. 1.Department of Chemical and Biomolecular EngineeringUniversity of Illinois at Urbana-ChampaignUrbanaUSA
  2. 2.Department of Chemical EngineeringCarnegie Mellon UniversityPittsburghUSA
  3. 3.Krannert School of ManagementPurdue UniversityWest LafayetteUSA

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