Journal of Global Optimization

, Volume 37, Issue 1, pp 75–84 | Cite as

A linear programming reformulation of the standard quadratic optimization problem

  • E. de Klerk
  • D. V. Pasechnik
Open Access
Original Paper


The problem of minimizing a quadratic form over the standard simplex is known as the standard quadratic optimization problem (SQO). It is NP-hard, and contains the maximum stable set problem in graphs as a special case. In this note, we show that the SQO problem may be reformulated as an (exponentially sized) linear program (LP). This reformulation also suggests a hierarchy of polynomial-time solvable LP’s whose optimal values converge finitely to the optimal value of the SQO problem. The hierarchies of LP relaxations from the literature do not share this finite convergence property for SQO, and we review the relevant counterexamples.


Linear programming Standard quadratic optimization Positive polynomials 


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Copyright information

© Springer Science+Business Media B.V. 2006

Authors and Affiliations

  1. 1.Department of Econometrics and Operations Research, Faculty of Economics and Business AdministrationTilburg UniversityTilburgThe Netherlands
  2. 2.School of Physical and Mathematical SciencesNanyang Technological UniversityBlk 5Singapore

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