On cones of nonnegative quadratic functions

Jos F. Sturm, Shuzhong Zhang

Research output: Contribution to journalArticlepeer-review

226 Scopus citations


We derive linear matrix inequality (LMI) characterizations and dual decomposition algorithms for certain matrix cones which are generated by a given set using generalized co-positivity. These matrix cones are in fact cones of nonconvex quadratic functions that are nonnegative on a certain domain. As a domain, we consider for instance the intersection of a (upper) level-set of a quadratic function and a half-plane. Consequently, we arrive at a generalization of Yakubovich's S-procedure result. Although the primary concern of this paper is to characterize the matrix cones by LMIs, we show, as an application of our results, that optimizing a general quadratic function over the intersection of an ellipsoid and a half-plane can be formulated as semidefinite programming (SDP), thus proving the polynomiality of this class of optimization problems, which arise, e.g., from the application of the trust region method for nonlinear programming. Other applications are in control theory and robust optimization.

Original languageEnglish (US)
Pages (from-to)246-267
Number of pages22
JournalMathematics of Operations Research
Issue number2
StatePublished - May 2003
Externally publishedYes


  • Co-positive cones
  • LMI
  • Matrix decomposition
  • Quadratic functions
  • S-procedure
  • SDP


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