Multivariate nonnegative quadratic mappings

Zhi Quan Luo, Jos F. Sturm, Shuzhong Zhang

Research output: Contribution to journalArticlepeer-review

134 Scopus citations


In this paper, we study several issues related to the characterization of specific classes of multivariate quadratic mappings that are nonnegative over a given domain, with nonnegativity defined by a prespecified conic order. In particular, we consider the set (cone) of nonnegative quadratic mappings, defined with respect to the positive semidefinite matrix cone, and study when it can be represented by linear matrix inequalities. We also discuss the applications of the results in robust optimization, especially the robust quadratic matrix inequalities and the robust linear programming models. In the latter application the implementational errors of the solution are taken into account, and the problem is formulated as a semidefinite program.

Original languageEnglish (US)
Pages (from-to)1140-1162
Number of pages23
JournalSIAM Journal on Optimization
Issue number4
StatePublished - 2004


  • Biquadratic functions
  • Convex cone
  • Linear matrix inequalities
  • Robust optimization


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