A quadratically constrained quadratic optimization model for completely positive cone programming

Naohiko Arima, Sunyoung Kim, Masakazu Kojima

Research output: Contribution to journalArticlepeer-review

17 Scopus citations


We propose a class of quadratic optimization problems which can be reformulated by completely positive cone programming with the same optimal values. The objective function can be any quadratic form. The constraints of each problem are described in terms of quadratic forms with no linear terms, and all constraints are homogeneous equalities, except one inhomogeneous equality where a quadratic form is set to be a positive constant. For the equality constraints, "a hierarchy of copositvity" condition is assumed. This model is a generalization of the standard quadratic optimization problem of minimizing a quadratic form over the standard simplex, and covers many of the existing quadratic optimization problems studied for exact copositive cone and completely positive cone programming relaxations. In particular, it generalizes the recent results on quadratic optimization problems by Burer and the set-semidefinite representation by Eichfelder and Povh.

Original languageEnglish
Pages (from-to)2320-2340
Number of pages21
JournalSIAM Journal on Optimization
Issue number4
StatePublished - 2013


  • A hierarchy of copositivity
  • Copositive programming
  • Quadratic optimization problem with quadratic constraints


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