Convex Hulls of Algebraic Sets

  • João Gouveia
  • Rekha Thomas
Part of the International Series in Operations Research & Management Science book series (ISOR, volume 166)


This article describes a method to compute successive convex approximations of the convex hull of the solutions to a system of polynomial equations over the reals. The method relies on sums of squares of polynomials and the dual theory of moment matrices. The main feature of the technique is that all computations are done modulo the ideal generated by the polynomials defining the set to the convexified. This work was motivated by questions raised by Lovász concerning extensions of the theta body of a graph to arbitrary real algebraic varieties, and hence the relaxations described here are called theta bodies. The convexification process can be seen as an incarnation of Lasserre’s hierarchy of convex relaxations of a real semialgebraic set. When the defining ideal is real radical the results become especially nice. We provide several examples of the method and discuss convergence issues. Finite convergence, especially after the first step of the method, can be described explicitly for finite point sets.


Convex Hull Semidefinite Programming Convex Relaxation Moment Matrice Finite Convergence 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



Both authors were partially supported by the NSF Focused Research Group grant DMS-0757371. J. Gouveia was also partially supported by Fundação para a Ciência e Tecnologia and R.R. Thomas by the Robert R. and Elaine K. Phelps Endowed Professorship.


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© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of WashingtonSeattleUSA
  2. 2.CMUC, Department of MathematicsUniversity of CoimbraCoimbraPortugal

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