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Convex Hulls of Algebraic Sets

In: Handbook on Semidefinite, Conic and Polynomial Optimization

Author

Listed:
  • João Gouveia

    (University of Washington
    University of Coimbra)

  • Rekha Thomas

    (University of Washington)

Abstract

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.

Suggested Citation

  • João Gouveia & Rekha Thomas, 2012. "Convex Hulls of Algebraic Sets," International Series in Operations Research & Management Science, in: Miguel F. Anjos & Jean B. Lasserre (ed.), Handbook on Semidefinite, Conic and Polynomial Optimization, chapter 0, pages 113-138, Springer.
    • Handle: RePEc:spr:isochp:978-1-4614-0769-0_5
    DOI: 10.1007/978-1-4614-0769-0_5
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    Cited by:

    1. Hamza Fawzi & James Saunderson & Pablo A. Parrilo, 2017. "Equivariant Semidefinite Lifts of Regular Polygons," Mathematics of Operations Research, INFORMS, vol. 42(2), pages 472-494, May.

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