Mathematical Programming

, Volume 109, Issue 1, pp 1–26 | Cite as

Semidefinite representations for finite varieties

  • Monique Laurent


We consider the problem of minimizing a polynomial over a set defined by polynomial equations and inequalities. When the polynomial equations have a finite set of complex solutions, we can reformulate this problem as a semidefinite programming problem. Our semidefinite representation involves combinatorial moment matrices, which are matrices indexed by a basis of the quotient vector space ℝ[x 1, . . . ,x n ]/I, where I is the ideal generated by the polynomial equations in the problem. Moreover, we prove the finite convergence of a hierarchy of semidefinite relaxations introduced by Lasserre. Semidefinite approximations can be constructed by considering truncated combinatorial moment matrices; rank conditions are given (in a grid case) that ensure that the approximation solves the original problem to optimality.


Lexicographic Order Monomial Basis Principal Submatrix Standard Monomial Residue Modulo 
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© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Monique Laurent
    • 1
  1. 1.CWIAmsterdamThe Netherlands

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