Journal
MATHEMATICAL PROGRAMMING
Volume 106, Issue 3, Pages 587-606Publisher
SPRINGER HEIDELBERG
DOI: 10.1007/s10107-005-0672-6
Keywords
polynomials; global optimization; sum of squares (SOS); semidefinite programming (SDP); radical ideal; variety; gradient ideal; algebraic geometry
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A method is proposed for finding the global minimum of a multivariate polynomial via sum of squares (SOS) relaxation over its gradient variety. That variety consists of all points where the gradient is zero and it need not be finite. A polynomial which is nonnegative on its gradient variety is shown to be SOS modulo its gradient ideal, provided the gradient ideal is radical or the polynomial is strictly positive on the real gradient variety. This opens up the possibility of solving previously intractable polynomial optimization problems. The related problem of constrained minimization is also considered, and numerical examples are discussed. Experiments show that our method using the gradient variety outperforms prior SOS methods.
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