Real Algebraic and Analytic Geometry |

Global optimization of polynomials using gradient tentacles and sums of squares.

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homepage: http://perso.univ-rennes1.fr/markus.schweighofer/

Submission: 2006, May 17.

*Abstract:
In this work, the combine the theory of generalized critical values
with the theory of iterated rings of bounded elements (real holomorphy rings).
We consider the problem of computing the global infimum of a real polynomial in several variables.
Every global minimizer lies on the gradient variety. If the polynomial attains minimum,
it is therefore equivalent to look for the greatest lower bound on its gradient variety. Nie, Demmel and
Sturmfels proved recently a theorem about the existence of sums of squares certificates for such lower bounds.
Based on these certificates, they find arbitrarily tight relaxations of the original problem that can be formulated
as semidefinite programs and thus be solved efficiently. We deal here with the more general case when
the polynomial is bounded from below but does not necessarily attain a minimum. In this case, the method of
Nie, Demmel and Sturmfels might yield completely wrong results. In order to overcome this problem,
we replace the gradient variety by larger semialgebraic sets which we call gradient tentacles.
It now gets substantially harder to prove the existence of the necessary sums of squares certificates.*

Mathematics Subject Classification (2000): 13J30, 90C26.

Keywords and Phrases: global optimization, polynomial, preorder, sum of squares, semidefinite programming.

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