Real Algebraic and Analytic Geometry |

Optimization of polynomials on compact semialgebraic sets.

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

Submission: 2004, August 24.

*Abstract:
We give a short introduction to Lasserre's method for minimizing a polynomial
on a compact basic closed semialgebraic set. It consists of successively
solving tighter and tighter convex relaxations of this problem which can be
formulated as semidefinite programs. We give a new short proof for the
convergence of the optimal values of these relaxations to the minimum which is
constructive and elementary. In the case that there is a unique minimizer, we
prove that every sequence of nearly optimal solutions of the successive
relaxations gives rise to a sequence of points converging to this minimizer.*

Mathematics Subject Classification (2000): 90C26, 13J30, 44A60, 14P10, 11E25.

Keywords and Phrases: nonconvex optimization, positive polynomial, sum of squares, moment problem, Positivstellensatz, semidefinite programming.

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