Abstract
We study ellipsoid bounds for the solutions \((x,\mu) \in \mathbb{R}^{n} \times \mathbb{R}^{r}\) of polynomial systems of equalities and inequalities. The variable μ can be considered as parameters perturbing the solution x. For example, bounding the zeros of a system of polynomials whose coefficients depend on parameters is a special case of this problem. Our goal is to find minimum ellipsoid bounds just for x. Using theorems from real algebraic geometry, the ellipsoid bound can be found by solving a particular polynomial optimization problem with sums of squares (SOS) techniques. Some numerical examples are also given.
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Nie, J., Demmel, J.W. Minimum Ellipsoid Bounds for Solutions of Polynomial Systems via Sum of Squares. J Glob Optim 33, 511–525 (2005). https://doi.org/10.1007/s10898-005-2099-2
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DOI: https://doi.org/10.1007/s10898-005-2099-2