Minimum Ellipsoid Bounds for Solutions of Polynomial Systems via Sum of Squares

AbstractWe 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.