Iterative Calculation of Sum Of Squares

Bruno Després (1) , Maxime Herda (2, 3)

1 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
2 LPP - Laboratoire Paul Painlevé - UMR 8524
3 RAPSODI - Reliable numerical approximations of dissipative systems

We propose an iterative algorithm for the calculations of sum of squares of polynomials, reformulated as positive interpolation. The method is based on the definition of a dual functional $G$. The domain of $G$, the boundary of the domain and the boundary at infinity are analyzed in details. In the general case, $G$ is closed convex. For univariate polynomials in the context of the Lukacs representation, $G$ is coercive and strictly convex which yields a unique critical point. Various descent algorithms are evoked. Numerical examples are provided, for univariate and bivariate polynomials.

Mots clés: Positive polynomials; sum of squares; convex programming; nonlinear programming

Domaines: Optimisation et contrôle [math.OC]; Analyse numérique [math.NA]

Bruno Després, Maxime Herda. Iterative Calculation of Sum Of Squares. 2018. ⟨hal-01946539v1⟩