A bound on the minimum of a real positive polynomial over the standard simplex.

Richard Leroy (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

We consider the problem of bounding below the minimum value $m$ taken by a positive polynomial $P \in \Z \left[X_{1},\dots,X_{k}\right]$ of degree $d$ over the standard simplex $\Delta \subset \R^{k}$. Using recent algorithmic developments in real algebraic geometry enables us to obtain a positive lower bound on $m$ in terms of the dimension $k$, the degree $d$ and the bitsize $\tau$ of the coefficients of $P$. The bound is explicit, and obtained without any extra assumption on $P$, in contrast with previous works reported in the literature.

Mots clés: minimum; lower bound; complexity; rational univariate representation

Domaines: Optimisation et contrôle [math.OC]; Calcul formel [cs.SC]

Richard Leroy. A bound on the minimum of a real positive polynomial over the standard simplex.. 2008. ⟨hal-00350115v1⟩