LMI Representations of the Convex Hulls of Quadratic Basic Semialgebraic Sets

U. Yildiran ; I. E. Kose

Journal of convex analysis, Tome 17 (2010) no. 2, pp. 535-551

Résumé

\newcommand{\closure}[1]{\overline{#1}} \newcommand{\chull}[1]{\text{{\bf co}}(#1)} \newcommand{\real}{{\mathbb R}} \newcommand{\set}[1]{{\cal #1}} We are motivated by the question of when a convex semialgebraic set in $\real^n$ is equal to the feasible set of a linear matrix inequality (LMI). Given a basic semialgebraic set, $\set{V}$, which is defined by quadratic polynomials, we restrict our attention to closure of its convex hull, namely $\closure{\chull{\set V}}$. Our main result is that $\closure{\chull{\set V}}$ is equal to the intersection of a finite number of LMI sets and the halfspaces supporting $\set V$ along a particular subset of the boundary of $\set V$. As a corollary, we show that in $\real^2$, the halfspaces of concern are finite in number, so that an LMI representation for $\closure{\chull{\set V}}$ always exists.