Journal of Convex Analysis 17 (2010), No. 2, 535--551
Copyright Heldermann Verlag 2010
LMI Representations of the Convex Hulls of Quadratic Basic Semialgebraic Sets
Ugur Yildiran
Dept. of Systems Engineering, Yeditepe University, Istanbul, Turkey
uyildiran@yeditepe.edu.tr
I. Emre Kose
Dept. of Mechanical Engineering, Bogazici University, Istanbul, Turkey
koseemre@boun.edu.tr
[Abstract-pdf]
\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.
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