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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

I. Emre Kose
Dept. of Mechanical Engineering, Bogazici University, Istanbul, Turkey


\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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