DOI: 10.68381/jca17036 ISSN: 0944-6532
LMI Representations of the Convex Hulls of Quadratic Basic Semialgebraic Sets
Uğur Yıldıran, İ. Emre Köse
We are motivated by the question of when a convex semialgebraic set in
{\mathbb R}^n
R
n
is equal to the feasible set of a linear matrix inequality (LMI). Given a basic semialgebraic set,
{\cal V}
V
, which is defined by quadratic polynomials, we restrict our attention to closure of its convex hull, namely
\overline{\text{{\bf co}}({\cal V})}
co
(
V
)
‾
. Our main result is that
\overline{\text{{\bf co}}({\cal V})}
co
(
V
)
‾
is equal to the intersection of a finite number of LMI sets and the halfspaces supporting
{\cal V}
V
along a particular subset of the boundary of
{\cal V}
V
. As a corollary, we show that in
{\mathbb R}^2
R
2
, the halfspaces of concern are finite in number, so that an LMI representation for
\overline{\text{{\bf co}}({\cal V})}
co
(
V
)
‾
always exists