DOI: 10.68381/jca15044 ISSN: 0944-6532
Computing Uniform Convex Approximations for Convex Envelopes and Convex Hulls
Rida Laraki, Jean Bernard Lasserre
We provide a numerical procedure to compute uniform convex approximations
\{f_{r}\}
{
f
r
}
of the convex envelope
\widehat{f}
f
^
of a rational fraction
f
f
defined on a compact basic semi-algebraic set
\mathbf{D}
D
. At each point
x
x
of the convex hull
\mathbf{K}=\mathrm{co}(\mathbf{D})
K
=
c
o
(
D
)
, computing
f_{r}(x)
f
r
(
x
)
reduces to solving a semidefinite program. We next characterize
\mathbf{K}
K
in terms of the projection of a semi-infinite LMI, and provide outer convex approximations
\{\mathbf{K}_{r}\}\downarrow \mathbf{K}
{
K
r
}
↓
K
. Testing whether
x\notin \mathbf{K}
x
∉
K
reduces to solving finitely many semidefinite programs