  The Monniaux Problem in abstract interpretation asks, roughly speaking,
whether the following question is decidable: given a program $P$, a safety
(\emph{e.g.}, non-reachability) specification $\varphi$, and an abstract domain
of invariants $\mathcal{D}$, does there exist an inductive invariant $I$ in
$\mathcal{D}$ guaranteeing that program $P$ meets its specification $\varphi$.
The Monniaux Problem is of course parameterised by the classes of programs and
invariant domains that one considers. In this paper, we show that the Monniaux
Problem is undecidable for unguarded affine programs and semilinear invariants
(unions of polyhedra). Moreover, we show that decidability is recovered in the
important special case of simple linear loops.
