The Monniaux Problem in abstract interpretation asks, given a program P, a safety (i.e., non-reachability) specification S, and an abstract domain of invariants D, whether the following problem is decidable:

Determine whether there exists an inductive invariant J in D guaranteeing that program P meets its specification S.

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.