  We give necessary and sufficient conditions for minimality of generalized
minimizers for linear-growth functionals of the form \[ \mathcal F[u] :=
\int_\Omega f(x,u(x)) \, \text{d}x, \qquad u:\Omega \subset \mathbb R^N\to
\mathbb R^d, \] where $u$ is an integrable function satisfying a general PDE
constraint. Our analysis is based on two ideas: a relaxation argument into a
subspace of the space of bounded vector-valued Radon measures $\mathcal
M(\Omega;\mathbb R^d)$, and the introduction of a set-valued pairing in
$\mathcal M(\Omega;\mathbb R^N) \times {\rm L}^\infty(\Omega;\mathbb R^N)$. By
these means we are able to show an intrinsic relation between minimizers of the
relaxed problem and maximizers of its dual formulation also known as the
saddle-point conditions. In particular, our results can be applied to
relaxation and minimization problems in BV, BD.
