  The Lagrange problem is established in the discrete field theory subject to
constraints with values in a Lie group. For the admissible sections that
satisfy a certain regularity condition, we prove that the critical sections of
such problems are the solutions of a canonically unconstrained variational
problem associated with the Lagrange problem (discrete Lagrange multiplier
rule). This variational problem has a discrete Cartan 1-form, from which a
Noether theory of symmetries and a multisymplectic form formula are
established. The whole theory is applied to the Euler-Poincar\'e reduction in
the discrete field theory, concluding as an illustration with the remarkable
example of the harmonic maps of the discrete plane in the Lie group $SO(n)$.
