  The free energy of quenched disordered systems is bounded above by the free
energy of the corresponding annealed system. This bound may be improved by
applying the annealing procedure, which is just Jensen inequality, after having
modified the Hamiltonian in a way that the quenched expressions are left
unchanged. This procedure is often viewed as a partial annealing or as a
constrained annealing, in the sense that the term that is added may be
interpreted as a Lagrange multiplier on the disorder variables.
  In this note we point out that, for a family of models, some of which have
attracted much attention, the multipliers of the form of empirical averages of
local functions cannot improve on the basic annealed bound from the viewpoint
of characterizing the phase diagram. This class of multipliers is the one that
is suitable for computations and it is often believed that in this class one
can approximate arbitrarily well the quenched free energy.
