  We compare the ground state of the random-field Ising model with Gaussian
distributed random fields, with its non-equilibrium hysteretic counterpart, the
demagnetized state. This is a low energy state obtained by a sequence of slow
magnetic field oscillations with decreasing amplitude. The main concern is how
optimized the demagnetized state is with respect to the best-possible ground
state. Exact results for the energy in d=1 show that in a paramagnet, with
finite spin-spin correlations, there is a significant difference in the
energies if the disorder is not so strong that the states are trivially almost
alike. We use numerical simulations to better characterize the difference
between the ground state and the demagnetized state. For d>=3 the random-field
Ising model displays a disorder induced phase transition between a paramagnetic
and a ferromagnetic state. The locations of the critical points R_c(DS),
R_c(GS) differ for the demagnetized state and ground state. Consequently, it is
in this regime that the optimization of the demagnetized stat is the worst
whereas both deep in the paramagnetic regime and in the ferromagnetic one the
states resemble each other to a great extent. We argue based on the numerics
that in d=3 the scaling at the transition is the same in the demagnetized and
ground states. This claim is corroborated by the exact solution of the model on
the Bethe lattice, where the R_c's are also different.
