  The typical behavior of optimal solutions to portfolio optimization problems
with absolute deviation and expected shortfall models using replica analysis
was pioneeringly estimated by S. Ciliberti and M. M\'ezard [Eur. Phys. B. 57,
175 (2007)]; however, they have not yet developed an approximate derivation
method for finding the optimal portfolio with respect to a given return set. In
this study, an approximation algorithm based on belief propagation for the
portfolio optimization problem is presented using the Bethe free energy
formalism, and the consistency of the numerical experimental results of the
proposed algorithm with those of replica analysis is confirmed. Furthermore,
the conjecture of H. Konno and H. Yamazaki, that the optimal solutions with the
absolute deviation model and with the mean-variance model have the same typical
behavior, is verified using replica analysis and the belief propagation
algorithm.
