  In the present paper, the primal-dual problem consisting of the investment
risk minimization problem and the expected return maximization problem in the
mean-variance model is discussed using replica analysis. As a natural extension
of the investment risk minimization problem under only a budget constraint that
we analyzed in a previous study, we herein consider a primal-dual problem in
which the investment risk minimization problem with budget and expected return
constraints is regarded as the primal problem, and the expected return
maximization problem with budget and investment risk constraints is regarded as
the dual problem. With respect to these optimal problems, we analyze a quenched
disordered system involving both of these optimization problems using the
approach developed in statistical mechanical informatics, and confirm that both
optimal portfolios can possess the primal-dual structure. Finally, the results
of numerical simulations are shown to validate the effectiveness of the
proposed method.
