  Previous studies into the budget constraint of portfolio optimization
problems based on statistical mechanical informatics have not considered that
the purchase cost per unit of each asset is distinct. Moreover, the fact that
the optimal investment allocation differs depending on the size of investable
funds has also been neglected. In this paper, we approach the problem of
investment risk minimization using replica analysis. This problem imposes cost
and return constraints. We also derive the macroscopic theory indicated by the
optimal solution and confirm the validity of our proposed method through
numerical experiments.
