  In the present work, the optimal portfolio minimizing the investment risk
with cost is discussed analytically, where this objective function is
constructed in terms of two negative aspects of investment, the risk and cost.
We note the mathematical similarity between the Hamiltonian in the
mean-variance model and the Hamiltonians in the Hopfield model and the
Sherrington{Kirkpatrick model and show that we can analyze this portfolio
optimization problem by using replica analysis, and derive the minimal
investment risk with cost and the investment concentration of the optimal
portfolio. Furthermore, we validate our proposed method through numerical
simulations.
