  In the present paper, the minimal investment risk for a portfolio
optimization problem with imposed budget and investment concentration
constraints is considered using replica analysis. Since the minimal investment
risk is influenced by the investment concentration constraint (as well as the
budget constraint), it is intuitive that the minimal investment risk for the
problem with an investment concentration constraint be larger than that without
the constraint (that is, with only the budget constraint). Moreover, a
numerical experiment shows the effectiveness of our proposed analysis.
