  In this paper, we revisit the portfolio optimization problems of the
minimization/maximization of investment risk under constraints of budget and
investment concentration (primal problem) and the maximization/minimization of
investment concentration under constraints of budget and investment risk (dual
problem) for the case that the variances of the return rates of the assets are
identical. We analyze both optimization problems by using the Lagrange
multiplier method and the random matrix approach. Thereafter, we compare the
results obtained from our proposed approach with the results obtained in
previous work. Moreover, we use numerical experiments to validate the results
obtained from the replica approach and the random matrix approach as methods
for analyzing both the primal and dual portfolio optimization problems.
