  We discuss the portfolio optimization problem with the obligatory deposits
constraint. Recently it has been shown that as a consequence of this nonlinear
constraint, the solution consists of an exponentially large number of optimal
portfolios, completely different from each other, and extremely sensitive to
any changes in the input parameters of the problem, making the concept of
rational decision making questionable. Here we reformulate the problem using a
quadratic obligatory deposits constraint, and we show that from the physics
point of view, finding an optimal portfolio amounts to calculating the
mean-field magnetizations of a random Ising model with the constraint of a
constant magnetization norm. We show that the model reduces to an eigenproblem,
with 2N solutions, where N is the number of assets defining the portfolio.
Also, in order to illustrate our results, we present a detailed numerical
example of a portfolio of several risky common stocks traded on the Nasdaq
Market.
