Generally, in the classical mean-variance portfolio selection approach several realistic features are not taken into account. Among these "forgotten" aspects, one of particular interest is the not infinite divisibility of the financial asset to select, i.e. the obligation to buy/sell only integer quantities of asset lots whose noumerousness is predetermined. In order to consider such a feature, we deal with and solve a suitably defined mixed-integer non-linear programming problem (MINPP). In particular, firstly, we consider a formulation of this problem in terms of quantities, i.e. integer number of asset lots to buy/sell, instead of starting capital percentage to invest/disinvest and, secondly, we propose an algorithm for finding a "good" feasible solution.

The mixed-integer nonlinear programming problem we propose considers 1 riskless asset and n risky ones, and it is formulated in terms of quantities of asset lots (whose noumerousness is predetermined) to buy/sell. Notice that in formulating this problem we reasonably assume that the riskless asset is (tipically) the money "going"/"coming" to/from a bank account. Because of that, "only" the n risky assets are constrained to be selected in non-negative and integer amount of lots. In particular, as commonly stated in most literature, we assume too that the loan rate is equal to the deposit one.

The procedure we develop for solving the introduced MINPP is articulated in two macro-steps. In the first one we consider a relaxed version of the introduced MINPP and we solve it. In the second macro-step, starting from the (optimal) solution of this relaxed problem, we find the optimal solution of the original MINPP.

Of course, because of the presence of non-negativity constraints, it is not possible to determine in a close form the optimal solution of the relaxed problem. In order to solve this problem, we use a programming techniques which is based on the so-called external penalty functions. From a qualitative standpoint, this algorithmic technique iteratively solves a sequence of programming problem which are characterized by the relaxation of the non-negativity constraints and by suitable "penalizations" of the variables violating the same corresponding constraint.

Generally, the searching algorithm for an optimal mixed-integer solution need a starting feasible mixed-integer one. Nevertheless, there is not a large amount of theoretical results concerning with the finding such a feasible solution. In particular, coping with this latest topic with regard to the MINNP we introduced, we found necessary and sufficient conditions for the existence of such a feasible mixed-integer solution.

Tipically, the "standard" searching algorithms for an optimal mixed-integer solution has an exponential complexity. So, in order to determine such a solution in a "reasonable" (computational) time it is quite important to start from a "good" feasible mixed-integer one. Notice that the numerical results we obtain from our experimentations (see section 7.) show that, in general, the feasible mixed-integer solutions identified by the necessary and sufficient conditions previously quoted are not particularly "good". Because of that, we have defined a methodology for looking for a feasible mixed-integer solution as close as possible to the optimal solution of the target MINPP.

There exist several "standard" methodologies for searching an optimal solution of a MINPP. Generally, each of them starts from a feasible mixed-integer solution (as "good" as possible) and, by following a suitable approach, determines an optimal one. In order to find the considered optimal mixed-integer solution, among the various "standard" programming techniques proposed in literature we suggest the use of the branch and bound (B&B) method based ones for, at least, the following reasons:

(6.a) our solving procedure, finding a feasible mixed-integer solution, allows to initialize the starting value of the objective function upper bound to a (positive) finite value instead of to the (positive) infinite one as in the classical B&B approach. So, the efficiency of this latest programming technique is increased because, a priori, we can stop to examinate all the branches of the decision tree to which are associated objective function values higher than the pre-determined starting one;

(6.b) the minimization of the objective function value of the starting MINPP suggests an interesting criterion for selecting, each time, the variable on which to effect the branching. In fact, one could select the variable better contributing to the reduction of the portfolio risk.

In this section we report the results of some simulated experimentations. In particular, column 2, 3 and 4 of each table is, respectively, devoted to the optimal solution of the programming problem without both the integrality constraints and the non-negativity ones, the optimal solution of the programming problem without the integrality constraints and with the non-negativity ones, and the feasible mixed-integer solution of the starting programming problem. Finally, in order to verify if the feasible mixed-integer solution our approach finds is "good" enough, we compare them with the one identified by the necessary and sufficient existence conditions (see column 5).

Notice that all the numerical simulations we realized (about 50) show that the variances associated, respectively, to the feasible mixed-integer solution and to the optimal relaxed programming problem one are quite close. Because of it, the feasible mixed-integer solution seems to be really a "good" feasible mixed-integer one from which to start for searching for an optimal mixed-integer one.

The solving approach we propose offers evidence for possible eneralizations, in fact it should be properly developed in order to take into account, besides the not infinite divisibility of the financial assets to select, other "operative" aspects like, for instance, the transaction costs and the taxation.