Heuristic algorithms for the portfolio selection problem with minimum transaction lots

The problem of selecting a portfolio has been largely faced in terms of minimizing the risk, given the return. While the complexity of the quadratic programming model due to Markowitz has been overcome by the recent progress in algorithmic research, the introduction of linear risk functions has given rise to the interest in solving portfolio selection problems with real constraints. In this paper we deal with the portfolio problem with minimum transaction lots. We show that in this case the problem of finding a feasible solution, is independently of the risk function, NP-complete. Moreover, given the mixed integer linear model, new heuristics are proposed which starting from the solution of the relaxed problem allow us to find a solution close to the optimal one. The algorithms are based on the construction of mixed integer subproblems (using only a part of the securities available) formulated using the information obtained from the solution of the relaxed problem. The heuristics have been tested with respect to two disjoint time periods, using real data from the Milan Stock Exchange.