Selection of optimal investment portfolios with cardinality constraints

We consider the problem of selecting an optimal portfolio within the standard mean-variance framework extended to include constraints of practical interest, such as limits on the number of assets that can be included in the portfolio and on the minimum and maximum investments per asset and/or groups of assets. The introduction of these realistic constraints transforms the selection of the optimal portfolio into a mixed integer quadratic programming problem. This optimization problem, which we prove to be NP-hard, is difficult to solve, even approximately, by standard optimization techniques. A hybrid strategy that makes use of genetic algorithms and quadratic programming is designed to provide an accurate and efficient solution to the problem.