Efficient Algorithms for Mean-variance Portfolio Optimization with Hard Real-world Constraints

The Markowitz mean-variance optimization model is a widely used tool for portfolio selection. However, in order to capture real world restrictions on actual investments, a Limited Asset Markowitz (LAM) model with the introduction of quantity and cardinality constraints has been considered. These two constraints have been modelled by adding binary variables to the Markowitz model, thus resulting in a Mixed Integer Quadratic Programming problem that is considerably more difficult to solve. We propose a new method for solving the LAM model based on a reformulation as a Standard Quadratic Program and on some recent theoretical results by the last two authors. We report optimal solutions of some previously unsolved benchmark problems used by several other authors and available from Beasley’s OR-Library. We also test our method on five new data sets involving real-world capital market indices from major stock markets. On these data sets we have been able to evaluate, on out-ofsample data, the performance of the portfolios obtained from the LAM model and to compare them to the classical Markowitz portfolio, and to the market index. This comparison seems to point in favour of the solutions obtained with the LAM model. We made our data sets and the solutions that we found publicly available for use by other researchers in this field.