  Several portfolio selection models take into account practical limitations on
the number of assets to include and on their weights in the portfolio. We
present here a study of the Limited Asset Markowitz (LAM), of the Limited Asset
Mean Absolute Deviation (LAMAD) and of the Limited Asset Conditional
Value-at-Risk (LACVaR) models, where the assets are limited with the
introduction of quantity and cardinality constraints. We propose a completely
new approach for solving the LAM model, based on reformulation as a Standard
Quadratic Program and on some recent theoretical results. With this approach we
obtain optimal solutions both for some well-known financial data sets used by
several other authors, and for some unsolved large size portfolio problems. We
also test our method on five new data sets involving real-world capital market
indices from major stock markets. Our computational experience shows that,
rather unexpectedly, it is easier to solve the quadratic LAM model with our
algorithm, than to solve the linear LACVaR and LAMAD models with CPLEX, one of
the best commercial codes for mixed integer linear programming (MILP) problems.
Finally, on the new data sets we have also compared, using out-of-sample
analysis, the performance of the portfolios obtained by the Limited Asset
models with the performance provided by the unconstrained models and with that
of the official capital market indices.
