This paper considers a portfolio selection problem in which portfolios with minimum number of active assets are sought. This problem is motivated by the need of inducing sparsity on the selected portfolio to reduce transaction costs, complexity of portfolio management, and instability of the solution. The resulting problem is a difficult combinatorial problem. We propose an approach based on the definition of an equivalent smooth concave problem. In this way we move the difficulty of the original problem to that of solving a concave global minimization problem. We present as global optimization algorithm a specific version of the Monotonic Basin Hopping method which employs, as local minimizer, an efficient version of the Frank-Wolfe method. We test our method on three data sets (of small, medium and large dimension) involving real-world capital market indices from major stock markets. The obtained results show the effectiveness of the presented methodology in terms of global optimization. Furthermore, also the out-of-sample performances of the selected portfolios, as measured by Sharpe ratio, appear satisfactory.
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