A concave optimization-based approach for sparse portfolio selection

Abstract

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 various data sets (of small, medium, and large dimensions) involving real-world capital market 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.

Keywords:

• zero-norm programming
• concave programming
• Frank–Wolfe method
• basin hopping method

Acknowledgements

The research of the authors from Università di Firenze has been partially supported by Progetto PRIN ‘Nonlinear Optimization, Variational Inequalities and Equilibrium Problems’.