An efficient Lagrange–Newton algorithm for long-only cardinality constrained portfolio selection on real data sets

Portfolio selection has always been a widely concerned issue in optimization and investment. Due to various forms of market friction, such as transaction costs and management fees, investors must choose a small number of assets from an asset pool. It naturally leads to the portfolio model with cardinality constraint. However, it is hard to solve this model accurately. Researchers generally use approximate methods to solve it, such as $l_1$ norm penalty. Unfortunately, these methods may not guarantee that the cardinality constraint is consistently met. In addition, short positions are challenging to implement in practice and are forbidden in some markets. Therefore, in this paper, we consider the long-only global minimum variance portfolio with cardinality constraint. We study the nonnegative cardinality constraint directly: defining the strong $\beta$-Lagrangian stationary point by nonnegative sparse projection operator, establishing the first-order optimality conditions in terms of the Lagrangian stationary point, as well as developing the Lagrange Newton algorithm to significantly reduce the scale of our problem and solve it directly. Finally, we conduct extensive experiments on real data sets. The numerical results show that the out-of-sample performances of our portfolio are better than some commonly used portfolio models for most data sets. Our portfolios usually lead to a higher Sharpe ratio and lower transaction costs with investment in fewer assets.