Projected gradient descent method for cardinality-constrained portfolio optimization

Cardinality-constrained portfolio optimization aims at determining the investment weights on given assets using the historical data. This problem typically requires three constraints, namely, capital budget, long–only, and sparsity. The sparsity restraint allows investment managers to select a small number of stocks from the given assets. Most existing approaches exploit the penalty technique to handle the sparsity constraint. Therefore, they require tweaking the associated regularization parameter to obtain the desired cardinality level, which is time-consuming. This paper formulates the sparse portfolio design as a cardinality-constrained nonconvex optimization problem, where the sparsity constraint is modeled as a bounded ${\ell }_0$-norm. The projected gradient descent (PGD) method is then utilized to deal with the resultant problem. Different from existing algorithms, the suggested approach, called ${\ell }_0$-PGD, can explicitly control the cardinality level. In addition, its convergence is established. Specifically, the ${\ell }_0$-PGD guarantees that the objective function value converges, and the variable sequences converges to a local minimum. To remedy the weaknesses of gradient descent, the momentum technique is exploited to enhance the performance of the ${\ell }_0$-PGD, yielding ${\ell }_0$-PMGD. Numerical results on four real-world datasets, viz. NASDAQ 100, S&P 500, Russell 1000, and Russell 2000 exhibit the superiority of the ${\ell }_0$-PGD and ${\ell }_0$-PMGD over existing algorithms in terms of mean return and Sharpe ratio.