  Sparse index tracking is one of the prominent passive portfolio management
strategies that construct a sparse portfolio to track a financial index. A
sparse portfolio is desirable over a full portfolio in terms of transaction
cost reduction and avoiding illiquid assets. To enforce the sparsity of the
portfolio, conventional studies have proposed formulations based on
$\ell_p$-norm regularizations as a continuous surrogate of the $\ell_0$-norm
regularization. Although such formulations can be used to construct sparse
portfolios, they are not easy to use in actual investments because parameter
tuning to specify the exact upper bound on the number of assets in the
portfolio is delicate and time-consuming. In this paper, we propose a new
problem formulation of sparse index tracking using an $\ell_0$-norm constraint
that enables easy control of the upper bound on the number of assets in the
portfolio. In addition, our formulation allows the choice between portfolio
sparsity and turnover sparsity constraints, which also reduces transaction
costs by limiting the number of assets that are updated at each rebalancing.
Furthermore, we develop an efficient algorithm for solving this problem based
on a primal-dual splitting method. Finally, we illustrate the effectiveness of
the proposed method through experiments on the S\&P500 and NASDAQ100 index
datasets.

Portfolio Management;Computational Engineering, Finance, and Science;Machine Learning