  Sparse index tracking is a prominent passive portfolio management strategy
that constructs a sparse portfolio to track a financial index. A sparse
portfolio is preferable to a full portfolio in terms of reducing transaction
costs and avoiding illiquid assets. To achieve portfolio sparsity, conventional
studies have utilized $\ell_p$-norm regularizations as a continuous surrogate
of the $\ell_0$-norm regularization. Although these formulations can construct
sparse portfolios, their practical application is challenging due to the
intricate and time-consuming process of tuning parameters to define the precise
upper limit of assets in the portfolio. 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. Moreover, our approach offers a choice between constraints on
portfolio and turnover sparsity, further reducing transaction costs by limiting
asset updates at each rebalancing interval. 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 Russell3000 index datasets.
