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.

翻译：稀疏指数追踪是一种主流的被动投资组合管理策略，通过构建稀疏投资组合来追踪金融指数。相较于全投资组合，稀疏组合在降低交易成本和规避非流动性资产方面更具优势。现有研究多采用基于 $\ell_p$-范数正则化的连续松弛形式替代 $\ell_0$-范数正则化以强制组合稀疏性。然而，此类方法在实际投资中操作困难，因为调整参数以精确指定组合中资产数量的上界既复杂又耗时。本文提出一种基于 $\ell_0$-范数约束的稀疏指数追踪问题新形式，能够便捷地控制组合中资产数量的上限。此外，该形式允许在组合稀疏性与换手率稀疏性约束之间进行选择，通过限制每次再平衡时更新的资产数量进一步降低交易成本。基于原始-对偶分裂方法，我们开发了一种高效算法求解此问题。最后，通过标普500和纳斯达克100指数数据集的实验验证了所提方法的有效性。